Abstract
We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.
1 Introduction
In this introduction we consider only piecewise expanding unimodal maps. However, all the following results can be extended to more general families of piecewise expanding interval maps (see Sect. 2). We call a map \(T:[0,1]\rightarrow [0,1]\) a piecewise expanding unimodal map or tent map if it is continuous and if there exists a turning point \(c\in (0,1)\) such that \(T|_{[0,c]}\) and \(T|_{[c,1]}\) are \(C^{1+\alpha }\), \(\Vert 1/T'\Vert _\infty <1\) and \(\Vert T'\Vert _\infty <\infty \), and \(T(1)=T(0)=0\). We assume that \(T\) is mixing, i.e., it is topologically mixing in the interval \([T^2(c),T(c)]\). Let \(\mu \) denote the unique (hence ergodic) absolutely continuous invariant probability measure (acip) for \(T\). By Birkhoff’s ergodic theorem, \(\mu \) almost every (or in this case also Lebesgue almost every) point \(x\in [0,1]\) is typical for \(\mu \), i.e.,
A natural question is how fast this convergence takes place. In order to answer this question one has to take a smaller set of observables: By [24] and [11], for any sequence \(\alpha _n\) such that \(\lim _{n\rightarrow \infty }\alpha _n=\infty \), there is a dense \(G_\delta \) set in \(C^0\) such that for all \(\varphi \) in this set one has
A suitable set of observables for which the question about the speed of convergence makes sense is for example the set of Hölder continuous functions (or more generally the set of functions of generalised bounded variation; see Definition 2.5 below). For \(\varphi \) Hölder, set
Since we have exponential decay of correlation (see, e.g., Proposition 4.3 below), \(\sigma (\varphi )\) is finite and since we can write \(\sigma (\varphi )=\lim _{n\rightarrow \infty }n^{-1}Var(S_n)\), where \(S_n\) is the \(n\)th Birkhoff sum, we see that \(\sigma (\varphi )\ge 0\). If \(\sigma (\varphi )=0\), then \(\varphi \) is a co-boundary and there exists an \(L^1\) function \(\psi \) so that \(\varphi =\psi \circ T-\psi \) almost surely. Henceforth, we exclude this (degenerate) case, i.e., we will always assume that \(\sigma (\varphi )>0\). Turning back to the question about the speed of convergence of (1), it is shown in [18] that if we restrict ourself to the set of Hölder continuous observables \(\varphi \) then the law of iterated logarithm (LIL) holds: For a.e. \(x\in [0,1]\), we have
For tent maps \(T\) the turning point \(c\) is of particular dynamical interest. A lot of information about the dynamics of \(T\) is contained in the forward orbit of \(c\), and it is natural to ask if (3) holds when we take \(x=c\). For a recent work where the assumption that the turning point satisfies the LIL is crucial, see [4]. However, even if we know that (1) and (3) hold for a.e. point \(x\), it is a very difficult question to say wether they hold for a particular point \(x\). So instead of asking for the LIL for \(c\) for a single tent map \(T\), we perturb this map by a one-parameter family of tent maps and ask if the LIL for \(c\) holds for almost every map in this family. Let \(T_a,\, a\in [0,1]\), be a one-parameter family of piecewise expanding unimodal maps through \(T=T_0\). We make some natural regularity assumptions on the parameter dependency as, e.g., the turning point \(c_a\) is Lipschitz continuous in \(a\) and if \(J\subset [0,1]\) is an interval on which \(x\ne c_a\), then \(a\mapsto T_a(x)\) is \(C^{1+\alpha }\) on \(J\) (for the precise conditions we refer to the beginning of Sect. 2). Of course in order that the question of this paragraph makes sense we have to exclude trivial one-parameter families as for example the constant one or families for which the turning point is eventually mapped to a periodic point for all parameters. The right condition here is transversality which is a common non-degeneracy condition for one-parameter families of interval maps (see, e.g., [2, 3, 5, 16, 26, 32, 33] for previous occurrences of this condition in the literature). We say that the family of tent maps \(T_a\) is transversal at \(T_0\) if there exists a constant \(C\ge 1\) such that
(If \(c_0\) is periodic for \(T_0\), then we take one-sided derivatives.) The transversality condition says that the \(a\)-derivative along the postcritical orbit is comparable to its \(x\)-derivate. Since the \(x\)-derivative is growing exponentially fast, this implies that if we change the parameter then the dynamics of the corresponding map will change fast which makes it possible to study the generic behaviour of the postcritical orbit. If the family \(T_a\) is transversal at \(T_0\), then it is shown in [32] that for a.e. parameter \(a\) close to \(0\) the turning point \(c_a\) is typical for the acip \(\mu _a\) (for related results see [8, 31], and [14]). Given almost sure typicality of the turning point we can now ask for the speed of convergence of (1) in this setting.
The main result of this paper can be stated as follows (see also Theorem 3.1 in Sect. 3 below). To the best of the authors knowledge, it is the first result which treats the question of a LIL for a specific point in a dynamical systems. Recall the notation \(\sigma \) in (2). We will use the notation \(\sigma _a\) when considering the map \(T_a\).
Theorem 1.1
Assume that \(T_0\) is mixing, its turning point \(c_0\) is not periodic, and the family \(T_a\) is transversal at \(T_0\). If \(\varphi \) is Hölder and \(\sigma _0(\varphi )>0\), then there exists \(\epsilon >0\) such that for almost every \(a\in [0,\epsilon ]\) the turning point \(c_a\) satisfies the LIL for the function \(\varphi \) under the map \(T_a\), i.e.,
In order to prove Theorem 1.1, we will show a stronger property, the so called almost sure invariance principle (ASIP), for the turning point. We say that the functions \(\xi _i:[0,\epsilon ]\rightarrow {\mathbb R},\, i\ge 1\), satisfy the ASIP with error exponent \(\gamma <1/2\) if there exists a probability space supporting a Brownian motion \(W\) and a sequence of variables \(\eta _i,\, i\ge 1\), such that
-
(i)
\(\{\xi _i\}_{i\ge 1}\) and \(\{\eta _i\}_{i\ge 1}\) have the same distribution;
-
(ii)
almost surely as \(n\rightarrow \infty \),
$$\begin{aligned} \left| W(n)-\sum _{i=1}^{n}\eta _i\right| =O(n^\gamma ). \end{aligned}$$
The following corollary is shown, e.g., in [29]. For other implications of the ASIP we refer to [17].
Corollary 1.2
If the functions \(\xi _i\) satisfy the ASIP then they satisfy also the LIL and the central limit theorem. More precisely, if \(\sigma ^2\) is the variance of the related Brownian motion, then
and, for all \(t\in {\mathbb R}\),
where \(m\) denotes the Lebesgue measure.
For \(\varphi \) Hölder and such that \(\sigma _0(\varphi )>0\), for \(i\ge 1\) and \(a\) small, set
Lemma 4.5 below guaranteesFootnote 1 that \(a\mapsto \sigma _a(\varphi )\) is continuous at \(0\) and, hence, the function \(\varphi _a\) is well-defined for \(a\) sufficiently close to \(0\). Due to this normalisation we have
for all \(a\) sufficiently close to \(0\). We are going to show an ASIP for the functions
Theorem 1.3
Assume that \(T_0\) is mixing, \(c_0\) is not periodic, and the family \(T_a\) is transversal at \(T_0\). Then there exists \(\epsilon >0\) such that the functions \(\xi _i:[0,\epsilon ]\rightarrow {\mathbb R},\, i\ge 1\), satisfy the ASIP for all error exponents \(\gamma >2/5\).
Remark 1.4
Theorem 1.3 and, hence, Theorem 1.1 hold also if \(c_0\) is periodic and \(T_0\) has a sufficiently high expansion (see Theorem 3.1 in Sect. 3 below). Because of the normalisation in the definition of the \(\xi _i\)’s, the variance of the Brownian motion in the ASIP is equal to \(1\). For a comment on the optimality of the error exponent \(\gamma \) see the beginning of Sect. 6.
Regarding the proof of Theorem 1.3 we go along a classical method from probability theory which consists in writing the Birkhoff sum approximatively as a sum of blocks of polynomial size, then in approximating these blocks by a martingale difference sequence, and finally in applying Skorokhod’s representation theorem which provides a link between a martingale and a Brownian motion. This strategy is illustrated on many examples in Philipp and Stout [29]. More precisely, using arguments from ergodic theory and perturbation theory, we go along the approach in [29, Section 3]. Note that the “usual” applications of [29] in dynamical systems refer to [29, Section 7] (see, e.g., [12, 13, 18], and [28]). The key property in these applications is a strong mixing condition which we do not have in our setting since loosely speaking the \(\xi _i\)’s are not iterations of a fixed map and, thus, there is no underlying invariant measure. However, we can more or less replace this strong mixing condition by uniformity of constants in the Lasota–Yorke inequality for the family \(T_a\) (see condition (II) in Sect. 2). By Keller–Liverani perturbation arguments [22], we have then uniformity of constants for the exponential decay of correlation (see Proposition 4.3). This in turn can be used to show a certain exponential decay of correlation for the maps \(\xi _i\)’s (see, e.g., the proof of Proposition 5.1) from which we are able to deduce similar estimates as in [29, Section 3]. In the recent work [15], Gouëzel uses spectral methods to show an almost sure invariance principle. His method is very powerful and it provides very good error estimates. However, we did not find an easy way to apply these spectral techniques to our setting.
We would like to highlight that the main technical novelty or difficulty of this paper is to treat processes which are (at least “locally”) close to processes generated by a dynamical system but for which there is no underlying invariant measure. Hence, various tools from ergodic theory cannot be applied directly and one has to “zoom in” in order to profit from ergodic theoretical facts. This explains the rather technical nature of this paper. The following classical example by Erdös and Fortet (see [19], p. 646; cf. also [10]) shows how careful one should be when one wants to show an ASIP for a process which is not but very close to a process generated by a dynamical system: Let \(\varphi (x)=\cos (2\pi x)+\cos (4\pi x)\) and consider the sequence \(\xi _i(x)=\varphi (2^ix)\), \(i\ge 1\). \(\xi _i\) is a process generated by the doubling map \(x\mapsto 2x\mod 1\). It is straightforward to check that \(\sigma (\varphi )>0\) and, for instance by the above cited “dynamical” paper [18], it follows that the process \(\xi _i\) satisfies the ASIP. However, if we change the process just slightly and consider instead \(\xi _i(x)=\varphi ((2^i-1)x)\) then, surprisingly, this new process does not satisfy anymore the central limit theorem (and, thus, not either the ASIP).
As mentioned in the beginning of this section, the above presented results for tent maps hold for more general piecewise expanding maps on the interval. First, it is not essential to take the turning point as the point of interest. Any other point works fine as long as the \(a\)- and \(x\)-derivatives along its forward orbit are comparable. If we consider other piecewise expanding maps on the interval than tent maps, then we have to add two more conditions. The first one is to have uniform constants in the Lasota–Yorke inequality (see condition (II) in Sect. 2.2). This is a natural condition when applying perturbation theory. The second condition (see condition (III) in Sect. 2.2) is a bit more technical but satisfied for many one-parameter families, as it is shown in Sect. 3. In Sect. 3 we mention also how to apply the main result of this paper, Theorem 2.6, to obtain almost sure typicality results similar to the ones in [32] but under alternative conditions (see Theorem 3.5).
The present paper deals exclusively with maps which are uniformly hyperbolic. It is a natural question if we can obtain a similar result in a non-uniformly hyperbolic setting. An interesting candidate for this question is the quadratic family \(f_a(x)=ax(1-x)\) with parameter \(a\in (0,4]\). Does the critical point \(c=1/2\) satisfy the LIL for Lebesgue almost every Collet–Eckmann (CE) map for sufficiently smooth observables? Despite a vast variety of results about the quadratic family, this question is still unsolved. To start with one should maybe content oneself with finding a positive Lebesgue measure set of CE parameters such that the critical points of the corresponding CE maps satisfy the LIL. Almost sure typicality of the critical point is known: By Avila and Moreira [1], the critical point for Lebesgue almost every CE map \(f_a\) in the quadratic family is typical for its SRB measure \(\mu _a\). (For the subset of CE parameters considered by Benedicks and Carleson this result was shown in [6].) An important ingredient in an attempt to find a positive measure set for which the turning point satisfies the LIL should be uniformity of constants in the set of CE parameters which one considers. For this one could follow the “start-up procedure” in Benedicks and Carleson [6] which yields, in addition to uniformity of constants, at each step nice “Markov partitions” on the parameter space. On the partition elements of these Markov partitions, which are intervals, one should be able to define functions \(\xi _i(a)\) as in (7). (Observe that at each step one excludes parameter intervals from the previous Markov partition and, finally, one ends up with a Cantor set of positive Lebesgue measure.) In [28] where the ASIP is shown for a fixed CE map, they use a tower construction to get an induced system with uniform hyperbolicity where more or less a straight forward application of [29, Section 7] implies an ASIP which projects then down to the ASIP for the original CE map. Since in the parameter space one has to exclude an open and dense set of regular parameters, the “start-up procedure” in [6] might provide a way to replace this tower construction in [28] when one deals only with one single CE map.
The paper is organised as follows. In Sect. 2, we formulate a general model and give the main notations for the one-parameter families of piecewise expanding maps considered in this paper. This is followed by the main statement, Theorem 2.6. Section 3 contains examples of one-parameter families, such as families of tent maps, to which the result of this paper applies. Section 4 deals with elementary facts as distortion estimates, uniform exponential decay of correlations, and the regularity of \(a\mapsto \sigma _a\). Sections 5 and 6 are dedicated to the proof of the main Theorem 2.6, i.e., the proof of an almost sure invariance principle.
2 Main statement
We begin this section with an introduction of the basic notation and a formulation of a suitable model for one-parameter families of piecewise expanding maps of the unit interval. A map \(T:[0,1]\rightarrow [0,1]\) will be called piecewise \(C^{1+\alpha }\), \(0<\alpha \le 1\), if there exists a partition \(0=b_0<b_1< \cdots <b_p=1\) of the unit interval such that for each \(1\le k\le p\) the restriction of \(T\) to the open interval \((b_{k-1},b_k)\) is a \(C^{1+\alpha }\) function. Let \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), be a one-parameter family of piecewise \(C^{1+\alpha }\) maps and let \(0=b_0(a)<b_1(a)< \cdots <b_{p(a)}(a)=1\) be the partition of the unit interval associated to \(T_a\). We assume that the Hölder constants are uniform in \(a\), i.e., there exist \(0<\alpha \le 1\) and a constant \(C\) so that
We assume that the maps are uniformly expanding, i.e., we assume that there are real numbers \(1<\lambda \le \Lambda <\infty \) such that for every \(a\in [0,1]\),
Remark 2.1
Regarding the frameworks in [21] and [30], it would be natural to skip the assumption that \(\Vert T_a'\Vert _\infty \le \Lambda \) and to replace the requirement that \(x\mapsto T_a'(x)\) is piecewise \(\alpha \)-Hölder by the requirement that \(x\mapsto 1/T_a'(x)\) is piecewise \(\alpha \)-Hölder. However, since our analysis on the parameter space seems to require some specific distortion estimates (see Lemma 4.4 below), we do at present not know how to make this improvement in our setting. For instance having in mind one-parameter families of one-dimensional Lorenz maps, it might be interesting to investigate such a more general setting.
We make the following natural assumptions on the parameter dependence.
-
(i)
The number of monotonicity intervals for the \(T_a\)’s is constant, i.e., \(p(a)\equiv p_0\), and the partition points \(b_k(a)\), \(0\le k\le p_0\), are Lipschitz continuous on \([0,1]\). It follows that there is a constant \(\delta _0>0\) such that
$$\begin{aligned} b_k(a)-b_{k-1}(a)\ge \delta _0, \end{aligned}$$for all \(1\le k\le p_0\) and \(a\in [0,1]\).
-
(ii)
If \(x\in [0,1]\) and \(J\subset [0,1]\) is a parameter interval such that \(b_k(a)\ne x\), for all \(a\in J\) and \(0\le k\le p_0\), then \(a\mapsto T_a(x)\) is \(C^{1+\alpha }\) and \(a\mapsto \partial _xT_a(x)\) is \(\alpha \)-Hölder where the Hölder constants are independent on \(x\). Further, the maps \(x\mapsto \partial _a T_a(x),\, x\in (b_{k-1}(a),b_k(a)),\, 1\le k\le p_0\), are \(\alpha \)-Hölder continuous (where the Hölder constants are uniform in \(a\)).
In order to obtain an acip, we refer to a paper by Keller [21] (see Theorems 3.3 and 3.5 therein) who extended the results in [25] on piecewise expanding \(C^2\) maps to a broader class of maps containing also piecewise expanding \(C^{1+\alpha }\) maps: For a fixed \(a\in [0,1]\) there exists a finite number of ergodic acip for \(T_a\). Further, by [27] combined with the remark in [36] after Definition 4 on page 514 (regarding property (III) therein cf. also [30, Proposition 5.1]), there exist at most \(p_0-1\) ergodic acip and the support of an ergodic acip is a finite union of intervals. Since we are always interested in only one ergodic acip, we can without loss of generality assume that for each \(T_a,\, a\in [0,1]\), there is a unique (hence ergodic) acip which we denote by \(\mu _a\). Let \(K(a)=\mathrm{supp }(\mu _a)\). We say that \(T_a\) is mixing if it is topologically mixing on \(K(a)\). For \(a\in [0,1]\), let \(\{D_1(a),\ldots ,D_{p_1(a)}(a)\}\) be the connected components of \(K(a){\setminus }\{b_0(a),\ldots ,b_{p_0}(a)\}\), i.e., the \(D_k(a)\)’s are the monotonicity intervals for \(T_a:K(a)\rightarrow K(a)\). We assume the following.
-
(iii)
The number of \(D_k(a)\)’s is constant in \(a\), i.e., \(p_1(a)\equiv p_1\) for all \(a\in [0,1]\). The boundary points of \(D_k(a),\, 1\le k\le p_1\), are \(\alpha \)-Hölder continuous in \(a\).
2.1 Partitions
For a fixed parameter value \(a\in [0,1]\), we denote by \({\mathcal {P}}_j(a)\), \(j\ge 1\), the partition on the dynamical interval consisting of the maximal open intervals of smooth monotonicity for the map \(T_a^j: K(a)\rightarrow K(a)\). More precisely, \({\mathcal {P}}_j(a)\) denotes the set of open intervals \(\omega \subset K(a)\) such that \(T_a^j:\omega \rightarrow K(a)\) is \(C^{1+\alpha }\) and \(\omega \) is maximal, i.e., for every other open interval \(\tilde{\omega }\subset K(a)\) with \(\omega \subsetneq \tilde{\omega },\, T_a^j:\tilde{\omega }\rightarrow K(a)\) is no longer \(C^{1+\alpha }\). Clearly, the elements of \({\mathcal {P}}_1(a)\) are the interiors of the intervals \(D_k(a),\, 1\le k\le p_1\).
We will define similar partitions on the parameter interval \([0,1]\). Let \(x_0:[0,1]\rightarrow [0,1]\) be a \(C^{1+\alpha }\) map from the parameter interval \([0,1]\) into the dynamical interval \([0,1]\) where we assume that
The points \(x_0(a),\, a\in [0,1]\), are the points of interest in this paper, i.e., we are interested in the properties of the forward orbit of these points under \(T_a\). The assumption (10) is only for convenience and it helps to make the partitions \({\mathcal {P}}_j\) below well-defined. (If a map \(x_0\) does not satisfy (10), then combining the fact that Lebesgue a.e. point \(x\in [0,1]\) is eventually mapped into \(K(a)\) under \(T_a\) with the transversality condition (I) below, one can derive that (10) is satisfied for some iteration of \(x_0\) restricted to some smaller intervals located around \(a=0\).) The forward orbit of a point \(x_0(a)\) under the map \(T_a\) we denote as
Observe that by assumption \(x_j(a)\in K(a)\), for all \(j\ge 0\) and \(a\in [0,1]\).
Remark 2.2
Since a lot of information for the dynamics of \(T_a\) is contained in the forward orbits of the partition points \(b_k(a),\, 0\le k\le p_0\), an interesting choice of the map \(x_0\) is
For example, in the case of tent maps we choose \(x_0(a)=T_a^{j_0}(c_a)\), for \(j_0\) sufficiently large (see Theorem 3.1 below).
Let \(J\subset [0,1]\) be an interval. By \({\mathcal {P}}_j|J,\, j\ge 1\), we denote the partition consisting of all open intervals \(\omega \) in \(J\) such that for each \(0\le i<j,\, x_i(a)\in K(a){\setminus }\{b_0(a),\ldots ,b_{p_0}(a)\}\), for all \(a\in \omega \), and such that \(\omega \) is maximal, i.e., for every other open interval \(\tilde{\omega }\subset J\) with \(\omega \subsetneq \tilde{\omega }\), there exist \(a\in \tilde{\omega }\) and \(0\le i<j\) such that \(x_i(a)\in \{b_0(a),\ldots ,b_{p_0}(a)\}\). Observe that this partition might be empty which is, e.g., the case when \(x_1(a)\) is equal to a boundary point \(b_k(a)\) for all \(a\in [0,1]\). However, such trivial situations (around \(a=0\)) are excluded by the transversality condition (I) formulated in the next Sect. 2.2. Knowing that condition (I) is satisfied, then the partition \({\mathcal {P}}_j|J\), \(j\ge 1\), around \(a=0\) can be thought of as the set of the (maximal) intervals of smooth monotonicity for \(x_j:J\rightarrow [0,1]\) (cf. Lemma 2.4 below). We set \({\mathcal {P}}_0|J=J\). Finally, in view of condition (I) below, observe that if a parameter \(a\in [0,1]\) is contained in an element of \({\mathcal {P}}_j|[0,1],\, j\ge 1\), then also the point \(x_0(a)\) is contained in an element of \({\mathcal {P}}_j(a)\) which implies that \(T_a^j\) is differentiable in \(x_0(a)\).
2.2 Main statement
We put two conditions on our sequence of maps \(x_j,\, j\ge 0\), around \(a=0\). The first one (see condition (I) below) is a common transversality condition for one-parameter families of interval maps which was already mentioned in the introduction. The second one (see condition (III) below) is more technical. It is used for controlling the measure of the set of partition elements with a too small image. Further, in order to apply perturbation results we require that we have uniform constants in the Lasota–Yorke inequalities for the different maps in the family (see condition (II) below). This condition does not depend on the choice of the map \(x_0\). Even if condition (III) is quite technical, assuming that conditions (I) and (II) hold, it is satisfied by many important one-parameter families of piecewise expanding maps, see Sect. 3. (Even if we suspect so, it is not clear to us if in general the transversality condition (I), possibly together with condition (II) and/or some weaker other conditions, implies condition (III).)
The transversality condition (I) requires that the derivatives of \(x_j\) and \(T_a^j\) at \(a=0\) are comparable. This is the very basic assumption in this paper. It says that locally the behaviour of the maps \(x_j\) are comparable to the behaviour of the maps \(T_a^j\). Since the LIL holds for the maps \(T_a^j\) one can therefore hope to obtain similar properties for the maps \(x_j\). Of course, in order to have transversality the choice of the map \(x_0:[0,1]\rightarrow [0,1]\) plays an important role. If, e.g., for every parameter \(a\in [0,1]\), \(x_0(a)\) is a periodic point for the map \(T_a\), then \(x_j\) will have bounded derivatives and the dynamics of \(x_j\) is completely different from the dynamics of \(T_a\). Henceforth, we will use the notations
-
(I)
The right-derivatives \(x_j'(0+),\, j\ge 1\), of \(x_j\) in \(0\) exist and there is a constant \(C\ge 1\) so that
$$\begin{aligned} \frac{1}{C}\le \left| \frac{x_j'(0+)}{(T_0^j)'(x_0(0+))}\right| \le C,\qquad \forall j\ge 1. \end{aligned}$$Further, for each \(j\ge 1\), there exists a neighbourhood \(V\subset [0,1]\) of \(0\) so that for all \(a\in V{\setminus }0\) and all \(0\le i<j\), we have \(x_i(a)\notin \{b_0(a),\ldots ,b_{p_0}(a)\}\).
Remark 2.3
Looking at the proof of the following Lemma 2.4, one can derive that condition (I) is satisfied if
where \(L\) is the Lipschitz constant of the partition points \(b_0(a),\ldots ,b_{p_0}(a)\). In other words, as soon as the initial derivative is sufficiently large we have transversality which makes it easy to verify this property numerically.
The following lemma ensures that if condition (I) holds then we can compare the \(a\)- and the \(x\)-derivatives along the forward orbit of \(x_0(a)\) on an entire, sufficiently small interval around \(0\). Its proof is given at the end of this section.
Lemma 2.4
Assume that the family \(T_a\) satisfies condition (I). Then, there exists \(\epsilon >0\) and a constant \(C\ge 1\) so that for \(\omega \in {\mathcal {P}}_j|[0,\epsilon ],\, j\ge 1\), we have
Furthermore, for each \(j\ge 1\), the number of \(a\in [0,\epsilon ]\) which are not contained in any element \(\omega \in {\mathcal {P}}_j|[0,\epsilon ]\) is finite.
Apart from transversality we require also to have uniform constants in the Lasota–Yorke inequality. Let \(\mathcal {L}_a:L^1([0,1])\rightarrow L^1([0,1])\) be the ordinary (Perron–Frobenius) transfer operator, i.e.,
The appropriate space of observables \(V_\alpha ,\, 0<\alpha \le 1\), for which \(\mathcal {L}_a\) has a spectral gap and which is convenient for our setting was introduced in [21] (see also [18, 35], and [30] which treats the higher dimensional case). \(V_\alpha \) is the space of functions of generalised bounded variation.Footnote 2
Definition 2.5
(Banach space \(V_\alpha \)) For \(\varphi \in L^1(m)\) and \(\delta >0\), we define
and, for \(0<\alpha \le 1\) and \(A>0\), set
The space \(V_\alpha \) consists of all \(\varphi \in L^1(m)\) such that \(|\varphi |_\alpha <\infty \). On \(V_\alpha \) we define the norm
(Observe that the norm \(\Vert \cdot \Vert _\alpha \) depends also on the constant \(A\).)
It follows immediately that \(V_\alpha \) contains all \(\alpha \)-Hölder functions. Further, by [21, Theorem 1.13] and [30, Proposition 3.4], the space \(V_\alpha \) together with the norm \(\Vert .\Vert _\alpha \) is a Banach space and there exists a constant \(C=C(\alpha )\) so that for all \(\varphi _1,\varphi _2\in V_{\alpha }\) we have
and
Having introduced our main Banach space \(V_\alpha \) we can now state our second condition. This condition is independent on the choice of the map \(x_0\).
-
(II)
\(T_0\) is mixing and there exist constants \(\epsilon >0\), \(C\ge 1\) and \(0<\tilde{\rho }<1\) such that for all \(\varphi \in V_\alpha \)
$$\begin{aligned} \Vert \mathcal {L}_a^n\varphi \Vert _\alpha \le C\tilde{\rho }^n\Vert \varphi \Vert _{\alpha }+C\Vert \varphi \Vert _{L^1},\quad \forall 0\le a<\epsilon . \end{aligned}$$(13)
As already mentioned above the last condition is a bit more technical. It is used to guarantee that images by \(x_n\) of “most” elements in \({\mathcal {P}}_n\) are not too small (see Lemma 4.1). For an alternative condition see Remark 4.2 below. For some better understanding of condition (III), see the example in the beginning of Sect. 3.
-
(III)
There exists \(\epsilon >0\) such that for all \(\delta _0>0\) there exists a constant \(C\) so that
$$\begin{aligned} \sum _{\omega \in {\mathcal {P}}_n|[0,\epsilon ]}\frac{1}{\Vert x_n|_\omega '\Vert _\infty }\le Ce^{n^{\delta _0}},\qquad \forall n\ge 1. \end{aligned}$$(14)
We can now state the main result of this paper. By Corollary 1.2, this result immediately implies the law of iterated logarithm. Recall the definition of \(\sigma \) in (2) (where the observable \(\varphi \) is now in the space \(V_\alpha \)).
Theorem 2.6
Let \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), be a piecewise expanding one-parameter family, satisfying properties (i)–(iii) and condition (II) for some \(0<\alpha \le 1\). If for a \(C^{1+\alpha }\) map \(x_0:[0,1]\rightarrow [0,1]\) property (10) and conditions (I) and (III) are satisfied, then for all \(\varphi \in V_\alpha \) such that \(\sigma _0(\varphi )>0\) there exists \(\epsilon >0\) so that the process \(\xi _i:[0,\epsilon ]\rightarrow {\mathbb R},\, i\ge 1\), defined by
satisfy the almost sure invariance principle for any error exponent \(\gamma >2/5\).
We conclude this section with the proof of Lemma 2.4.
Proof of Lemma 2.4
Recall that the boundary points \(b_0(a),\ldots ,b_{p_0}(a)\) are Lipschitz continuous and let \(L\) be their Lipschitz constant. By condition (I), we can take \(j_0\ge 1\) be so large that \(|x_{j_0}'(0+)|\ge \sup _{a\in [0,1]}\sup _{x\in K(a)}|\partial _aT_a(x)|/(\lambda -1)+2L+1\). Recall that by condition (I) there exists a neighbourhood \(V\subset [0,1]\), so that \(x_i(a)\notin \{b_0(a),\ldots ,b_{p_0}(a)\}\), for all \(a\in V{\setminus }0\) and \(0\le i<j_0\). Hence, by continuity, we find \(\epsilon >0\) (where \([0,\epsilon )\subset V\)) so that
Let \(j\ge 1\) and assume in the following formulas that, for the parameter values \(a\in [0,\epsilon ]\) under consideration, \(x_j\) and \(T_a^j\) are differentiable in \(a\) and \(x_0(a)\), respectively. For \(0\le k<j\) we have
which implies
For \(j>j_0\), choosing \(k=0\) and \(k=j_0\), respectively, we get the following upper and lower bounds:
where for the lower bound we used the assumption (16). It is only left to show that for each \(j\ge j_0\) the number of \(a\in [0,\epsilon ]\) which are not contained in any element \(\omega \in {\mathcal {P}}_j|[0,\epsilon ]\) is finite. This is easily done by induction over \(j\). Observe first that, by the assumption on \(x_0,\,x_j(a)\in K(a)\) for all \(j\ge 0\) and \(a\in [0,1]\). So the only case that prevents \(a\) to be contained in any element of \(\omega \in {\mathcal {P}}_j|[0,\epsilon ]\) is when \(x_i(a)\in \{b_0(a),\ldots b_{p_0}(a)\}\), for some \(i<j\). By the choice of \(\epsilon \) above inequality (16), only \(0\) and \(\epsilon \) might not be contained in any element of \(\omega \in {\mathcal {P}}_{j_0}|[0,\epsilon ]\). Assume that \(j\ge j_0\) and consider the partition \({\mathcal {P}}_{j+1}|[0,\epsilon ]\). From the lower bound in (19), we derive that \(|x_j'(a)|\ge \lambda ^{j-j_0}2L>L\) for all \(a\) contained in an element of \({\mathcal {P}}_j|[0,\epsilon ]\). Since the boundary points \(b_k(a)\) are \(\mathrm{Lip }(L)\), we have that \(x_j(a)\in K(a){\setminus }\{b_1(a),\ldots ,b_k(a)\}\) for all but finitely many \(a\in [0,\epsilon ]\). Hence, by the induction assumption we conclude that the number of \(a\in [0,\epsilon ]\) which are not contained in any element \(\omega \in {\mathcal {P}}_{j+1}|[0,\epsilon ]\) is finite. This concludes the proof of Lemma 2.4. \(\square \)
3 Tent maps and other examples
In this section we give some examples of piecewise expanding one-parameter families to which Theorem 2.6 can be applied.
We start with a trivial example which provides a good insight regarding the technical condition (III). Let \(T_0:[0,1]\rightarrow [0,1]\) be a mixing piecewise expanding map admitting a unique acip \(\mu _0\) with support, say, \([0,1]\). Let \(\varphi \in V_\alpha \) so that \(\sigma _0(\varphi )>0\). We will deduce the well-known fact that the functions \(\varphi (T_0^j(x))-\int \varphi d\mu _0\), \(j\ge 1\), satisfy the ASIP (see, e.g., [18]) from Theorem 2.6: As the one parameter family we take the constant family \(T_a\equiv T_0\), for all \(a\in [0,1]\). The map \(x_0\) is the identity, i.e., \(x_0(a)=a\). Obviously the transversality condition (I) is satisfied. The Lasota–Yorke inequality for the map \(T_0\) which we need follows from [21, Theorem 3.2]. In order to apply Theorem 2.6, the remaining condition to verify is condition (III). Let \(h=d\mu _0/dm\) be the density of \(\mu _0\). By [20] and [23], there exists a constant \(C\) so that
Since \(h\) is a fixed point of the transfer operator \(\mathcal {L}_0\), we derive that
for a.e. \(x\). Recall that the elements \({\mathcal {P}}_1(0)\) are of the form \((b_{i-1},b_i)\), for \(1\le i\le p_0\), and observe that the set of boundary points \(\{\partial T_0^n(\omega )\mid \omega \in {\mathcal {P}}_n(0)\}\) consists of maximally \(2np_0\) points. This implies that we can find points \(x_1,\ldots ,x_k,\, k\le 2np_0\), which lie close to this set, so that
(In the first inequality we used also a standard distortion estimate for piecewise expanding maps; see, e.g., (32) below.) Since, by definition, \(x_n'(a)=(T_0^n)'(a)\) and \({\mathcal {P}}_n|[0,1]={\mathcal {P}}_n(0)\), this concludes the verification of condition (III). Observe that in this trivial setting the right hand side of (14) is only increasing linearly in \(n\).
We continue by studying some non-trivial examples, first the tent maps which is the main purpose of this paper and then \(\beta \)-transformations and Markov partition preserving families. In the end of this section, we give an application of our results in order to obtain almost sure typicality results similar to the ones in [32].
3.1 Tent maps
Let \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), be a one-parameter family of tent maps, i.e., there exist \(1<\lambda \le \Lambda <\infty \) and \(0<\alpha \le 1\) so that, for each \(a\in [0,1]\), the map \(T_a:[0,1]\rightarrow [0,1]\) is continuous and there exists a turning point \(c_a\in (0,1)\) such that \(T_a|_{[0,c_a]}\) and \(T_a|_{[c_a,1]}\) are \(C^{1+\alpha }\), \(0<\alpha \le 1\), (where the Hölder constants, see (8), are uniform in \(a\)), \(\lambda \le |T_a'(x)|\le \Lambda \), for all \(x\ne c_a\), and \(T_a(1)=T_a(0)=0\). Regarding the parameter dependency we assume that properties (i) and (ii) in the beginning of Sect. 2 are satisfied. Recall the definition (4) of a transversal family of tent maps \(T_a\).
Theorem 3.1
Assume that the family \(T_a\) is transversal at \(T_0\). Further, assume that \(T_0\) is mixing and that the turning point \(c_0\) is either not periodic or if \(p\) is its period then
If \(\varphi \in V_\alpha \) so that \(\sigma _0(\varphi )>0\), then there exists \(\epsilon >0\) such that for a.e. \(a\in [0,\epsilon ]\) the turning point \(c_a\) satisfies the LIL for the function \(\varphi \) under the map \(T_a\).
Proof
In order to prove Theorem 3.1, we will verify conditions (I)–(III). Then we can apply Theorem 2.6 and Corollary 1.2 which concludes the proof. (We have also to make sure that property (iii) in Sect. 2 is satisfied. This will follow, as a by-product, from the second last paragraph in this proof.)
Regarding condition (I), we define the map \(x_0:[0,1]\rightarrow [0,1]\) as \(x_0(a)=T_a^{j_0}(c_a)\), where \(j_0\ge 1\) is so large that (4) holds for all \(j\ge j_0\). Observe that, since \(T_0\) is piecewise expanding and by (4), we find a constant \(\delta >0\) so that \(x_0(a)\notin \{0,c_a,1\}\), for all \(a\in (0,\delta ]\) (otherwise, in a neighbourhood of \(a=0\), \(c_a\) would be pre-periodic and hence \(|x_j'(0)|\) would be bounded in \(j\) contradicting the transversality (4)). Hence, property (10) is satisfied for \(x_0\) on the interval \([0,\delta ]\). As before, using once more (4), for each \(j\ge 1\), we find a neighbourhood \(V\subset [0,\delta ]\) of \(0\) so that \(x_j(a)\ne c_a\), for all \(a\in V{\setminus } 0\) (otherwise \(|x_j'(a)|\) would be bounded). We conclude that \(x_0\) satisfies condition (I) (and we can assume that \(x_0\) satisfies (10) on the interval \([0,1]\)).
We continue with the verification of condition (II) which is a condition on the family and which does not involve the map \(x_0\). The problem in verifying condition (II) is to get uniform constants in the Lasota–Yorke inequality. If \(c_0\) is not periodic let \(p\) be so large so that also in this non-periodic case inequality (22) is satisfied. [21, Theorem 3.2] and its proof shows that for all \(\delta >0\) and all \(a\in [0,1]\) we find a constant \(C=C(\delta ,a)\) and \(A=A(\delta ,a)>0\) (recall that the norm \(\Vert \cdot \Vert _\alpha \) depends also on \(A\)) so that, setting \(\rho =(2+\delta )/\lambda ^{\alpha p}\), we have
By (22), we can fix \(\delta >0\) so small that \(\rho <1\). Hence, if we show that in a neighbourhood of \(0\) we can choose the constants \(C\) and \(A\) uniformly in \(a\), then (23) combined with the assumption that \(T_0\) is mixing implies condition (II). In order to verify this uniformity of \(C\) and \(A\), we have to show that the constants \(K\) and \(A\) in [21, Lemma 3.1] can be chosen independently on \(a\) in an neighbourhood of \(0\). Set \(M=\delta \lambda ^{-\alpha }(\delta /(16+2\delta ))^{1-\alpha }\). By continuity we find an \(\epsilon >0\) so that \(T_a^i(c_a)\ne c_a\), for all \(a\in [0,\epsilon ]\) and all \(1\le i\le p-1\). Hence, we find a constant \(\kappa >0\) so that for all \(a\in [0,\epsilon ]\) the sizes of the intervals of monotonicity for \(T_a^p:[0,1]\rightarrow [0,1]\) are larger than \(\kappa \). This and the fact that \(x\mapsto |(T_a^p)'(x)|^{-1}\) is \(\alpha \)-Hölder continuous on these monotonicity intervals imply that there is an integer \(k\) and a constant \(0<\kappa '\le \kappa \) so that, for each \(a\in [0,\epsilon ]\), there is a refinement \(\{I_1(a),I_2(a),\ldots ,I_k(a)\}\) of the partition of \([0,1]\) into monotonicity intervals of \(T_a^p\) so that, for all \(1\le j\le k\), we have \(\kappa '\le |I_j(a)|\le 2\kappa '\) and
By this choice of \(\{I_1(a),\ldots ,I_k(a)\}\), we easily see that properties (16) and (17) in the proof of [21, Lemma 3.1] are satisfied. Further, setting \(A=\kappa '\delta /(16+2\delta )\) corresponds to (17) in [21]. The remaining part of the proof of [21, Lemma 3.1] immediately shows then that the constant \(K\) therein only depends on the constants \(M,\, \delta \), and \(\kappa '\) which are by construction independent on \(a\in [0,\epsilon ]\).
It is left to verify condition (III). Let \(h_a=d\mu _a/dm\) denote the density of the acip for \(T_a\). We show first that there is a positive lower bound of \(h_a\) on its support which is uniform in \(a\) close to \(0\), i.e., there exists a constant \(H<\infty \) so that
We claim that there exist \(\epsilon >0\) and an integer \(N\ge 1\) so that, for all \(a\in [0,\epsilon ]\), there is an interval \(I\subset K(a)\) of length \(1/N\) so that \(\mathrm{ess\ inf }_{x\in I}h_a(x)\ge 1/2\). We show this claim by contradiction. By condition (II) (see (44) below), we find constants \(\epsilon >0\) and \(C\) so that, for all \(a\in [0,\epsilon ]\), we have the bound \(\Vert h_a\Vert _\alpha \le C\). For \(N\ge 1\), divide the unit interval into \(N\) disjoint intervals \(I_1,\ldots ,I_N\) of length \(1/N\). For \(1\le \ell \le N\), let \(M_\ell (a)\) and \(m_\ell (a)\) denote the essential supremum and the essential infimum of \(h_a\) on \(I_\ell \), respectively. Observe that \(\sum _{\ell =1}^NM_\ell (a)/N\ge \int _0^1h_a(x)dx=1\). Now, if the claim was not true, we find \(a\in [0,\epsilon ]\) so that \(m_\ell (a)\le 1/2\), for all \(1\le \ell \le N\). From this we deduce
Since the right hand side tends to zero, uniformly in \(a\) for \(N\rightarrow \infty \), we get a contradiction. Henceforth, fix \(\epsilon >0\) and \(N\ge 1\) so that the just proven claim holds and, for \(a\in [0,\epsilon ]\), let \(I(a)\) be the interval of length \(1/N\) so that \(\mathrm{ess\ inf }_{x\in I(a)}h_a\ge 1/2\). We turn to the proof of (24). By the expansion of \(T_a\), it follows that there exists an integer \(0\le k_0\le \ln N/\ln \lambda \) such that \(c_a\in T_a^{k_0}(I(a))\). Let \(0<\epsilon '\le \epsilon \), be so that \(T_a\) is mixing for all \(a\in [0,\epsilon ']\) (this is possible by condition (II); see the beginning of the proof of Proposition 4.3). Note that \(T_a\) mixing implies that the support \(K(a)\) of the acip is equal to \([T^2_a(c_a),T_a(c_a)]\). From this we derive that property (iii) in Sect. 2 is satisfied. By [34] and since \(T_a\) is mixing, we have that \(T_a:K(a)\rightarrow K(a)\) is exact, i.e., for each set \(S\subset K(a)\) of positive Lebesgue measure it follows that \(\lim _{j\rightarrow \infty }|K(a){\setminus } T_a^j(S)|=0\). Observe that, since \(T_a\) is a tent map, if \(J\) is an interval of length close to \(K(a)(=\![T^2_a(c_a),T_a(c_a)])\) then we have \(T_a^2(K(a))=K(a)\). Thus, exactness implies that there is an integer \(k_1\) such that \(T_a^{k_1}([c_a-1/2N,c_a])=T_a^k([c_a,c_a+1/2N])=K(a)\). Since the image of an interval by \(T_a^j,\, j\ge 1\), changes continuously in \(a\) we can choose the integer \(k_1\) independently on \(a\in [0,\epsilon ]\). Hence, we conclude that \(T_a^{k_0+k_1}(I(a))=K(a)\), for all \(a\in [0,\epsilon ]\). Using the equality
the desired property (24) follows.
Let \(\epsilon >0\) be the constant in Lemma 2.4. We recall one important fact from [32, Section 6.3]: Under the assumption that the transversality condition (I) is satisfied, it is shown that there exists \(0<\epsilon '\le \epsilon \) so that without loss of generality (otherwise inverse the order) if \(0\le a_1\le a_2\le \epsilon '\) then for all \(\omega _1\in {\mathcal {P}}_n(a_1)\), \(n\ge 1\), there exists (exactly) one \(\omega _2\in {\mathcal {P}}_n(a_2)\) so that \(\omega _1\) and \(\omega _2\) have the same combinatorics up to the iteration \(n-1\). In order to apply the distortion estimate (32) below, we divide the interval \([0,\epsilon ']\) into smaller intervals. For \(n\ge 1\), let \(\mathcal {I}_n\) be a partition of \([0,\epsilon ']\) into intervals \(I\) of length approximately equal to \(\epsilon '/n^{1/\alpha }\). For \(I\in \mathcal {I}_n\), let \(a_I\) denote the right boundary point of \(I\). By the proof of Lemma 2.4, it immediately follows that each two disjoint elements in \({\mathcal {P}}_n|[0,\epsilon ']\) have different combinatorics up to \(n-1\). Further, observe that, by the definition of the partitions \({\mathcal {P}}_n|I\) and \({\mathcal {P}}_n(a)\), if \(\omega \in {\mathcal {P}}_n|I\) and \(a_1\in \omega \), then \(\omega \) and the element \(\omega _1\) in \({\mathcal {P}}_n(a_1)\) containing the point \(x_0(a_1)\) have the same combinatorics up to time \(n-1\). By the cited fact above, it follows that there exists an element \(\omega _2\in {\mathcal {P}}_n(a_I)\) having the same combinatorics as \(\omega _1\), up to the iteration \(n-1\). Hence, \(\omega \) and \(\omega _2\) have the same combinatorics up to time \(n-1\). Altogether we get that, for \(I\in \mathcal {I}_n\), there exists an injective map from \({\mathcal {P}}_n|I\) to \({\mathcal {P}}_n(a_I)\) which maps each element in \({\mathcal {P}}_n|I\) to the element in \({\mathcal {P}}_n(a_I)\) with the same combinatorics up to \(n-1\). Using Lemma 2.4 and the distortion estimate (32) below, we derive
where the last inequality follows by (24), (20), and (21) ((24) guarantees that the constant \(C\) does not depend on \(a\)). Now, we can sum over the intervals in \(\mathcal {I}_n\) which concludes the verification of condition (III) (where the right hand side in (14) increases in this setting like \(n^{1+1/\alpha }\)).\(\square \)
Instead of taking the turning points \(c_a\) as the points of interest we can choose arbitrary points \(x_0(a)\in [0,1]\), as long as the transversality condition (I) is satisfied. However, in order to verify condition (III), we will still assume that the family itself is transversal at \(T_0\). (It is quite likely that with some additional work this assumption can be dropped.)
Theorem 3.2
Assume that the family \(T_a\) is transversal at \(T_0\). Further, assume that \(T_0\) is mixing and that the turning point \(c_0\) is either not periodic or if \(p\) is its period then (22) is satisfied. Let \(x_0:[0,1]\rightarrow [0,1]\) be a \(C^{1+\alpha }\) map so that the transversality condition (I) is satisfied. If \(\varphi \in V_\alpha \) so that \(\sigma _0(\varphi )>0\), then there exists \(\epsilon >0\) such that for almost every \(a\in [0,\epsilon ]\) the point \(x_0(a)\) satisfies the LIL for the function \(\varphi \) under the map \(T_a\).
Proof
Condition (II) for the family \(T_a\) is already verified in the proof of Theorem 3.1.
Observe that in Theorem 3.2 we do not assume that \(x_0\) satisfies (10). However, we can make the following reasoning. Observe that, for all \(a\in [0,1]\), all points in \((0,1)\) are mapped after a finite number of iteration into \([T_a^2(c_a),T_a(c_a)]\). As explained in the beginning of the proof of Proposition 4.3 below, the fact that condition (II) is satisfied gives a constant \(0<\epsilon '\le \epsilon \) so that \(T_a\) is mixing for all \(a\in [0,\epsilon ']\). Hence, \(K(a)=[T_a^2(c_a),T_a(c_a)]\), for all \(a\in [0,\epsilon ']\). Since condition (I) is satisfied, we find \(0<\epsilon ''\le \epsilon '\) and an iteration \(k\ge 0\) so that \(x_k(a)\in [T_a^2(c_a),T_a(c_a)]{\setminus }\{0,c_a,1\}\), for all \(a\in [0,\epsilon '']\). Hence, renaming \(x_k\) by \(x_0\) (and considering the smaller interval \([0,\epsilon '']\)), without loss of generality, we can assume that \(x_0\) satisfies (10).
Regarding condition (III) we note that property (24) also holds in the setting of Theorem 3.2. Then we can follow word by word the last paragraph in the proof of Theorem 3.1 which concludes the verification of condition (III).\(\square \)
3.2 Generalised \(\beta \)-transformations and Markov partition preserving families
First we consider a generalised form of \(\beta \)-transformations. Let \(T:[0,\infty )\rightarrow [0,1]\) be piecewise \(C^{1+\alpha },\, 0<\alpha \le 1\), and \(0=b_0<b_1<...\) be the associated partition, where \(b_k\rightarrow \infty \) as \(k\rightarrow \infty \). We assume that \(T\) is right continuous and \(T(b_k)=0\), for each \(k\ge 0\). Further, for each \(a>1\), we have \(\Vert T'(a\, \cdot )^{-1}\Vert _{L^\infty ([0,1])}<1\) and \(\Vert T'(a\, \cdot )\Vert _{L^\infty ([0,1])}<\infty \). For \(a_0>1\), we define the one-parameter family \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), by \(T_a(x)=T((a_0+a)x)\). It is shown in [32, Lemma 5.1] that each \(T_a\) admits a unique acip \(\mu _a\) whose support \(K(a)\) is an interval adjacent to \(0\). Further, the length of K(a) is an increasing, piecewise constant function in \(a\) where the discontinuities are isolated point. Let \(\lambda (a)=\mathrm{ess\ inf }_{x\in [0,1]}|T_a'(x)|\). Regarding the verification of condition (II), we make sure that a similar condition as in (22) is satisfied: We assume that \(b_j/a_0\ne 1\), for all \(j\ge 0\), and there exists \(p\ge 1\) such that
for all \(1\le i\le p-1\) and \(k\ge 1\). Furthermore, we assume that \(|K(a)|\) is constant in a neighbourhood of \(a=0\).
Theorem 3.3
Let \(x_0:[0,1]\rightarrow [0,1]\) be a \(C^{1+\alpha }\) map satisfying condition (I). If \(\varphi \in V_\alpha \) so that \(\sigma _0(\varphi )>0\), then there exists \(\epsilon >0\) such that for almost every \(a\in [0,\epsilon ]\) the point \(x_0(a)\) satisfies the LIL for the function \(\varphi \) under the map \(T_a\).
We continue with one-parameter families preserving a Markov structure. Assume that we have a one-parameter family \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), as described in the beginning of Sect. 2 with a partition \(0\equiv b_0(a)<b_1(a)< \cdots <b_{p_0}(a)\equiv 1\) and satisfying properties (i)–(iii). We require additionally that the family \(T_a\) fulfils the following Markov property. Set \(B_k(a)=(b_{k-1}(a),b_k(a))\), \(1\le k\le p_0\).
-
(M)
For each \(1\le k\le p_0\) the image \(T_a(B_k(a))\), \(a\in [0,1]\), is a union of monotonicity intervals \(B_\ell (a)\), \(1\le \ell \le p_0\) (modulo a finite number of points).
Theorem 3.4
Let \(T_a\) be a family satisfying the Markov property (M) and let \(x_0:[0,1]\rightarrow [0,1]\) be a \(C^{1+\alpha }\) map satisfying condition (I). If \(\varphi \in V_\alpha \) so that \(\sigma _0(\varphi )>0\), then there exists \(\epsilon >0\) such that for almost every \(a\in [0,\epsilon ]\) the point \(x_0(a)\) satisfies the LIL for the function \(\varphi \) under the map \(T_a\).
Proof of Theorems 3.3 and 3.4
Due to the Markov structure, the proof of Theorem 3.4 is much easier than the proofs of Theorems 3.1, 3.2, and 3.3. We leave it as an exercise to the reader. The proof of Theorem 3.3 is very similar to the proof of Theorem 3.1. Regarding property (10) we can argue as in the proof of Theorem 3.2. The fact that \(T_0\) is mixing is shown in the last paragraph in [32, Section 5.2]. Property (25), ensures that we can go word by word along the verification of condition (II) in the proof of Theorem 3.1. Knowing that condition (II) is satisfied ensures that \(\sigma _a(\varphi )>0\) in an neighbourhood of \(0\) (see Lemma 4.5 below). It remains to verify condition (III). Observe that, by the construction of the family \(T_a\), if \(0\le a_1\le a_2\le 1\) then for all \(\omega _1\in {\mathcal {P}}_n(a_1)\), \(n\ge 1\), there exists \(\omega _2\in {\mathcal {P}}_n(a_2)\) so that \(\omega _1\) and \(\omega _2\) have the same combinatorics up to the iteration \(n-1\). Hence, if we show that the densities are uniformly bounded below on their support (see (24)), we can follow the last paragraph in the proof of Theorem 3.1 which concludes the verification of condition (III). The only obstacle in showing (24) might be the case when \(K(0)\) is smaller than \(K(a)\) but this case is excluded by our assumption on the family \(T_a\). The proof of (24) in a neighbourhood of \(a=0\) is done in detail in [32, inequality (30)]. \(\square \)
3.3 Almost sure typicality
Let \(T_a:[0,1]\rightarrow [0,1],\, a\in [0,1]\), be a one-parameter family of piecewise expanding maps as described in Sect. 2 and satisfying properties (i)–(iii) therein. Let \(x_0:[0,1]\rightarrow [0,1]\) be a \(C^{1+\alpha }\) map satisfying (10). As above let \(h_a\) denote the density of \(\mu _a\). As a corollary of Theorem 2.6 we get the following typicality result. Recall the definition of typical in (1).
Theorem 3.5
If conditions (I)–(III) are satisfied and if there exists \(\epsilon >0\) and a constant \(C\) so that
then there exists \(0<\epsilon '\le \epsilon \) so that \(x_0(a)\) is typical for \(\mu _a\) for a.e. \(a\in [0,\epsilon ']\).
Proof
For \(\kappa >0\) small, let \(\mathcal {B}=\{(q-r,q+r)\cap [0,1]\mid q\in {\mathbb Q},\, r\in {\mathbb Q}\cap [0,\kappa ]\}\). Observe that in order to prove Theorem 3.5, it is sufficient to show that there exists an \(\epsilon '>0\) so that, for each \(B\in \mathcal {B}\), \(x_0(a)\) satisfies the LIL for \(\chi _B\) under the map \(T_a\), for a.e. \(a\in [0,\epsilon ']\). From the proof of Theorem 2.6, we see that the constant \(\epsilon \) in the assertion of Theorem 2.6 does only depend on the constant \(\epsilon '\) in Proposition 4.3 and the length of the interval of parameters \(a\) on which \(\sigma _a(\varphi )>0\). Since \(\epsilon '\) in Proposition 4.3 does only depend on the family \(T_a\) and not on the observable \(\varphi \), it is enough to show that there exists \(\delta >0\) so that \(\sigma _a(\chi _B)>0\), for all \(B\in \mathcal {B}\) and all \(a\in [0,\delta ]\). By Proposition 4.3 and (44) below, and (11), we find \(\delta >0,\, C\), and \(0<\rho <1\) so that, for all \(a\in [0,\delta ]\), we have \(\Vert h_a\Vert _\infty \le C\Vert h_a\Vert _\alpha /2\le C\) and, for all \(B\in \mathcal {B}\) and \(a\in [0,\delta ]\), we have
for all \(n\ge 1\), where in the last inequality we used also (12). Altogether, for \(a\in [0,\delta ]\), we derive
Now, by taking \(\kappa >0\) in the definition of \(\mathcal {B}\) sufficiently small, we can choose \(N\) so that \(\sigma _a(\chi _B)^2\ge |B|/2C\), for all \(B\in \mathcal {B}\) and all \(a\in [0,\epsilon ]\). This concludes the proof of Theorem 3.5. \(\square \)
Remark 3.6
The question of typicality of a point \(x_0(a)\) for almost every parameter \(a\) in a general setting, was already studied in [32] (see also [8, 14], and [31] for more specific cases). Theorem 3.5 provides some alternative conditions. The method in [32] is inspired by a technique developed in [6] (see also [7] for another application of this technique). This method is very different from the one used in the present paper.
4 Preliminaries regarding the proof of Theorem 2.6
In this section, we fix an \(\epsilon >0\) which is at least so small as in Lemma 2.4 and conditions (II) and (III). When the meaning is clear, we will write \({\mathcal {P}}_j\) instead of \({\mathcal {P}}_j|[0,\epsilon ]\).
We start with an elementary but important statement about the size of exceptionally small partition elements. Since we are far away from having Markov partitions, the image \(x_j(\omega )\) of a partition element \(\omega \) in \({\mathcal {P}}_j\) might be very small (despite the expansion of the map \(x_j:\omega \rightarrow [0,1]\)). If this image is too small it contains not sufficient information in order to use it in our analysis. From condition (III) we can derive a good control of the total size of partition elements having too small images for our purpose.
Lemma 4.1
Assume that condition (III) is satisfied. Let \(d_j>0,\, j\ge 1\), be a sequence decaying at least stretched exponentially fast, i.e., there exists \(\delta >0\) so that
There exists a constant \(C\) such that, for all \(j\ge 1\), the size of the exceptional set \(E_j:=\{\omega \in {\mathcal {P}}_j\mid |x_j(\omega )|\le d_j\}\subset {\mathcal {P}}_j\), has the upper bound
Proof
Take \(\delta _0\) in condition (III) strictly less than a \(\delta \) satisfying (27). By the distortion estimate (31) below, for \(\omega \in {\mathcal {P}}_j\) such that \(|x_j(\omega )|\le d_j\), we have \(|\omega |\le C\frac{d_j}{\Vert x_j|_\omega '\Vert _\infty }\). We conclude that
Remark 4.2
Lemma 4.1 is the only place where we need condition (III). As an alternative condition to (III) it would be sufficient to require the following:
-
(III)’
For each \(\delta >0\) there are constants \(C\) and \(\beta >0\) so that
$$\begin{aligned} |\{\omega \in {\mathcal {P}}_j\mid |x_j(\omega )|\le e^{-j^\delta }\}|\le Ce^{-j^\beta }. \end{aligned}$$
We preferred to put the slightly stronger condition (III) in Sect. 2 since it is the condition which we actually verify in the examples considered in Sect. 3.
Since the sequence of maps \(x_j\) is not the iteration of a fixed dynamical system admitting an invariant measure, in order to gain information about this sequence we have to switch locally from \(x_j\) to \(T_{a_0}^j\) for some fixed parameter value \(a_0\). After having switched we can profit from the abundant existing results for such a fixed mixing piecewise expanding map \(T_{a_0}\). Very frequently we will use the exponential decay of correlations of \(T_{a_0}\). Since we can only switch locally, we need that the constants in the decay of correlation for different \(T_a\) in the family are uniform.
Proposition 4.3
(Uniform decay of correlations) Assume that the family \(T_a\) satisfies condition (II). Then, the family \(T_a\) has uniform exponential decay of correlations for \(a\) close to \(0\), i.e., there exist constants \(0<\epsilon '\le \epsilon ,\, C\ge 1\), and \(0<\rho <1\) such that for all \(a\in [0,\epsilon ']\), for all functions \(\varphi \in V_\alpha \), and all \(\psi \in L^1\) we have
Proof
The proof is a direct application of the perturbation results of Keller and Liverani [22] using the estimates in Keller [21] and Saussol [30]. Observe that \(\Vert \mathcal {L}_a^n\varphi \Vert _{L^1}=\Vert \varphi \Vert _{L^1}\). At the end of this proof we will show that for all \(\varphi \in V_{\alpha }\)
Combined with (13) in condition (II) and since \(T_0\) is mixing, by [22], we find \(0<\epsilon '\le \epsilon \) so that, for all \(a\in [0,\epsilon ']\), \(T_a\) is mixing and \(\mathcal {L}_a\) can be written as \(\mathbb {P}_a+\mathcal {Q}_a\) where \(\mathbb {P}_a\varphi =h_a\int \varphi dm\) is a one-dimensional projection and where there are constants \(C\ge 1\) and \(0<\rho <1\) (both independent on \(a\)) so that \(\Vert \mathcal {Q}_a^n\varphi \Vert _{\alpha }\le C\rho ^n\Vert \varphi \Vert _{\alpha }\), for all \(n\ge 1\). Furthermore, for the later use we note that by [22] we get a constant \(\kappa >0\) such that
For \(\varphi \in V_{\alpha }\) and \(\psi \in L^1\), we get
Hence, using (11), we derive
It remains to show (28). Recall the notation \(b_0,\ldots ,b_{p_0}\) for the partition points (the \(b_i\)’s depend on \(a\) and are Lipschitz in \(a\), say with Lipschitz constant \(L\)). Observe that \(\Vert (\mathcal {L}_a-\mathcal {L}_0)\varphi \Vert _{L^1}\) is bounded above by
Let \(J_i\) be the interval \(T_0([b_i,b_{i+1}])\) from which we subtract at each boundary point an interval of length \(\Lambda L|a|\). Since the partition points \(b_i\) are Lipschitz in \(a\), it follows that if \(y\in (T_0|_{[b_i,b_{i+1}]})^{-1}(J_i)\) then \(y\in (b_i(a'),b_{i+1}(a'))\), for all \(a'\in [0,a]\). Furthermore, we have that \((T_a|_{[b_i,b_{i+1}]})^{-1}(J_i)\subset (b_i(0),b_{i+1}(0))\). Recall property (ii) in the beginning of Sect. 2, in particular, recall that \(a'\mapsto T_{a'}'(y)\) is \(\alpha \)-Hölder. Restricting the integral above to the interval \(J_i\), we apply the triangle inequality and we split the integral into two integrals where the first one is (recall (11))
and the second one is
where we used the first inequality in (39) below (therein set \(x_{i+1}^1=x_{i+1}^2=x\)). In order to derive (28), it remains only to consider the integrals over \(T_a([b_i,b_{i+1}]){\setminus } J_i\) and \(T_0([b_i,b_{i+1}]){\setminus } J_i\), respectively. However, one easily sees that the measures of these sets are bounded by a constant times \(|a|\). Using once more (11), this concludes the proof. \(\square \)
The next lemma is a collection of various distortion estimates. Recall the notations of the partitions in Sect. 2.1. In particular, recall that \({\mathcal {P}}_j(a)\) is the partition in the phase space, while \({\mathcal {P}}_j(=\!{\mathcal {P}}_j|[0,\epsilon ])\) denotes the partition in the parameter space.
Lemma 4.4
(Distortion) There exists a constant \(C\) such that the following holds.
For \(a_1,a_2\in [0,\epsilon ]\) and \(k\ge 1\), if \(x\in [0,1]\), has the same combinatorics under \(T_{a_1}\) and \(T_{a_2}\) up to the \((k-1)\)th iteration, then
Let \(\tilde{\omega }\in {\mathcal {P}}_k\). If \(\omega \subset \tilde{\omega }\) is an interval, then
Let \(k\ge 1\) and \(a_1,a_2\in [0,\epsilon ]\) so that \(|a_1-a_2|\le 1/k^{1/\alpha }\). If \(\omega _1\in {\mathcal {P}}_k(a_1)\) and \(\omega _2\in {\mathcal {P}}_k(a_2)\) have the same combinatorics up to the \((k-1)\)th iteration then
Let \(1\le k\le \ell \). For \(\omega \in {\mathcal {P}}_\ell \) and \(a\in \omega \), we have
Proof
Property (33) follows immediately from Lemma 2.4.
We next show property (30). Set \(x^1_i=T_{a_1}^i(x)\) and \(x^2_i=T_{a_2}^i(x),\, 0\le i\le k-1\). We assume that the constant \(C\) in the assertion of Lemma 4.4 satisfies \(C\gg \delta _0^{-1}\) where \(\delta _0\) is the constant in property (i) in Sect. 2. By this choice, regarding the proof of (30) the only non-trivial situation is when \(|a_1-a_2|\ll \delta _0\). Recall that the partition points \(b_0(a)<\cdots <b_{p_0}(a)\) are Lipschitz, say with constant \(L\). Let \(0\le i\le k-1\) and take \(\ell =\ell (i)\) so that \(x_i^1\in (b_{\ell -1}(a_1),b_\ell (a_1))\). Since \(|a_1-a_2|\ll \delta _0\), we find \(y\in (0,1)\) so that \(|x_i^1-y|<L|a_1-a_2|/2\) and \(y\in (b_{\ell -1}(a),b_\ell (a))\), for all \(a\in [a_1,a_2]\). By property (ii) in Sect. 2, it follows then that \(|T_{a_1}(y)-T_{a_2}(y)|\le C|a_1-a_2|\). Hence, we derive
This estimate immediately implies (30).
Regarding property (31) observe first that by (18) (when \(k=0\) therein) we get
As in proving (30), we can assume that \(|\omega |\ll \delta _0\) (otherwise we can compensate by possibly increasing the constant \(C\)). We proceed similarly as in deriving (34). Let \(0\le i\le k-1\) and take \(\ell =\ell (i)\) so that \(x_i(a_1)\in (b_{\ell -1}(a_1),b_\ell (a_1))\). Since \(|a_1-a_2|\ll \delta _0\), we find \(y\in (0,1)\) so that \(|x_i(a_1)-y|<L|a_1-a_2|/2\) and \(y\in (b_{\ell -1}(a),b_\ell (a))\), for all \(a\in [a_1,a_2]\). By property (ii) in Sect. 2, it follows that \(|T_{a_1}'(y)-T_{a_2}'(y)|\le C|a_1-a_2|^\alpha \). Hence, by a similar calculation as in (34), we get
Thus,
from which follows that
Recall that, by property (ii) in Sect. 2, \(a\mapsto \partial _aT_a(x)\) and \(x\mapsto \partial _aT_a(x)\) are \(\alpha \)-Hölder continuous. Hence, using a “help” point \(y\) as above, we get
Combined with the \(\alpha \)-Hölder continuity of \(x_0'\), by comparing each term on the right hand side of (35) for \(a=a_1\) and \(a=a_2\), it follows
Altogether, we have
where in the last inequality we used also the fact that \(|x_k'(a_2)/(T_{a_2}^k)'(x_0(a_2))|\ge C^{-1}\), by Lemma 2.4.
It is left to prove the distortion estimate (32). Choose two points \(x_0^1\in \omega _1\) and \(x_0^2\in \omega _2\) and, for \(1\le i\le k\), let \(x_i^1=T_{a_1}^i(x_0^1)\) and \(x_i^2=T_{a_2}^i(x_0^2)\). We claim that there is a constant \(C\) so that
In order to show (38), we proceed similarly as in showing (34). Let \(0\le i\le k-1\) and take \(\ell =\ell (i)\) so that \(x_i^1\in (b_{\ell -1}(a_1),b_\ell (a_1))\). By possible increasing the constant \(C\) in the assertion of Lemma 4.4 we can assume that \(|a_1-a_2|\ll \delta _0\) and we find \(y\in (0,1)\) so that \(|x_i^1-y|<L|a_1-a_2|/2\) and \(y\in (b_{\ell -1}(a),b_\ell (a))\), for all \(a\in [a_1,a_2]\). Since \(|y-x_i^2|\le \lambda ^{-1}|T_{a_2}(y)-T_{a_2}(x_i^2)|\), we obtain
As in (34), we have \(|T_{a_2}(y)-T_{a_1}(y)|\le C|a_2-a_1|\), and note that
Altogether, recalling that \(|a_1-a_2|\le 1/k^{1/\alpha }\), we find a constant \(C\) so that
From this estimate we easily deduce (38).
By (36) and (38), for all \(0\le i\le k-1\), we obtain
which implies
Since the right hand side is bounded by a constant independent on \(k,\, a_1\) and \(a_2\), this concludes the proof of (32). \(\square \)
Recall the definition of \(\sigma _a(\varphi )\) in (2) (where \(\varphi \) here is in the space \(V_\alpha \)). In order to ensure that the functions \(\xi _j(a),\, j\ge 1\), defined in (15) depend nicely on \(a\), we have to investigate the \(a\)-dependence of \(\sigma _a\).
Lemma 4.5
(Regularity of \(a\mapsto \sigma _a\)) Assume that the family \(T_a\) satisfies condition (II). Let \(\epsilon '>0\) be the constant in Proposition 4.3. For each \(\varphi \in V_\alpha \) there exist constants \(C\) and \(\kappa >0\) such that
Proof
For simplicity we assume that \(a'=0\) and \(\int \varphi d\mu _0=0\). The general case is proven similarly (cf. the last paragraph in this proof). For a constant \(\kappa '>0\) to be determined later in the proof, let \(k_0=k_0(a,0)\) be minimal such that, for \(\bar{a}=0\) and \(\bar{a}=a\), we have
By Proposition 4.3, the absolute value of the integral in the sum is bounded by a constant (independent on \(a\)) times \(\rho ^k\) which implies that
Observe that, for all \(k\ge 0\),
We get
By (29), we immediately get that the absolute value of the first two terms on the right hand side and of the last term in the sum are bounded above by a constant (depending only on \(\Vert \varphi \Vert _\infty \)) times \(a^\kappa \). Regarding the remaining two integrals, again by (29), we have
In order to bound the integral on the right hand side, we need the following sublemma.
Sublemma 4.6
For all \(a,a'\in [0,\epsilon ']\) and \(k\ge 1\), there exists a set of intervals \({\mathcal {P}}_k(a,a')\) such that for each \(J\in {\mathcal {P}}_k(a,a')\) there exist \(\omega \in {\mathcal {P}}_k(a)\) and \(\omega '\in {\mathcal {P}}_k(a')\) such that \(J=\omega \cap \omega '\) and \(\omega \) and \(\omega '\) have the same combinatorics (up to iteration \(k-1\)). Furthermore,
(Recall that \(p_1\) is the number of elements in \({\mathcal {P}}_1(a)\).)
Proof
We show inductively in \(k\) that
This immediately implies (42).
Recall that the properties (i) and (iii) in the beginning of Sect. 2 asserts that the boundary points of the elements in \({\mathcal {P}}_1(a)\) are \(\alpha \)-Hölder continuous and the partition points \(b_j(a),\, 0\le j\le p_0\), are Lipschitz continuous in \(a\). This immediately shows (43) for \(k=1\). Let \(k>1\) and assume the assertion holds for \(k-1\). For \(J_0\in {\mathcal {P}}_{k-1}(a,a')\), let \(\omega \in {\mathcal {P}}_k(a)|J_0\) and \(j=j(\omega )\) such that \(T_a^{k-1}(\omega )\subset (b_{j-1}(a),b_j(a))\). By (30) in Lemma 4.4 and by the Lipschitz continuity of \(b_{j-1}(a)\) and \(b_j(a)\), we derive
If the right hand side is positive, then we find \(\omega '\in {\mathcal {P}}_k(a')|J_0\) with the same combinatorics as \(\omega \). Furthermore, by the distortion estimate (32) in Lemma 4.4 (where we set \(a_1=a_2=a'\)), we find a constant \(C\) (independent on \(\omega \)) such that
Since there are maximal \(p_1^{k-1}\) elements in \({\mathcal {P}}_{k-1}(a,a')\) and maximal \(p_1\) elements in \({\mathcal {P}}_k(a)|J_0\), we derive that
By the induction assumption, this concludes the proof of (43).\(\square \)
Observe that (41) implies that \(a\) is bounded above by a constant times \(e^{-k_0/\kappa ''}\). Combined with (30) in Lemma 4.4, if \(\kappa ''\le 1/2\log \Lambda \), we derive that for all \(J\in {\mathcal {P}}_k(0,a),\, k\le k_0\),
If in addition \(\kappa ''\le \min (\alpha /4\log (p_1),\alpha /2\log (p_1\Lambda ))\), it follows, for \(k\le k_0\),
Since the right hand side is bounded above by a constant times \(p_1^{k_0}a^{\alpha /2}\) which is in turn bounded above by a constant times \(a^{\alpha /4}\), altogether we derive that, for \(\kappa '\) sufficiently small, there exists a constant \(C\) (depending on \(\Vert \varphi \Vert _\infty \) and \(\Vert \varphi \Vert _\alpha \)) such that
where in the last inequality we possibly have to decrease \(\kappa >0\).
If \(a'\ne 0\) observe that by (13) it follows \(\Vert h_a\Vert _{\alpha }=\lim _{n\rightarrow \infty }\Vert \mathcal {L}_a^nh_a\Vert _{\alpha }\le C\Vert h_a\Vert _{L^1}\), and by (29) we conclude that
Combined with (11), this ensures that the constant \(C\) is uniform in \(a'\). This concludes the proof of Lemma 4.5.\(\square \)
5 Switching locally from the parameter to the phase space
The aim of this section is to prove the following Proposition 5.1 which is the main estimate needed in verifying a law of large numbers for the squares of the blocks defined in the following Sect. 6 (see Lemma 6.2 therein). Its proof is given in the end of this section. Recall that in Theorem 2.6, we assume \(\sigma _0(\varphi )>0\). Hence, by Lemma 4.5, we find a constant \(\epsilon >0\) so that \(\sigma _a(\varphi )>0\), for all \(a\in [0,\epsilon ]\). (Note that by the assumption in Lemma 4.5 the present constant \(\epsilon \) is smaller than the constant \(\epsilon '\) in Proposition 4.3 which ensures that we can apply this proposition in the following.) Let
where \(\kappa >0\) is so small as in (29) and Lemma 4.5. Fix \(\eta >0\) so small that
The expectation \(E(\xi )\) of a function \(\xi (a)\) is the integral \(\epsilon ^{-1}\int _0^\epsilon \xi (a) da\).
Proposition 5.1
There exists a constant \(C\) (depending essentially only on \(\varphi \) and the constants in the uniform exponential decay of correlation of the family \(T_a\)) such that
Furthermore, for \(m\) and \(n\) as in (47) and \(v=m-m^{1/4}\), if \(\omega \in {\mathcal {P}}_v\) such that \(\lambda _0^{-m^{1/4}}\le |x_v(\omega )|\le n^{-3/\alpha }\) then
Remark 5.2
With some more effort inequalities (47) and (48) can be proven to hold for all \(n\ge 1\) (the argument uses a similar construction as in (66) below; the upper bound of \(|x_v(\omega )|\) in (48) can be replaced, e.g., by \(2\lambda _0^{-m^{1/4}}\)).
The following lemma provides us with a tool to switch locally from the parameter space to the phase space. This can then be used in the proof of Proposition 5.1 to gain informations about the sequence \(x_j\) on the parameter space by considering the iterations \(T_{a_0}^j\) on the phase space for a fixed parameter value \(a_0\). Recall the definition (5) of \(\varphi _a\).
Lemma 5.3
(Switching locally from parameter to phase space) There exists a constant \(C\) such that the following holds. Let \(v,v_1,\ldots ,v_{\ell _0},\, 1\le \ell _0\le 4\), be integers satisfying
Let \(\omega \) be an interval such that there exists \(\tilde{\omega }\in {\mathcal {P}}_v\) with \(\omega \subset \tilde{\omega }\) and \(|x_v(\omega )|\ge \lambda _0^{-v}\). For all \(a_0\in \omega \) we have
Proof
In order to prove Lemma 5.3, we need an ingredient similar to the one provided by Sublemma 4.6. The difference here is that we compare the partitions on the parameter space with the partitions on the phase space.
Sublemma 5.4
Let \(v,\, \omega \), and \(a_0\) be as in the assertion of Lemma 5.3, and let \(v\le \nu \le v+\eta v\). There exists a set of intervals \({\mathcal {P}}_{v,\nu }(\omega ,a_0)\) such that for each \(J\in {\mathcal {P}}_{v,\nu }(\omega ,a_0)\) there exist \(\omega _1\in {\mathcal {P}}_\nu |\omega \) and \(\omega _2\in {\mathcal {P}}_{\nu -v}(a_0)|x_v(\omega )\) such that \(J=x_v(\omega _1)\cap \omega _2\) and \(x_v(\omega _1)\) and \(\omega _2\) have the same combinatorics, i.e., for \(a\in \omega _1\) and \(x\in \omega _2\), \(x_{v+i}(a)\) and \(T_{a_0}^i(x)\) have the same combinatorics for \(0\le i<\nu -v\). Furthermore,
Proof
The proof is similar to the proof of Sublemma 4.6. For \(\nu =v\) there is nothing to show (by definition \(x_v(\omega )={\mathcal {P}}_0|x_v(\omega )\)). Henceforth, we assume \(\nu >v\). Given \(v\), we show inductively in \(\nu >v\) that
If \(\nu \le v+\eta v\), inequality (51) implies then
where in the last inequality we used the condition (46) on \(\eta \) and the fact that \(|x_v(\omega )|\ge \lambda _0^{-v}\) (we used also that \(|\omega |\le C\lambda ^{-v}\) which follows from Lemma 2.4). This concludes the proof of Sublemma 4.6.
Since, by properties (i) and (iii) in Sect. 2, the boundary points of \(\omega \in {\mathcal {P}}_1(a_0)|x_v(\omega )\) are \(\alpha \)-Hölder continuous and the partition points \(b_0(a),\ldots ,b_{p_0}(a)\) are Lipschitz continuous in \(a\), this immediately shows (51) for \(\nu =v+1\). Let \(\nu >v+1\) and assume the assertion holds for \(\nu -1\). For \(J_0\in {\mathcal {P}}_{v,\nu -1}(\omega ,a_0)\), let \(\omega _2\in {\mathcal {P}}_{\nu -v}(a_0)|J_0\) and \(j=j(\omega _2)\) such that \(T_{a_0}^{\nu -v-1}(\omega _2)\subset (b_{j-1}(a_0),b_j(a_0))\). By (30) in Lemma 4.4 and by the Lipschitz continuity of \(b_{j-1}(a)\) and \(b_j(a)\), we derive
If the right hand side is positive for an appropriate choice of \(a,a'\in \omega \), then we find \(\omega _1\in {\mathcal {P}}_\nu |\omega \), where \(x_v(\omega _1)\) and \(\omega _2\) have the same combinatorics. Furthermore, by the distortion estimate (32) in Lemma 4.4, we find a constant \(C\) such that
where in the last inequality we used (33). Since there are maximal \(p_1^{\nu -v-1}\) elements in \({\mathcal {P}}_{v,\nu -1}(\omega ,a_0)\) and maximal \(p_1\) elements in \({\mathcal {P}}_{\nu -v}(a_0)|J_0\), we derive that
By the induction assumption, this concludes the proof of (43).\(\square \)
Recall that \(\xi _{v_1}(a)=\varphi _a(x_{v_1}(a))\) (see (15)). For \(a=x_v|_\omega ^{-1}(y)\) and \(a_0\in \omega \), we write
By Lemma 4.5 and (29), we easily see that the difference of the first two terms on the right hand side is bounded from above by a constant times \(|\omega |^\kappa \). To estimate the integral over the difference of the last two terms we use the partition given by Sublemma 5.4. First, observe that, by Lemma 4.5 and (11), we find a constant \(C\) only dependent on \(\varphi \) (and, in particular, not on \(a\)) so that
If \(J\in {\mathcal {P}}_{v,v_1}(\omega ,a_0)\) and \(y\in J\), then by (30) in Lemma 4.4 we have
which implies
Altogether, we obtain (recall (52))
where in the last inequality we used the assumption that \(|x_v(\omega )|\ge \lambda _0^{-v}\), the definition (45) of \(\lambda _0\), and the condition (46) on \(\eta \). Then, similarly we derive
and so on. This concludes the proof of Lemma 5.3. \(\square \)
Corollary 5.5
There exists a constant \(C\) such that the following holds. Let \(v,v_1,\ldots ,v_{\ell _0}\), \(1\le \ell _0\le 4\), be positive integers as in the assertion of Lemma 5.3 and let \(\omega \) be an interval such that there exists \(\tilde{\omega }\in {\mathcal {P}}_v\) with \(\omega \subset \tilde{\omega }\) and \(|x_v(\omega )|\ge \lambda _0^{-v}\). For all \(a_0\in \omega \) we have
Proof
Doing the change of variables \(y=x_v(a),\, a\in \omega \), by the distortion estimate (31) in Lemma 4.4, we derive
Applying Lemma 5.3 concludes the proof. \(\square \)
5.1 Proof of Proposition 5.1
We are going to show (48). Let \(\omega \) be as in the assertion. We write
Hence, in order to prove (48), it is sufficient to show that there is a constant \(C\) such that
Let \(a_0\in \omega \). By Corollary 5.5, we have
Proposition 4.3 gives (recall also (52)), for all \(\ell \ge k\),
By the normalisation (6) and applying once more Proposition 4.3, we have
Hence, we conclude
Regarding (54), the first and last term on the right hand side are fine, and also the second term since by assumption \(|x_v(\omega )|\le n^{-3/\alpha }\). For the remaining term we use the lower bound \(|x_v(\omega )|\ge \lambda _0^{-m^{1/4}}\) which gives (recall the definition of \(\lambda _0\) in (45))
In order to prove Proposition 5.1, it is only left to prove (47) which follows now easily from (48) combined with Lemma 4.1 in which we take \(d_v=Ce^{-v^{1/5}}\) (where \(C\) is taken so that \(d_v\ge \lambda _0^{-m^{1/4}}\)). Recall that by Lemma 4.1, for each \(v\ge 1\), there is an exceptional set \(E_v\subset {\mathcal {P}}_v\) so that \(|E_v|\le Cd_v^{1/2}\) and \(|x_v(\omega )|\ge d_v\) for all \(\omega \in {\mathcal {P}}_v{\setminus } E_v\). Let \({\mathcal {P}}_v^*\) be a refinement of the partition \({\mathcal {P}}_v{\setminus } E_v\) so that for \(\omega \in {\mathcal {P}}_v^*\) we have \(d_v\le |x_v(\omega )|\le 2d_v\). Since \(d_v\ge \lambda _0^{-m^{1/4}}\), by (48), we obtain
Since \(d_v^{1/2}n\le Ce^{-m^{1/5}/C}m=o(1)\), this concludes the proof of (47) and, thus, the proof of Proposition 5.1.
6 Proof of Theorem 2.6 via Skorokhod’s representation theorem
As mentioned in the introduction, in order to prove Theorem 2.6, we go along the classical, probabilistic approach in [29]. It consists in rearranging the Birkhoff sum as a sum of blocks of polynomial size where we then approximate the blocks by a martingale and apply Skorokhod’s representation theorem to it. The optimal power of the polynomial size of the blocks in our setting is \(2/3\) which gives then an error exponent \(\gamma >2/5\) in the almost sure invariance principle in Theorem 2.6. Being familiar with the technique in [29], it is natural to ask if the error exponent could be decreased to \(\gamma >1/3\): If one considers a fixed dynamical system as, e.g., in [18], then one could take \(1/2\) as the power of the polynomial size of the block and when separating these blocks by small blocks of logarithmic (or very small polynomial) size then this would lead to an error exponent \(\gamma >1/3\). However, in our setting the estimate (69) below is not good enough to be able to establish an error exponent \(\gamma >1/3\), and we do not know how to improve this estimate. In the recent work [15], Gouëzel uses spectral methods to show an almost sure invariance principle and he obtains remarkable error estimates which are independent on the dimension of the process. For example for the maps studied in [18] he gets the error exponent \(\gamma >1/4\). However, we didn’t find an easy way to apply these spectral methods to our setting. The strategy via Skorokhod’s representation theorem is also convenient here because of its simplicity. Nevertheless, since our setting is rather special, we have to go step by step through the method of building blocks and approximating by martingales. In particular, we cannot apply directly the main statement in [29, Chapter 7] since the functions \(\xi _i\) are maps on the parameter space where the concept of invariant measures does not make any sense and we are not able to verify nor to formulate an analog of a strong mixing condition (cf. [29, 7.1.2]) in our setting. However, there are statements in [29] which we can take over more or less one to one. This will keep this section of a reasonable length.
6.1 Building the blocks
Fix a constant \(\epsilon >0\) as in the beginning of Sect. 5. This ensures that we can apply all the results in Sects. 4 and 5. Take \(\delta >0\) sufficiently small (to be determined later on; see, e.g., the proof of Lemma 6.4 below). We approximate the functions \(\xi _i:[0,\epsilon ]\rightarrow [0,1],\, i\ge 1\), by stepfunctions \(\chi _i\). In order to do that, we introduce the \(\sigma \)-fields \(\mathcal {F}_i\) which are generated by the intervals in \({\mathcal {P}}_{r_i}(={\mathcal {P}}_{r_i}|[0,\epsilon ])\) where \(r_i=i+[i^{\delta }]\). Observe that, by (33) and (9),
The stepfunctions \(\chi _i\) are defined as \(\chi _i=E(\xi _i\mid \mathcal {F}_i)\). Recall the constants \(\rho \) in Proposition 4.3, \(\lambda _0\) in (45), and \(\eta \) in (46). We introduce a constant \(\rho _0\) defined as
We have the following basic properties.
Lemma 6.1
For almost every \(a\in [0,\epsilon ]\), we have
Furthermore, there exists a constant \(C\) such that for all \(i\ge 1\) and \(j\ge 0\) there exists an exceptional set of intervals \(E_{i,j}\) so that for a.e. \(a\in [0,\epsilon ]{\setminus } E_{i,j}\)
where \(E_{i,j}=\emptyset \), for \(j\le 2i^{\delta }\), and \(|E_{i,j}|\le C\rho _0^{(j-i^{\delta })/2}\), otherwise.
Proof
We show first (57). Let \(\omega \in {\mathcal {P}}_i\) and fix an arbitrary parameter \(a_\omega \) in \(\omega \). For \(\tilde{\omega }\in {\mathcal {P}}_{r_i}|\omega \) and \(a_0\in \tilde{\omega }\), by the definition of \(\varphi _a\), (29), and Lemma 4.5, we have
which implies that, for a.e. \(a_0\in \tilde{\omega }\),
Recall the estimate (55). Let \(E_i=\{\omega \in {\mathcal {P}}_i\mid |x_i(\omega )|\le \lambda ^{-\alpha i^\delta /2}\}\). We get
The integral is bounded by a constant times \(\lambda ^{-\alpha i^{\delta }}\) (recall (52)). By Lemma 4.1, we have \(|E_i|\le C\lambda ^{-\alpha i^\delta /4}\). It follows
By Borel–Cantelli this concludes the proof of (57).
We turn to the proof of (58). If \(j\le 2i^{\delta }\), there is nothing to prove. If \(j\ge 2i^{\delta }\), let \(k=\max (r_i,(i+j)/(1+\eta ))\). Denoting by \(\widetilde{{\mathcal {P}}}_k\) the \(\sigma \)-field generated by the intervals in \({\mathcal {P}}_k\), observe that we have
Hence, in order to prove (58), it is sufficient to consider the terms
For \(\omega \in {\mathcal {P}}_k\), we have, by (31),
Regarding Lemma 5.3, we can only give a good estimate of the right hand side, if the image of \(\omega \) under \(x_k\) is sufficiently large. Hence, we define the exceptional set
By the definition of \(k\), we derive that \(\rho _0^{j-i^\delta }\) is smaller than \(\rho _0^{k^\delta /2}\) for \(j\ge 2 i^\delta \) small and smaller than \(\rho _0^{\eta k}\) for \(j\) large. In particular, this implies that \(\rho _0^{j-i^\delta }\) is decaying stretched exponentially fast in \(k\). Applying Lemma 4.1, we derive that \(|E_{i,j}|\le C\rho _0^{(j-i^{\delta })/2}\) for some constant \(C\) (since the constants in the above two upper bounds for \(\rho _0^{j-i^\delta }\) are uniform, the proof of Lemma 4.1 easily shows that this constant \(C\) can be chosen uniformly in \(i\) and \(j\)). On the other hand, by the definition of \(k\), \(\rho _0^{j-i^\delta }\) is greater than \(\rho _0^{(1+\eta )k}\) which in turn is, by the definition (56) of \(\rho _0\), greater than \(\lambda _0^{-k}\). In other words \(|x_k(\omega )|\ge \lambda _0^{-k}\), for \(\omega \in {\mathcal {P}}_k{\setminus } E_{i,j}\), and we can apply (49) in Lemma 5.3 which gives (observe that by the definition of \(k\) we have \(k\le i+j\le k+\eta k\))
By Proposition 4.3 and (52), we get
Since \(i+j-k\ge \eta (j-i^\delta )/(1+\eta )\) for all \(j\ge 2i^\delta \) and for \(k\) as defined above, by the definition (56) of \(\rho _0\), we get
which concludes the proof of (58). \(\square \)
We define blocks of integers \(I_j\), \(j\ge 1\), inductively where \(I_1=\{1\}\) and \(I_j\) contains \([j^{2/3}]\) consecutive integers and there are no gaps between the blocks. For \(j\ge 1\), we set
Let \(M=M(N)\) denote the index of \(y_j\) containing \(\chi _N\). Observe that there exists a constant \(C\) so that
By (57), for a.e. \(a\in [0,\epsilon ]\), we find a constant \(C(a)\) so that
for all \(N\ge 1\). Hence, in order to prove Theorem 2.6 it is sufficient to consider the sum \(\sum _{j=1}^My_j\).
6.2 Law of large numbers for \(y_j^2\)
In this section we will prove the following key lemma. It is the main technical ingredient in the proof of Theorem 2.6.
Lemma 6.2
For any \(\gamma >2/5\) the following holds (\(\gamma \) corresponds to the error exponent in Theorem 2.6). For a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) such that
Before we start with the proof of Lemma 6.2, we recall a version of the strong law of large numbers by Gal and Koksma. Its proof is, e.g., given in [29, Theorem A.1].
Theorem 6.3
(Gal–Koksma’s strong law of large numbers) Let \(z_j,\, j\ge 1\), be zero-mean random variables and assume that there exist a real number \(q\ge 1\) and a constant \(C\) such that
Then for all \(\iota >0\), we have \(\frac{1}{n^{q/2+\iota }}\sum _{j=1}^nz_j\rightarrow 0\) almost surely.
Proof of Lemma 6.2
In this proof we will mainly work with the original \(\xi _i\) instead of their approximations \(\chi _i\). Let
Writing \(y_j^2-w_j^2=(y_j+w_j)(y_j-w_j)\), by (57), we derive that \(\sum _{j\ge 1}|y_j^2-w_j^2|\) is almost surely finite. Hence, it is sufficient to prove (62) where \(y_j\) is replaced by \(w_j\). Regarding the \(w_j\)’s we claim that it is sufficient to show that for all \(\iota >0\) there is a constant \(C\) such that
Indeed, by the estimate (47) in Proposition 5.1, we have
Hence, applying Theorem 6.3 to (63) and recalling (60), concludes the proof of (62) (where \(y_j\) is replaced by \(w_j\)).
In the following we will prove (63). Observe that \((Ew_j^2)^2\le Ew_j^4\). We have
We consider first \(Ew_j^4\). For \(\iota >0\) small, let \(S=\{(v_1,v_2,v_3,v_4)\in I_j^4\mid v_1\le v_2\le v_3\le v_4\text { and either} \, v_2-v_1\ge j^\iota \, or \, v_4-v_3\ge j^\iota \}\). We have
Let \((v_1,\ldots ,v_4)\in S\). We consider first the case when \(v_4-v_3\ge j^\iota \). In order to apply Lemma 5.3 we have to get rid of partition elements with a too small image. Let \(E_{v_3}=\{\omega \in {\mathcal {P}}_{v_3}\mid |x_{v_3}(\omega )|\ge \rho ^{j^\iota /2}\}\). By (60) and Lemma 4.1, the measure of \(E_{v_3}\) is decaying stretched exponentially fast in \(j\). For \(\omega \in {\mathcal {P}}_{v_3}{\setminus } E_3\) and \(a_0\in \omega \), by equality (53) (for \(\ell _0=4\)) combined with Lemma 5.3 (for \(\ell _0=1\)), we derive
Let \(L_\ell =x_{v_\ell }\circ x_{v_3}|_\omega ^{-1}\), \(1\le \ell \le 3\). For \(y\in x_{v_3}(\omega )\), we have \(\xi _{v_\ell }(x_{v_3}|_\omega ^{-1}(y))=\varphi _a(L_\ell (y))\) where \(a=x_{v_3}|_\omega ^{-1}(y)\). Hence, by Lemma 4.5 and (29), we derive that \(|\xi _{v_\ell }(x_{v_3}|_\omega ^{-1}(y))-\varphi _{a_0}(L_\ell (y))| \le C|\omega |^\kappa \). It follows
Observe that, by the distortion estimate (33), we have \(|L_\ell '(y)|\le C\lambda ^{-(v_3-v_1)}\), which implies \(|\chi _{x_{v_3}(\omega )}\varphi _{a_0}\circ L_\ell |_\alpha \le C|\chi _{x_{v_\ell }(\omega )} \varphi _{a_0}|_\alpha \). Since \(\Vert \chi _{x_{v_3}(\omega )}\varphi _{a_0}\circ L_\ell \Vert _\infty =\Vert \chi _{x_{v_\ell }(\omega )}\varphi _{a_0}\Vert _\infty \), by (11),
where in the last inequality we used (12). Hence, by Proposition 4.3 and (52), the absolute value of the integral on the right hand side in (66) is bounded from above by a (uniform) constant times \(\rho ^{v_4-v_3}\le \rho ^{j^\iota }\). By the definition of \(|E_{v_3}|\), we get that \(|x_{v_3}(\omega )|\rho ^{j^\iota }\le \rho ^{j^\iota /2}\), for all \(\omega \in {\mathcal {P}}_{v_3}{\setminus } E_{v_3}\). The second and last term in the sum on the right hand side of (66) decays exponentially fast in \(v_3\). Altogether, we conclude that \(|\int \prod _{\ell =1}^4\xi _{v_\ell }da|\) is decaying stretched exponentially fast in \(j\) whenever \((v_1,\ldots ,v_4)\in S\) and \(v_4-v_3\ge j^\iota \).
The case when \(v_2-v_1\ge j^\iota \) is easier. Instead of considering the functions \(L_\ell \), we can apply directly Lemma 5.3 with \(\ell _0=4\). Then, a similar reasoning gives also the stretched exponential decay of \(|\int \prod _{\ell =1}^4\xi _{v_\ell }da|\) in this case. Altogether, recalling (65) and observing that \(|S|\) is growing only polynomially fast in \(j\), for each \(\iota >0\) we find a constant \(C\) so that
Regarding the term \(E w_j^2w_k^2\), we can assume that \(k\ge j+2\) since for \(k=j+1\) we just can apply Cauchy’s inequality and (67) for estimating \(Ew_j^2w_{j+1}^2\) and Proposition 5.1 for estimating \(Ew_{j}^2Ew_{j+1}^2\) which yields the upper bound \(j^{4/3+2\iota }\) for \(|Ew_j^2w_{j+1}^2-Ew_j^2Ew_{j+1}^2|\). (For the other terms we have to give a better bound otherwise the bound we get when summing over \(k\) is not good enough.) Henceforth, let \(k\ge j+2\). We first give a good upper bound for \(|Ey_j^2w_k^2-Ey_j^2Ew_k^2|\). This is convenient, since \(y_j\) is constant on elements of the partition \({\mathcal {P}}_{m-m^{1/4}}\), where \(m\) denotes the smallest integer in \(I_k\). Let \(v=m-m^{1/4}\). Since \(d_v:=\lambda _0^{-m^{1/4}}\) is decaying stretched exponentially fast in \(v\), we can apply Lemma 4.1 and we find an exceptional set \(E_v\subset {\mathcal {P}}_v\) so that \(|E_v|\le Cd_v^{1/2}\) and \(|x_v(\omega )|\ge d_v\) for all \(\omega \in {\mathcal {P}}_v{\setminus } E_v\). Let \({\mathcal {P}}_v^*\) be a refinement of the partition \({\mathcal {P}}_v{\setminus } E_v\) so that for \(\omega \in {\mathcal {P}}_v^*\) we have \(d_v=\lambda _0^{-m^{1/4}}\le |x_v(\omega )|\le |I_k|^{-3/\alpha }\). Applying the local estimate (48) in Proposition 5.1, we obtain
Recall that \(y_j\) is constant on elements of \({\mathcal {P}}_v^*\). We get
On the other hand, by the global estimate (47) in Proposition 5.1, we have \(Ey_j^2Ew_k^2\ge Ey_j^2(|I_k|-C)\). Altogether, we derive
Writing \(E|w_j^2-y_j^2|=E|w_j+y_j||w_j-y_j|\le Cj^{2/3}E|w_j-y_j|\), by (57) and (59), we derive that \(E|w_j^2-y_j^2|\) is stretched exponentially decreasing in \(j\). Hence, by (68) and once more by (47), it follows
Recalling (64), we can now easily derive (63). This concludes the proof of Lemma 6.2.
\(\square \)
6.3 Martingale representation and embedding procedure
In this section we will follow closely Sections 3.4 and 3.5 in [29]. Let \(\mathcal {L}_j,\, j\ge 1\), be the \(\sigma \)-field generated by \((y_1,y_2,\ldots ,y_j)\), and set
Then \(\{Y_j,\mathcal {L}_j\}\) defined by \(Y_j=y_j+u_{j+1}-u_j\) is a martingale difference sequence. Recalling the definition of the \(\sigma \)-fields \(\mathcal {F}_i\) in the beginning of Sect. 6.1, we see that \(\mathcal {L}_{j-1}\subset \mathcal {F}_{i(j)}\) where \(i(j)=\max \{i\in I_{j-1}\}\). Hence, we can write
Recall (58) in Lemma 6.1 and the to it related notations. Recall also that \(i(j)\le Cj^{5/3}\) (see, e.g., (60)). Setting \(E_j=\cup _{k\ge 0} E_{i(j),k}\), we have \(|E_j|\le C\rho _0^{i(j)^{\delta }/2}\le C^2e^{-j^{5\delta /3}/C}\), and there exists a constant \(C\) so that for a.e. \(a\in [0,\epsilon ]{\setminus } E_j\) we have
Further, for \(\ell \ge 0\), we derive
for a.e. \(a\in E_{i(j),\ell }{\setminus }\cup _{k>\ell }E_{i(j),k}\). Since
for \(\delta >0\) sufficiently small, we see that \(|\{|u_{M+1}|\ge N^{\gamma }\}|\) is summable over \(N\ge 1\) (recall (60)). We conclude that, for a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
In other words, in the following we can work with the martingale difference sequence \(Y_j\) instead of \(y_j\). The \(Y_j\) inherit the law of large numbers shown for \(y_j\):
Lemma 6.4
For any \(\gamma >2/5\) the following holds. For a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
Proof
Put \(v_j=u_j-u_{j+1}\). Since \(Y_j^2=y_j^2-2y_jv_j+v_j^2\), by Lemma 6.2 and Cauchy’s inequality, it is sufficient to show that, for a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
Observe that by (60) we have \(N^{4\gamma -1}/M\ge CN^{4\gamma -8/5}\), and since \(\gamma >2/5\) we have \(4\gamma -8/5>0\). By (70) and (71), there exists a constant \(C\) so that for all \(\delta >0\) sufficiently small and all \(M\) sufficiently large we have, for \(1\le j\le M\),
and, for all \(\ell \ge 0\) and \(1\le j\le M\), we have
for a.e. \(a\in (E_{i(j),\ell }\cup E_{i(j+1),\ell }){\setminus }(\cup _{k>\ell }E_{i(j),k}\cup E_{i(j+1),k})\). Combined with (72), we get
where the right hand side is summable in \(N\). This concludes the proof of (75) and, thus, the proof of the lemma. \(\square \)
Lemma 6.5
For any \(\gamma >2/5\) the following holds. For a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
Proof
Set \(R_j=Y_j^2-E(Y_j^2\mid \mathcal {L}_{j-1})\) and observe that \(\{R_j,\mathcal {L}_j\}\) is a martingale difference sequence. By the definition of \(Y_j\) and by Minkowski’s inequality, we have
By (67), for all \(\iota >0\) we find a constant \(C\) so that \(Ew_j^4\le Cj^{4/3+\iota }\). Since \(w_j^4-y_j^4=(w_j^2+y_j^2)(w_j+y_j)(w_j-y_j)\), we can apply (59) and we derive that \(E|w_j^4-y_j^4|\) is uniformly bounded in \(j\). By (70) and (71), we derive that \(Ev_j^4\le Cj^{4\delta }\). Hence, for all \(\iota >0\), we have
and by a martingale result (see, e.g., [9]) we get that \(\sum _{j\ge 1}j^{-7/6-\iota }R_j\) converges almost surely. By Kronecker’s Lemma we conclude that, for a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
where we used (60) in the last inequality. Since \(21/30<4/5<2\gamma \) this concludes the proof of the lemma.\(\square \)
Now we apply the following martingale embedding result to the martingale difference sequence \(Y_j\). For a proof see, e.g., [17, Theorem A.1].
Theorem 6.6
(Skorokhod’s representation theorem) Let \(\{ \sum _{j=1}^MY_j,\ \mathcal {L}_M\},\, M\ge 1\), be a zero-mean and square-integrable martingale. Then there exists a probability space which supports a zero-mean and square-integrable martingale \(\{ \sum _{j=1}^M\widetilde{Y}_j,\ \widetilde{\mathcal {L}}_M\},\, M\ge 1\), a Brownian motion \(W\), and a sequence of nonnegative variables \(T_j,\, j\ge 1\), such that
-
\(\{ Y_j \}_{j\ge 1}\) and \(\{ \widetilde{Y}_j \}_{j\ge 1}\) have the same distribution;
-
\(\sum _{j=1}^M \widetilde{Y}_j = W(\sum _{j=1}^M T_j)\) almost surely;
-
\(E(T_j\mid \mathcal {G}_{j-1})=E(\widetilde{Y}_j^2\mid \mathcal {G}_{j-1})\) almost surely, where \(\mathcal {G}_j\) is the \(\sigma \)-field generated by \(\{W(t),\ 0\le t\le \sum _{\ell \le j}T_\ell \}\).
We will keep the same notation, i.e., instead of writing \(\widetilde{Y}_j\) and \(\widetilde{\mathcal {L}}_j\), we keep writing \(Y_j\) and \(\mathcal {L}_j\). Since \(\mathcal {L}_j\subset \mathcal {G}_j\), for all \(j\ge 1\), we have
almost surely. We can now show a strong law of large numbers for the sequence \(T_j\).
Lemma 6.7
For any \(\gamma >2/5\) the following holds. For a.e. \(a\in [0,\epsilon ]\), there exists a constant \(C\) so that
Proof
By (79) we get
almost surely. By Lemmas 6.4 and 6.5, the first two terms are almost surely bounded by a constant times \(N^{2\gamma }\). Write \(R_j=E(T_j\mid \mathcal {G}_{j-1})-T_j\). By (79), \(\{R_j,\mathcal {G}_j\}\) is a martingale difference sequence satisfying \(ER_j^2\le 4EY_j^4\). Hence, we can go along the proof of Lemma 6.5 and we get the same upper bound for this term.\(\square \)
Now we can go word by word along the proof of [29, Lemma 3.5.3] replacing \(1/2-\alpha /2+\gamma \) and Lemma 3.5.1 therein by \(\gamma \) and Lemma 6.7 from our setting, respectively, and we obtain
Recalling (61) and (73), this concludes the proof of Theorem 2.6.
Notes
In order that condition (II) in Lemma 4.5 is satisfied, we assume that \(T_0\) is mixing and \(c_0\) is not periodic (see proof of Theorem 3.2).
The present paper will not simplify if one considers a space of observables with more regularity (e.g., one could consider only the space \(V_1\) which corresponds to the space of functions of bounded variations). Moreover, regarding, e.g., the Hölder continuity in the estimate (40) below, which will in general not improve by taking more regular observables, the space \(V_\alpha \) seems to be indeed the natural choice.
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Acknowledgments
The author is grateful to Viviane Baladi for encouraging him to study the LIL for the critical point of piecewise expanding unimodal maps. This work was accomplished at DMA, ENS, Paris. The author was supported by the Swiss National Science Foundation.
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Schnellmann, D. Law of iterated logarithm and invariance principle for one-parameter families of interval maps. Probab. Theory Relat. Fields 162, 365–409 (2015). https://doi.org/10.1007/s00440-014-0575-7
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DOI: https://doi.org/10.1007/s00440-014-0575-7
Keywords
- Almost sure invariance principle
- Law of iterated logarithm
- One-parameter families
- Interval maps
- Transversality
- Tent maps
Mathematics Subject Classification (2010)
- 37A10
- 37E05
- 60F17