Erratum to: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Vitor Araújo; Stefano Galatolo; Maria José Pacifico

doi:10.1007/s00209-015-1613-6

Mathematische Zeitschrift

February 2016, Volume 282, Issue 1–2, pp 615–621 | Cite as

Erratum to: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Authors
Authors and affiliations

Vitor Araújo
Stefano Galatolo
Maria José PacificoEmail author

Erratum

First Online: 28 December 2015

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1 Erratum to: Math. Z. (2014) 276:1001–1048 DOI 10.1007/s00209-013-1231-0

In [1, Sect. 4.3, pp. 1029–1033], the construction of a global Poincaré map with suitable properties, for a flow containing a singular hyperbolic attractor, was presented and summarized in [1, Theorem 5] as follows:

Theorem 1

[1, Theorem 5, Sect. 4, p. 1021] For an open and dense subset of $C^2$ vector fields X having a singular hyperbolic attractor $\Lambda $ on a 3-manifold, there exists a finite family $\Xi $ of cross sections and a global (n-th return) Poincaré map $R:\Xi _0\rightarrow \Xi $, $R(x)=X_{\tau (x)}(x)$ such that

(1)

the domain $\Xi _0=\Xi \!\setminus \!\Gamma $ is the entire cross sections with a family $\Gamma $ of finitely many smooth arcs removed, and $\tau :\Xi _0\rightarrow [\tau _0,+\infty )$ is a smooth function bounded away from zero by some uniform constant $\tau _0>0$.
(2)

We can choose coordinates on $\Xi $ so that the map R can be written as $F:\tilde{Q}\rightarrow Q$, $F(x,y)=(T(x),G(x,y))$, where $Q={\mathbb {I}}\times {\mathbb {I}}$, ${\mathbb {I}}=[0,1]$, and $\tilde{Q}=Q\!\setminus \!\Gamma _0$ with $\Gamma _0={\mathcal {C}}\times {\mathbb {I}}$ and ${\mathcal {C}}=\{c_1,\dots ,c_n\}\subset {\mathbb {I}}$ a finite set of points.
(3)

The map $T:{\mathbb {I}}\!\setminus \!{\mathcal {C}}\rightarrow {\mathbb {I}}$ is $C^{1+\alpha }$ piecewise monotonic with $n+1$ branches defined on the connected components of ${\mathbb {I}}\!\setminus \!{\mathcal {C}}$ and has a finite set of a.c.i.m., $\mu ^i_T$. Also $\inf |T'|>1$ where it is defined, $1/|T'|$ has universal bounded p-variation, and then $d\mu ^i_T/dm$ has bounded p-variation.
(4)

The map $G:\tilde{Q}\rightarrow {\mathbb {I}}$ preserves and uniformly contracts the vertical foliation ${\mathcal {F}}=\{\{x\}\times {\mathbb {I}}\}_{x\in {\mathbb {I}}}$ of Q: There exists $0<{\uplambda }<1$ such that $ {\text {dist}}(G(x,y_1),G(x,y_2)) \le {\uplambda }\cdot |y_1-y_2|$ for each $y_1, y_2 \in {\mathbb {I}}$. In addition, the map G satisfies a type of bounded variation regularity condition; see [1, Sect. 3].
(5)

The map F admits a finite family of physical probability measures $\mu ^{i}_{F}$ which are induced by $\mu ^i_T$ in a standard way. The Poincaré time $\tau $ is integrable both with respect to each $\mu ^{i}_{F}$ and with respect to the two-dimensional Lebesgue area measure of Q.
(6)

Moreover if, for all singularities $\sigma \in \Lambda $, we have the eigenvalue relation $-{\uplambda }_2(\sigma )>{\uplambda }_1(\sigma )$, then the second coordinate map G of F has a bounded partial derivative with respect to the first coordinate, i.e., there exists $C>0$ such that $|\partial _x G(x,y)|<C$ for all $(x,y)\in ({\mathbb {I}}\!\setminus \!\{c_1,\dots , c_n\})\times {\mathbb {I}}$.

This construction was a modification of a similar construction in [2]. The modification was done to provide injectivity. Injectivity of the global Poincaré map is important for the arguments in [1] where the physical measure is shown to be exact dimensional.

As pointed to the authors by Fan Yang, the construction presented in the paper has a problem: Although the resulting global Poincaré map R is injective, the smoothness domains of this global Poincaré map might not be strips, that is, union of stable leaves crossing the cross section. This is needed for the rest of the arguments in the section, which rely on quotienting the dynamics of R over the stable leaves to obtain a one-dimensional piecewise expanding map of class $C^{1+\alpha }$, for some $\alpha >0$.

The problem does not affect the first part of the paper, namely [1, Theorem A] on the decay of correlations of maps preserving a contracting foliation, but some changes are necessary for the application to singular hyperbolic flows in [1, Corollaries 1 & 2, p. 1006].

Corollary 1

There exists an open dense set ${\mathcal {A}}$ of vector fields (satisfying a nonresonance condition) in $SH^2(M^3)$ such that, for each $X\in {\mathcal {A}}$, we can find a finite family $\Xi $ of cross sections to the flow $X_t$ of X such that an iterate of the Poincaré first return map $F:\text {dom}(F)\subset \Xi \rightarrow \Xi $ has a finite set of SRB measures $\mu ^{i} _{F}$, each of them has exponential decay of correlations with respect to Lipschitz observables: There are $C,\Lambda \in \mathbb {R}^{+},~\Lambda <1$ satisfying for every pair $f,g:\Xi \rightarrow {\mathbb {R}}$ of Lipschitz functions

$$\begin{aligned} \left| \int f\cdot (g\circ F^{n})~d\mu ^{i} _{F} -\int g~d\mu ^{i} _{F} \int f~d\mu ^{i} _{F} \right| \le C\Lambda ^{n}||g||_{Lip}||f||_{Lip}, \quad n\ge 1. \end{aligned}$$

Corollary 2

If $X_t$ is a flow over a singular hyperbolic attractor in the setting of Corollary 1 and if, in addition, the eigenvalues of every equilibrium point $\sigma $ of X in $\Lambda $ satisfy ${\uplambda }_1(\sigma )+{\uplambda }_2(\sigma )<0$ (which includes the classical Lorenz system of ODEs), then for each regular point $x_{0}\in \Lambda $ such that $d_{\mu _X}(x_{0})$ exists, we have

$$\begin{aligned} \lim _{r\rightarrow 0}\frac{\log \tau _{r}^{X_t}(x,x_{0})}{-\log r}=d_{\mu _X}(x_{0})-1 \end{aligned}$$

for $\mu _X$-almost each $x\in \Lambda .$

In this note, we explain how the construction of the global Poincaré map R can be adjusted, in a similar way to the one originally presented in [2], to obtain [1, Corollary 1] (where the n-th return map F considered in the statement is replaced by the map R defined here) and [1, Corollary 2], which are the main results of the second part of [1].

2 Adjusting the construction of the global Poincaré map and recovering Corollary 1

To adjust the construction of the global Poincaré map, we keep Steps 1–3 in [1, Sect. 4.3], where we obtained a family $\Xi _0$ of adapted cross sections to the flow with flow-boxes giving an open cover of the attractor so that the Poincaré first return time between elements of $\Xi _0$ is bigger than some uniform positive constant. Consequently, we also have the properties stated in [1, Remark 12, p. 1031].

Now we change [1, Definition 7, p. 1032] of the global Poincaré map to be the same as in [2]:

$$\begin{aligned} R(z)=X_{\tau _0(X_{T}(z))}(X_{T}(z)), \quad z\in \Xi , \end{aligned}$$

(1.1)

where $\tau _0(x)$ is the first hitting time of x to a cross section of $\Xi $, and $T>0$ is a large threshold time (defined in the same Sect. 4.3 of [1]) ensuring Poincaré maps after this time are hyperbolic (more precisely, Proposition 7 and Lemma 6 from [1] simultaneously hold for R). The function

$$\begin{aligned} \tau (z)=\tau _0(X_T(z))+T \end{aligned}$$

is the global Poincaré time replacing the definitions in equation (4.19) of [1, p. 1032].

In this way, as shown in [2, Sect. 5], the global Poincaré map R satisfies all the properties in [1, Theorem 5] with the exception of R being a n-th return map, which is enough to prove [1, Corollary 1] using [1, Theorem A], as already written there.

3 Recovering Corollary 2

However, this adjustment does not guarantee that R is injective in general, since the orbit segment $[z,X_T(z)]$ can intersect several cross sections of $\Xi $, and this is the only way for injectivity to fail.

If injectivity does not hold, then the Steinberger relation from [1, Theorem 9, p. 1043] cannot be applied. Consequently, the proof of exact dimensionality of the singular hyperbolic flows presented in [1, Sect. 6, pp. 1042–1043] would not hold.

Moreover, the application to the results given in [1, Sect. 7] has the following problem: The almost everywhere existence of local dimension for the invariant measure might fail. We note, however, that [1, Corollary 2] deals only with target points $x_0$ where the local dimension exists.

We now show how to change [1, Sect. 7] adapting it to the return map which is considered here, in a way that [1, Corollary 2] is proved in the same form as it is stated.

The first step of the strategy implemented in [1, Sect. 7, p. 1043] is to obtain the logarithm law for the global first return map P of the cross section from the logarithm law of the (long) return map R.

We know that this latter logarithm law holds by applying [1, Proposition 11] since R has superpolynomial decay of correlations. We recall the statement of this result which is a general criteria to obtain the Logarithm law. In what follows, let $\tau _{R}(x,B_{r}(\overline{x_{0}}))$ be the number of R iterations needed for the orbit of x to enter in the target $B_{r}(\overline{x_{0}})$.

Proposition 1

[1, Proposition 11, p. 1044] For each $x_0$

$$\begin{aligned} \lim \sup _{r\rightarrow 0}\frac{\log \tau _{R}(x,B_r (x_0))}{-\log r} \ge \overline{d}_{\mu _R} (x_0)~,~\lim \inf _{r\rightarrow 0}\frac{\log \tau _{R}(x,B_r (x_0))}{-\log r} \ge \underline{d}_{\mu _R} (x_0) \end{aligned}$$

(2.1)

hold for $\mu _R$-almost every x.

Moreover, if the system has superpolynomial decay of correlations under Lipschitz observables and $d_{\mu } (x_0)$ exists, then for $\mu $-almost every x, it holds

$$\begin{aligned} \lim _{r\rightarrow 0}\frac{\log \tau _{R}(x,B_r (x_0))}{-\log r}=d_{\mu _R} (x_0). \end{aligned}$$

(2.2)

This first step was done in [1, Remark 17, p. 1044], where the properties of the long return map are important. We separate our corrections into three parts, which apply for the definition of R here presented in (1.1) and lead to the same goal of the above-mentioned Remark 17 in [1].

3.1 Invariant measures associated with the first return map P and to R

Let $\Xi =\cup _{i}\Sigma _{i}$ be the global cross section and $P:\Xi \rightarrow \Xi $ be the its first return map.

Consider the suspension flow $S_t^P$ over the cross section $\Xi $ with roof function $\tau _0$ (first return time function) and base transformation P with invariant measure $\mu _P$ such that the measure $\widetilde{\mu _P}=(\mu _P\times {\text {Leb}})/\mu _P(\tau _0)$ on the suspension is conjugated to the physical measure $\mu $ of the original flow $X_t$.

More precisely, in the space $\Xi \times [0,+\infty )$ with the distance given by the maximum of the distances in $\Xi $ (induced by the Riemannian distance from M) and in ${\mathbb {R}}$ (the Euclidean distance), we consider the relation $(x,\tau _0(x))\sim _P(P(x),0)$, extend it to be symmetric and transitive, and on the quotient space, we define $S_t^P(x,s)=(x,s+t),t\in {\mathbb {R}}$. Now, $\Phi _P:\Xi \times [0,+\infty )/\sim _P\rightarrow M, (x,s)\mapsto X_s(x)$ is a diffeomorphism with its image such that $\Phi _P(S_t^P(x,s))=X_t(\Phi _P(x,s))$. If $\mu _X$ is the physical measure for the flow, then we define $\widetilde{\mu _P}=(\Phi _P^{-1})_*\mu _X$ which is an $S_t^P$-invariant and ergodic probability measure. This measure induces a P-invariant ergodic probability $\mu _P$ such that $\widetilde{\mu _P}=(\mu _P\times {\text {Leb}})/\mu _P(\tau _0)$.

Consider also the suspension flow $S_t^R$ over the same space $\Xi $ with roof function $\tau $ and associated return map R as base transformation with the physical invariant measure $\mu _R$. On this suspension, the measure $\widetilde{\mu _R}=(\mu _R\times {\text {Leb}})/\mu _R(\tau )$ is semiconjugated to the physical measure $\mu _X$ of the flow $X_t$, that is, $\mu _X=(\Phi _R)_*\widetilde{\mu _R}$. Indeed, $\Phi _R(S_t^R(x,s))=X_t(\Phi _R(x,s))$ where $\Phi _R:\Xi \times [0,+\infty )/\sim _R\rightarrow M$ has the same expression as before, the only difference being the equivalence relation $(x,\tau (x))\sim _R(R(x),0)$, which identifies $(R(x_1),0)\sim _R(R(x_2),0)\sim _R(x_1,\tau (x_1))\sim _R(x_2,\tau (x_2))$ for $x_1,x_2$ in different cross sections with $R(x_1)=R(x_2)$. Hence, $\Phi _R$ is in general not injective: Points may have finitely many pre-images, as observed before.

From [1, Remark 12, item (2)], we know that $\Xi $ can be constructed in a way that there exists $\varepsilon _1>0$ such that

$$\begin{aligned} \underline{\tau }=\inf \{t>0: X_t(x)\in \Xi ,x\in \Xi \}\ge \varepsilon _1. \end{aligned}$$

that is, the minimum time needed to go from one cross section $\Sigma _{i}\in \Xi $ to another $\Sigma _{j}\in \Xi $ by the flow is bounded away from zero. Consequently, $0<\varepsilon _0<\inf \tau _0<\tau _0<\tau $.

Hence, fixing $0<t<\varepsilon _0$ and a measurable subset $A\subset \Xi $, the set $A \times [0,t]$ satisfies

$$\begin{aligned} \widetilde{\mu _P}(A\times [0,t])&= \mu _X\big (\Phi _P(A\times [0,t])\big ) = \mu _X\big (X_{[0,t]}(A)\big )\\&= \widetilde{\mu _R}\Big (\Phi _R^{-1}\big (X_{[0,t]}(A)\big )\Big ) \ge \widetilde{\mu _R}\big (A\times [0,t]\big ) \end{aligned}$$

and the inequality above is due to the lack of injectivity of R (and consequently of $\Phi _R$). This means that

$$\begin{aligned} \frac{\mu _P (A) \cdot t}{\mu _P(\tau _0)} \ge \frac{ \mu _R (A) \cdot t }{\mu _R(\tau )} \quad \text {thus}\quad \mu _P(A)\ge \mu _R(A)\frac{\mu _P(\tau _0)}{\mu _R(\tau )}. \end{aligned}$$

Because A was arbitrary, we see that $\mu _P\ge \kappa \cdot \mu _R$ for some constant $\kappa >0$.

3.2 Consequences for local dimension

From this, a P-invariant subset with full $\mu _R$-measure must have full $\mu _P$-measure. Indeed, let $A\subset \Xi $ be P-invariant such that $\mu _R(A)=1$ . Then, $\mu _P(A)>0$ and $\mu _P(A)=1$ since $\mu _P$ is P-ergodic. Also, from the inequality obtained in the last remark, we have that for all sufficiently small $\delta >0$

$$\begin{aligned} \frac{\log (\mu _R(B_{\delta }(x)))}{\log \delta } \ge \frac{\log (\mu _P(B_{\delta }(x))/\kappa )}{\log \delta } \end{aligned}$$

and so $\overline{d}_{\mu _R}(x)\ge \overline{d}_{\mu _P}(x)$ and $\underline{d}_{\mu _R}(x)\ge \underline{d}_{\mu _P}(x)$ for all $x\in \Xi $.

3.2.1 Local dimensions of $\mu _X, \mu _P$ and $\mu _R$

Let $x_0\in M$ be such that $d_{\mu _X}(x_0)=\lim _{r\rightarrow 0}\frac{\log \mu _X(B_r(x_0))}{\log r}$ be well defined, and let $(x,s)\in \Xi \times [0,+\infty )/\sim _P$ be such that $\Phi _P(x,s)=x_0$ with $0\le s<\tau _0(x)$. Since both $\overline{d}_{\mu _X}$ and $\underline{d}_{\mu _X}$ are $X_t$-invariant, then the points where $d_{\mu _X}$ exists form another invariant subset. Hence, we can assume without loss that $s>0$.

Lemma 1

If $d_{\mu _X}$ exists at some point $x_0$ for the flow, then both $d_{\mu _R}$ and $d_{\mu _P}$ exist at some point $\overline{x_0}$ in the intersection of the orbit of $x_0$ with $\Xi $ and are equal to $d_{\mu _X}(x_0)-1$.

Proof

Since locally the distance in $\Xi \times [0,+\infty )/\sim _P$ is the maximum of the distances in $\Xi $ and in ${\mathbb {R}}$, then we get $B_\delta (x,s)=B_\delta (x)\times (s-\delta ,s+\delta )$ and

$$\begin{aligned} \widetilde{\mu _P}(B_\delta (x,s)) =\frac{1}{\mu _P(\tau _0)}\cdot (2\delta )\cdot \mu _P(B_\delta (x)). \end{aligned}$$

Note that the distance in $\Xi $ is induced by the distance in the manifold. Moreover, $\mu _X(B_r(x))=\widetilde{\mu _P}\big (\Phi ^{-1}_P(B_r(x))\big )$ and $\Phi _P$ is injective and locally a diffeomorphism. Hence, we can find $\kappa >0$ such that

$$\begin{aligned} B_{r/\kappa }(x,s) \subset \Phi ^{-1}_P(B_r(x)) \subset B_{\kappa r}(x,s) \end{aligned}$$

for all small enough $r>0$ (we can take $\kappa = \Vert D\Phi _P(x,s)\Vert +\Vert D\Phi _P(\Phi _P(x,s))^{-1}\Vert $). This ensures that

$$\begin{aligned} d_{\mu _X}(x_0) \ge \limsup _{r\rightarrow 0}\frac{\log \big ((2\kappa r)\cdot \mu _P(B_{\kappa r}(x))/\mu _P(\tau _0)\big )}{\log r} \ge 1+\overline{d}_{\mu _P}(x); \end{aligned}$$

and also

$$\begin{aligned} d_{\mu _X}(x_0) \le \liminf _{r\rightarrow 0}\frac{\log \big ((2r/\kappa ) \cdot \mu _P(B_{r/\kappa }(x))/\mu _P(\tau _0)\big )}{\log r} \le 1+\underline{d}_{\mu _P}(x). \end{aligned}$$

Hence, $d_{\mu _P}(x)=d_{\mu _X}(x_0)-1$ where x is the first hit of $x_0$ to $\Xi $ in negative time.

Therefore, if $d_{\mu _X}(x_0)$ exists, then $d_{\mu _P}(x)$ exists with the value $d_{\mu _X}(x_0)-1$ for all points $x\in \Xi $ in the same orbit of $x_0$.

Now for $\mu _R$: since $\Phi _R:\Xi \times [0,+\infty )/\sim _R\rightarrow M$ is a finite-to-one local diffeomorphism almost everywhere, there are $i=i(x_0)\in {\mathbb {Z}}^+$, $(x_j,s_j)\in \Xi \times (0,+\infty )/\sim /R$ such that $\Phi _R(x_j,s_j)=x$ and $0<s_j<\tau (x_j); j=1,\dots ,i$.

Hence, we can find $\kappa '>0$ such that we obtain the disjoint unions

$$\begin{aligned} \cup _{j=1}^iB_{r/\kappa '}(x_1,t_j) \subset \Phi ^{-1}_R(B_r(x)) \subset \cup _{j=1}^iB_{\kappa ' r}(x_1,t_j) \end{aligned}$$

for all small enough $r>0$ (we can take $\kappa '= \sum _{j=1}^i\Vert D\Phi _R(x_j,s_j)\Vert +\Vert D\Phi _R(\Phi _R(x_j,s_j))^{-1}\Vert $) implying the inequality

$$\begin{aligned} d_{\mu _X}(x_0) \ge \limsup _{r\rightarrow 0}\frac{\log \big (\sum _{j=1}^i(2\kappa ' r)\cdot \mu _R(B_{\kappa ' r}(x_j))/\mu _R(\tau )\big )}{\log r} \ge 1+\min _{1\le j\le i}\overline{d}_{\mu _R}(x_j). \end{aligned}$$

But, as already observed, we also have $\overline{d}_{\mu _R}(x_j)\ge \overline{d}_{\mu _P}(x_j)$ and $\underline{d}_{\mu _R}(x_j)\ge \underline{d}_{\mu _P}(x_j)$ for all $j=1,\dots ,i$. Hence, there exists $1\le h\le i$ such that

$$\begin{aligned} \overline{d}_{\mu _R}(x_h) = \min _{1\le j\le i}\overline{d}_{\mu _R}(x_j) \le \underline{d}_{\mu _R}(x_h) \end{aligned}$$

allowing us to conclude $\overline{d}_{\mu _R}(x_h)=\overline{d}_{\mu _P}(x_h)=\underline{d}_{\mu _R}(x_h)=\underline{d}_{\mu _P}(x_h)$. $\square $

3.3 The logarithm law for the first return map

We will now prove a logarithm law for the first return map P for an exact dimensional target point.

3.3.1 The logarithm law for R implies the logarithm law for P

Let us suppose that $\overline{x_0}$ is a point as in the statement of Lemma 1 at the previous Sect. 2.2, for which the local dimension of the measures ${\mu _R}$ and ${\mu _P}$ coincides. Applying [1, Proposition 11], we know that there exists a full ($\mu _R$ and $\mu _P$)-measure subset $A\subset \Xi $ such that

$$\begin{aligned} \limsup _{r\rightarrow 0}\frac{\log (\tau _{R}(x,B_{r}(\overline{x_{0}}))}{-\log r} \le d_{\mu _R }(\overline{x_{0}}), \quad x\in A. \end{aligned}$$

Let also $\tau _{P}(x,B_{r}(\overline{x_{0}}))$ be the number of P iterations needed for the orbit of x to enter in the target $B_{r}(\overline{x_{0}})$. Note that if $y=R^{n}(x)$, then $y=P^{m}(x)$ with $m\le n\left\lceil \frac{T}{\underline{\tau } }\right\rceil $ (one iteration of R corresponds at most to $\left\lceil \frac{T}{\underline{\tau }}\right\rceil $ iterations of P, where $\left\lceil z\right\rceil $ is the least integer equal or greater than $z\in {\mathbb {R}}$). Then,

$$\begin{aligned} \tau _{P}(x,B_{r}(\overline{x_{0}})) \le \left\lceil \frac{T}{\underline{\tau }}\right\rceil \tau _{R}(x,B_{r}(\overline{x_{0}})) \quad \text {and}\quad \limsup _{r\rightarrow 0}\frac{\log (\tau _{P}(x,B_{r}(\overline{x_{0}}))}{-\log r} \le d_{\mu _R }(\overline{x_{0}}). \end{aligned}$$

By Sect. 2.1, this holds for $\mu _P$-almost each x, and by the choice of $\overline{x_0}$, we have $d_{\mu _R }(\overline{x_{0}})=d_{\mu _P }(\overline{x_{0}})$. Then, the logarithm law also holds for P.

3.4 The logarithm law for the flow

After the changes described in the previous Sects. 2.1, 2.2 and 2.3 of this erratum, we can continue the construction as in [1, Sect. 7, p. 1044]: By applying [1, Proposition 12] and [1, Proposition 13], we get the logarithm law for the flow for each target point where the local dimension exists, leading to [1, Corollary 2].

References

1.
Araujo, V., Galatolo, S., Pacifico, M.J.: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors. Math. Zeitschrift 276(3–4), 1001–1048 (2014)MATHMathSciNetCrossRefGoogle Scholar
2.
Araújo, V., Pujals, E.R., Pacifico, M.J., Viana, M.: Singular-hyperbolic attractors are chaotic. Trans. A.M.S. 361, 2431–2485 (2009)MATHCrossRefGoogle Scholar

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Authors and Affiliations

Vitor Araújo
- 1
Stefano Galatolo
- 2
Maria José Pacifico
- 3
Email author

1.Instituto de MatemáticaUniversidade Federal da BahiaSalvadorBrazil
2.Dipartimento di Matematica ApplicataUniversita di PisaPisaItaly
3.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

Erratum to: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

1 Erratum to: Math. Z. (2014) 276:1001–1048 DOI 10.1007/s00209-013-1231-0

Theorem 1

Corollary 1

Corollary 2

2 Adjusting the construction of the global Poincaré map and recovering Corollary 1

3 Recovering Corollary 2

Proposition 1

3.1 Invariant measures associated with the first return map P and to R

3.2 Consequences for local dimension

3.2.1 Local dimensions of \(\mu _X, \mu _P\) and \(\mu _R\)

Lemma 1

Proof

3.3 The logarithm law for the first return map

3.3.1 The logarithm law for R implies the logarithm law for P

3.4 The logarithm law for the flow

References

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