Singular stationary measures for random piecewise affine interval homeomorphisms

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.


Introduction
For the last forty years there has been an intensive interest in the study of non-autonomous real one-dimensional dynamical systems, especially in the context of the theory of groups of smooth diffeomorphisms acting on the unit circle (see e.g. [Ghy01,Nav11] and the references therein). In a probabilistic approach, such a system equipped with an appropriate probability distribution generates in a natural way a Markov process on the circle (see e.g. [Arn98,Kif86] as general references on random dynamical systems).
Recently, a continuously growing interest in random dynamics has led to an intensive study of random systems given by groups or semigroups of one-dimensional non-smooth maps, for instance interval or circle homeomorphisms (see e.g. [AM14,SZ16,GH16,GH17,Mal17,GS17]). In this paper we consider the properties of stationary measures for a certain class of such systems.
Let f 1 , . . . , f m , m ≥ 2, be homeomorphisms of a 1-dimensional compact manifold X (the closed interval or the unit circle). Such a system of maps generates a semigroup consisting of iterates f in • · · · • f i 1 for i 1 , . . . , i n ∈ {1, . . . , m}, n ∈ {0, 1, 2, . . .}. Let (p 1 , . . . , p m ) be a probability vector. A Borel probability measure on X is called stationary, if for every Borel set A ⊂ X. The Krylov-Bogolyubov Theorem shows that such a measure always exists (but is non-necessarily unique). However, in most cases little is known about its properties. Assuming some regularity of the system (e.g. forward and backward nonsingularity of the transformations) and the uniqueness of the stationary measure, which occur for a wide class of systems (see e.g. [DKN07]), we know that the stationary measure is either absolutely continuous or singular with respect to the Lebesgue measure. Determining which of the two possibilities occur is a well-known problem, especially in the context of groups of smooth diffeomorphisms acting on the circle (see e.g. [Nav17,Question 18]). Up to now, an answer has been given only in some particular cases. For instance, a conjecture by Y. Guivarc'h, V. Kaimanovich and F. Ledrappier (see [DKN09,Conjecture 1.21] states that for a finitely generated subgroup of PSL(2, R) acting smoothly on the circle, the stationary measure is singular. The conjecture was proved by Y. Guivarc'h and Y. Le Jan in [GLJ90] for non-cocompact subgroups and by B. Deroin, V. Kleptsyn and A. Navas in [DKN09] for some minimal actions of the Thompson group and subgroups of PSL(2, R) by C 2 -diffeomorphisms.
On the other hand, the absolute continuity of the stationary measure was proved to hold for a number of random systems of non-homeomorphic maps of the interval (usually expanding at least at average), see e.g. [Pel84,Buz00,AS14]. Let us note that the question of determining singularity or absolute continuity of the stationary measure is non-trivial even in the apparently simple case of two contracting similarities f 1 , f 2 of the unit interval [0, 1], given by f 1 (x) = λx, f 2 (x) = λx + 1 − λ for λ ∈ (0, 1). Then the unique stationary measure ν λ for the probability vector (1/2, 1/2) is called the symmetric Bernoulli convolution and is always either singular or absolutely continuous. It is known (see [Shm14]) that the set of parameters λ > 1/2 for which ν λ is singular has Hausdorff dimension zero, and the only known values of "singular" parameters are the reciprocals of the Pisot numbers, as proved in [Erd39]. It is a long-standing open question whether these are the only examples of singular Bernoulli convolutions. Despite many results in this direction, a complete answer is still unknown and stimulates an active research. See e.g. [PSS00,Var16] for comprehensive surveys on the subject.
In this paper we consider a random system of two piecewise affine orientation-preserving homeomorphisms of the circle with a unique common fixed point. We look at it as a system of two piecewise affine increasing homeomorphisms f − , f + of the interval [0, 1]. We assume that f i (0) = 0, f i (1) = 1 for i = −, +, each f i has one point of non-differentiability x i ∈ (0, 1) and f − (x) < x < f + (x) for x ∈ (0, 1). See Definition 2.1 and Figure 1 for a precise description. Since systems of this type were introduced in [AM14] by Alsedà and Misiurewicz, we call them Alsedà-Misiurewicz systems, or AM-systems.
Note that on the intervals (0, min(x − , x + )) and (max(x − , x + ), 1) the system {f − , f + } is equivalent, respectively, to two (typically different and non-symmetric) one-dimensional random walks, which are glued in a continuous way. This makes such systems interesting from a probabilistic point of view and we believe they can serve as models for many stochastic phenomena which appear in random one-dimensional dynamics.
For instance, if Λ(0), Λ(1) are negative, then the endpoints of the interval are attracting in average, so a typical trajectory converges to one of them, which can give rise to two intermingled basins for the step skew product F + (see e.g. [Kan94,BM08,GH17]). In this paper we assume that the Lyapunov exponents Λ(0), Λ(1) are positive. Then for almost all paths i = (i n ) n ∈ Σ + 2 , any two trajectories defined by i converge to each other, i.e. |f in • · · · • f i 1 (x) − f in • · · · • f i 1 (y)| → ∞ as n → ∞ for x, y ∈ [0, 1]. This phenomenon is called synchronization (see e.g. [Ant84,Bax89,PRK01,KV14]). Moreover, apart from purely atomic stationary measures supported at the common fixed points 0 and 1, there exists a (unique) stationary measure µ on [0, 1] such that µ({0, 1}) = 0. In this paper we study the properties of the measure µ, which we call the stationary measure for the AM-system.
In [AM14] L. Alsedà and M. Misiurewicz showed that for some parameters of an AMsystem the stationary measure µ is equal to the Lebesgue measure and conjectured that µ should be singular for typical parameters. In this paper we provide a precise condition under which the stationary measure is equal to the Lebesgue measure (Theorem 2.4) and verify the conjecture on singularity for some set of parameters, showing that for a class of AM-systems with resonant boundary derivatives (i.e. with ln f + (0)/ ln f − (0) = ln f + (1)/ ln f − (1) = −k/l ∈ Q) the measure µ is indeed singular and supported on an exceptional minimal set, which is a Cantor set of dimension smaller than 1. See Theorem 2.10 for details. We also determine the value of the Hausdorff dimension of µ in the case l = 1 (see Theorem 2.12). Furthermore, we present an interesting example of an AM-system with a singular stationary measure of full support [0, 1] (Theorem 2.16). Finally, we show that the considered systems with the same resonance are topologically conjugate (Theorem 2.15).
To our knowledge, these are the first examples of singular stationary measures for nonexpanding random systems generated by semigroups of homeomorphisms of the circle of that type. The fact that the maps are piecewise affine is especially interesting, since such systems are studied intensively and often serve as models for smooth systems (see e.g. [Nav17,Questions 12 and 16]). In a forthcoming paper [BS19] we prove that the stationary µ for an AM-system is singular and has Hausdorff dimension smaller than 1 for an open set of parameters, including also non-resonant cases.
Notice that in the resonant case mentioned above, the stationary measure is supported on an exceptional minimal set (i.e. invariant Cantor sets where the systems is minimal), while in the non-resonant one, its support is equal to the entire interval [0, 1] (see Proposition 2.6). It should be noted that the properties of exceptional minimal sets are a well-known subject of interest, especially in the context of the groups of diffeomorphisms. For instance, a conjecture of Ghys and Sullivan says that exceptional minimal sets for groups of C 2 -diffeomorphisms have Lebesgue measure zero. The hypothesis has been recently verified by B. Deroin, V. Kleptsyn and A. Navas [DKN18] for real-analytic diffeomorphisms, while the question remains open in the smooth case. Our paper contributes to the study of such sets for piecewise-linear systems.
The plan of the paper is as follows. In Section 2 we describe the AM-systems and state the results of the paper in a precise way. Section 3 contains preliminaries, while Section 4 is devoted to the proofs of the minor results and Theorem 2.4. The proofs of the main results (Theorems 2.10 and 2.12) are split into Section 5 (case l = 1) and Section 6 (case l > 1). Sections 7 and 8 contain, respectively, the proofs of Theorems 2.15 and 2.16.
Definition 2.1. An AM-system is the system {f − , f + } of increasing homeomorphisms of the interval [0, 1] of the form See Figure 1. Figure 1. An example of an AM-system.
We consider an AM-system as a random system with probabilities p − , p + , where p − , p + > 0, p − + p + = 1.
Definition 2.2. The Lyapunov exponents of an AM-system {f − , f + } with probabilities p − , p + are defined as It is known (see [AM14,GH16,GH17]) that if the Lyapunov exponents are positive, then there exists a unique stationary measure without atoms at the endpoints of [0, 1], i.e. a Borel probability measure µ on [0, 1], such that with µ({0, 1}) = 0. For details, see Theorem 3.6. Throughout the paper, by a stationary measure for an AM-system with positive Lyapunov exponents we will mean the measure µ. It is known that if the Lyapunov exponents are positive, then the measure µ is non-atomic and is either absolutely continuous or singular with respect to the Lebesgue measure (see Propositions 3.10 and 3.11).
Definition 2.3. We say that an AM-system {f − , f + } is of: Note that in the case x + < x − (which will be assumed in most of the paper, see Lemma 4.1), the system is of overlap. In [AM14, Theorem 6.1] Alsedà and Misiurewicz showed that if a − = a + = a, b − = b + = b, 1/a + 1/b = 2, p − = p − = 1/2, then the measure µ is the Lebesgue measure on [0, 1]. The first result of our paper, presented below, gives an exact condition for an AM-system to have a stationary Lebesgue measure.
Theorem 2.4. Let {f − , f + } be an AM-system with probabilities p − , p + , such that the Lyapunov exponents Λ(0), Λ(1) are positive. Then the stationary measure µ is the Lebesgue measure on [0, 1] if and only if the system is of border type and In this case we also have p − b − + p + a + = 1.
In [AM14] the authors conjectured that the stationary measure µ for an AM-system with positive Lyapunov exponents is typically singular. The main result of this paper verifies this conjecture for some set of the system parameters. First, we split the AM-systems into two kinds: resonant and non-resonant, which have different kinds of behaviour.
Definition 2.5. We say that that an AM-system {f − , f + } with probabilities p − , p + exhibits a resonance at the point 0, if More precisely, a (k : l)-resonance at 0 occurs for k, l ∈ N if which is equivalent to a − = f − (0) = ρ l , b + = f + (0) = ρ −k for some ρ ∈ (0, 1) and also to Without loss of generality, we always assume that k, l are relatively prime.
We will show that in the resonant case the (topological) support of the stationary measure µ for some parameters is a Cantor set in [0, 1] of Hausdorff dimension smaller than 1 (see Theorems 2.10 and 2.12). A different situation occurs in the non-resonant case, as shown in the following proposition (for the proof see Proposition 4.3 and Corollary 4.5).
Proposition 2.6. If an AM-system with positive Lyapunov exponents has no resonance at one of the endpoints 0, 1, then it is minimal in (0, 1) and the support of µ is equal to [0, 1].
Before stating the main results of this paper, we need to present some definitions. Let be the symmetry of [0, 1] with respect to its center.
Obviously, a system {f − , f + } is symmetric if and only if a − = a + and b − = b + . It is straightforward that for symmetric systems we have x + = I(x − ) and f + (x + ) = I(f − (x − )). Moreover, for symmetric systems the existence of (k : l)-resonance at 0 is equivalent to the existence of (k : l)-resonance at 1. Note also that if a symmetric systems exhibits (k : l)resonance, then the condition k > l is equivalent to the positivity of the exponents Λ(0), Λ(1) for p − = p + = 1/2 (see the proof of Lemma 4.1).
Definition 2.8. For an AM-system of disjoint type, we call the interval (f − (x − ), f + (x + )) the central interval of the system {f − , f + }.
The main results of this paper shows the singularity of the stationary measure µ for some symmetric AM-systems of disjoint type, which exhibit a resonance.
Theorem 2.10. Let {f − , f + } be a symmetric AM-system of disjoint type with positive Lyapunov exponents. If the system exhibits (k : l)-resonance for some relatively prime k, l ∈ N, k > l, and satisfies ρ < η, where and η ∈ (1/2, 1) is the unique solution of the equation η k+l − 2η k+1 + 2η − 1 = 0, then the stationary measure µ is singular with where supp µ denotes the topological support of µ. Moreover, supp µ is a nowhere dense perfect set consisting of all limit points of trajectories of any point x ∈ (0, 1) under {f − , f + }, which jump over the central interval infinitely many times.
Remark 2.11. The condition ρ < η is equivalent to ρx − < 1 2 and implies that the system is of disjoint type. In the case l = 1 it holds for all systems of disjoint type.
In the case l = 1 we give a more precise description of the measure µ.
Theorem 2.12. Let {f − , f + } be a symmetric AM-system of disjoint type with probabilities p − , p + , such that the Lyapunov exponents are positive. If the system exhibits (k : 1)-resonance for some k ∈ {2, 3, . . .}, then where ρ is defined as above and η − , η + ∈ (0, 1) are, respectively, the unique solutions of the equations Remark 2.13. Under the assumptions of Theorem 2.10, if l = 1 or l > 1, p − = p + = 1/2, then the stationary measure µ is a countable sum of (geometrically) similar copies, with disjoint supports, of a self-similar measure of an iterated function system with the Strong Separation Condition. In the case l = 1 this iterated function system consists of k maps, while in the case l > 1, p − = p + = 1/2 it is infinite. See Propositions 5.14 and 6.15.
The next result shows that the considered resonant systems are uniquely determined (up to topological conjugacy) by their resonance data.
Theorem 2.15. Let {f − , f + }, {g − , g + } be symmetric AM-systems of disjoint type. If both system exhibit (k : l)-resonance for some relatively prime k, l ∈ N, k > l, and satisfy ρ < η, then they are topologically conjugated, i.e. there exists an increasing homeomorphism h : [0, 1] → [0, 1] such that The last result shows that there exist symmetric resonant AM-systems with singular stationary measure of full support.
Theorem 2.16. If a symmetric AM-system with probabilities p − = p + = 1/2 and positive Lyapunov exponents exhibits (5 : 2)-resonance and satisfies ρ = η, with ρ, η defined as above, then µ is singular with Note that in this case the condition ρ = η is equivalent to which gives ρ ≈ 0.513649.
Remark 2.17. The resonance (5 : 2) was chosen because the proof is relatively short in this case. Similar arguments work also for some other values of the resonance (k : l) with l > 1.

Preliminaries
Notation. We write Z * = Z \ {0}. For j ∈ Z * we set For x ∈ R, A ⊂ R we use the notation The convex hull of a set A is denoted by conv A. We write |I| for the length of an interval I. The symbol Leb denotes the Lebesgue measure. By dim H (resp. dim B ) we denote the Hausdorff (resp. box) dimension. The Hausdorff dimension of a Borel measure ν in R n is defined as dim H ν = inf{dim H A : A is a Borel set and ν(R n \ A) = 0}.
We study {f 1 , . . . , f m } as the random systems of maps, given by the step skew product where i = (i n ) n∈N and σ : Σ + m → Σ + m is the left-side shift, i.e. σ((i n ) n∈N ) = (i n+1 ) n∈N . Let M be the space of all Borel probability measures on [0, 1]. For ν ∈ M we denote by supp ν the topological support of ν, i.e. the intersection of all closed sets in [0, 1] of full measure ν.
Definition 3.2. A measure µ ∈ M is called a stationary measure of the system {f 1 , . . . , f m } with probabilities p 1 , . . . , p m , if Analogously, the transfer operator T : L 1 ([0, 1], Leb) → L 1 ([0, 1], Leb) is defined as  Note that since the maps f i fix the endpoints of the interval, the Dirac measures at 0 and 1 are stationary for any probabilities p i . If we assume that the endpoints are repelling in average, then there exists a stationary measure with no atoms at 0, 1. More precisely, we have the following. Remark 3.7. Actually, in [GH16,GH17] the theorem was proved for systems of C 1 -diffeomorphisms, but the proof goes through if we only assume that the map are smooth in some neighbourhoods of 0, 1.
It is well-known (see e.g. [DF66, Theorem 2.5]) that whenever the operator T preserves absolute continuity and singularity of measures (with respect to the Lebesgue measure) and the stationary measure is unique, then it is of pure type (i.e. is either absolutely continuous or singular with respect to the Lebesgue measure). It is easy to see that the same holds for the measure µ from Theorem 3.6. Hence, we have the following.
Proposition 3.10. The stationary measure µ is either absolutely continuous or singular with respect to the Lebesgue measure.
Another standard fact is that µ cannot have atoms.
Proposition 3.11. The stationary measure µ is non-atomic.
The following lemma is useful in determining singularity of the measure.

Preliminary results and proof of Theorem 2.4
In this section we prove Theorem 2.4 together with other preliminary results on the AMsystems. We begin with the following observation.
Proof. The inequality x + < x − can be written as which is equivalent to By the positivity of the Lyapunov exponents for p − = p + = 1/2, which gives (1). As already noted, if the system is symmetric and exhibits a (k : l)-resonance for k > l, then the assumption on the positivity of the Lyapunov exponents for p − = p + = 1/2 is satisfied. Indeed, in this case we have a − = a + = a ∈ (0, 1) and The following lemma is used in the proof of Theorem 2.4.
Proof. An elementary calculation shows that the system is of border type if and only if which is equivalent to Suppose that the system is of border type and Then so by (2), Then which gives (2).
The following proposition, which gives the first part of Proposition 2.6, is essentially proved in [Ily10, Lemma 3] and [GH17, Proposition 2.1] (formally, in the case of diffeomorphisms). For completeness, we present the proof suited to our setup.
Proposition 4.3. If an AM-system {f − , f + } has no resonance at one of the endpoints 0, 1, then it is minimal in (0, 1). Proof. To fix notation, assume that the system has no resonance at 0 (in the other case the proof is analogous). Choose ) with some chosen n 0 ∈ Z and every ε > 0 there exist n ∈ N and i 1 , . . . , i n ∈ {−, +} such that To show (3), we choose n 0 so that K ⊂ (0, x + ) and let Since we assume that {f − , f + } has no resonance at 0, we have α ∈ R + \ Q. Hence, for any y ∈ K and δ > 0 we can find k, l ∈ N such that , if δ is chosen sufficiently small. In particular, This together with (5) shows (3) and ends the proof.
Assume now that an AM-system {f − , f + } with probabilities p − , p + has positive Lyapunov exponents, which is equivalent to Then, by Theorem 3.6, there exists a unique probability stationary measure µ for the system, such that µ({0, 1}) = 0. By Propositions 3.10 and 3.11, we have the following.
Proposition 4.4. The stationary measure µ is non-atomic. Moreover, it is either absolutely continuous or singular with respect to the Lebesgue measure.
Propositions 3.4 and 4.3 imply the following corollary, which completes the proof of Proposition 2.6.
We end the section by proving Theorem 2.4.
Proof of Theorem 2.4. The transfer operator T on L 1 ([0, 1], Leb) has the form for the constant unity function 1. If the system is of border type, then Conversely, if (6) holds, then applying it to points x ∈ [0, 1] close to the endpoints of [0, 1] we get (7). To end the proof, it is enough to use Lemma 4.2.
Remark 4.6. As noted in the introduction, for the case 5. Proofs of Theorems 2.10 (case l = 1) and 2.12.
In Theorems 2.10 and 2.12 we consider a symmetric AM-system of disjoint type {f − , f + } with probabilities p − , p + , positive Lyapunov exponents and a (k : l)-resonance for some relatively prime k, l ∈ N, k > l. In this section we prove the results in the case l = 1. The proof is divided into several parts concerning consecutive assertions of the theorems.
Since the system is symmetric, in fact we have A simple computation shows that the condition of the positivity of the Lyapunov exponents is equivalent to Note that the above considerations prove Remark 2.14.
Construction of the set Λ. Now we construct a set Λ ∈ (0, 1) which will be shown later to be the support of the measure µ in (0, 1). Our strategy is the following. First, we construct a family of disjoint closed intervals I j , j ∈ Z * , with the union I = j∈Z * I j being forward-invariant under {f − , f + }. The disjointness of I j follows from the assumption that the system is of disjoint type. We check that the intervals I −k , . . . , I −1 are mapped by f + into I 1 with separation gaps, i.e. f + (I −k ), . . . , f + (I −1 ) are disjoint subsets of I 1 (see Lemma 5.1 and Figure 3). Further iterates of these images and their similar copies generate an infinite collection of disjoint Cantor sets, whose union Λ is fully invariant and minimal under the action of {f − , f + } (see Proposition 5.10). As we wish to calculate the dimension of Λ, it is convenient to describe Λ as the union of the attractor Λ −1 of a self-similar iterated function system {φ r } k r=1 on I −1 and its similar copies. Moreover, as the successive levels of the Cantor set Λ −1 are produced during jumps over the central interval, we obtain a characterization of Λ in terms of limit points of trajectories jumping over the central interval infinitely many times (see Proposition 5.9). Let and for j ∈ Z * define The following lemma is elementary and describes the combinatorics of the intervals I j , j ∈ Z * .
Lemma 5.1. The following statements hold.
The sets I j , j ∈ Z * are pairwise disjoint and situated in (0, 1) in the increasing order with respect to j. Figure 3.
Proof. The assertion (a) follows directly from the definition of I j . To show (b), we first check sup I −2 < inf I −1 . This is equivalent to which boils down to (8). By (9), sup I −1 < inf I 1 . The rest of the assertion (b) follows directly from the above facts and the definition of I j .
The assertions (c)-(e) are easy consequences of the definition of I j , the symmetry of the system and the fact which follows from the definition of f ± .
Note that Lemma 5.1 implies f (I) ⊂ I. More precisely, for every i ∈ {−, +} and j ∈ Z * we have Proof. Enumerate the components of (0, 1) \ I by U j , j ∈ Z, such that U j is the gap between I j−1 and I j for j < 0, U 0 is the gap between I −1 and I 1 , and U j is the gap between I j and I j+1 for j > 0. Take x ∈ (0, 1) \ I. Since the system is symmetric, we can assume x ∈ U j , j ≤ 0. Then to prove the lemma it is enough to notice that by Lemma 5.1, we have Consider the maps Note that for x ∈ I −1 . Obviously, the maps φ r are contracting similarities with φ r = ρ r . Let be the limit set of the iterated function system generated by {φ r } k r=1 on I −1 . Recall that it is the unique non-empty compact set in I −1 satisfying Obviously, Λ j are pairwise disjoint compact sets and Λ j ⊂ I j . Furthermore, for n ≥ 0, r 1 , . . . , r n ∈ {1, . . . , k} let where for n = 0 we set I j;r 1 ,...,rn = I j , φ r 1 • · · · • φ rn = id. Since |φ r | = ρ r , for every j ∈ Z * and an infinite sequence r 1 , r 2 , . . . ∈ {1, . . . , k} the segments I j;r 1 ,...,rn , n ≥ 0, form a nested sequence of sets, such that Description of trajectories. Lemma 5.1 and (11) imply immediately the following.
Lemma 5.4. The following statements hold.
, for x ∈ I, jumps over the central interval at the time s, for s ≥ 0, if and only if In particular, for given j ∈ Z * and i 1 , jump over the central interval at the same times.
n=0 of x ∈ I j does not jump over the central interval at any time 0 ≤ s < n, then by Lemmas 5.1 and 5.4, (12). The other implication follows directly from Lemmas 5.1 and 5.4.
Proof. Follows directly from Lemmas 5.4, 5.5 and the definitions of the maps F j,j , G ± r .
Definition 5.8. For x ∈ (0, 1) let ω ∞ (x) be the set of limit points of all trajectories of x under {f − , f + }, which jump over the central interval infinitely many times, i.e.
Proposition 5.10. We have Proof. The first assertion follows directly from Lemma 5.3, while Proposition 5.9 implies minimality.
Propositions 5.11 and 5.12 imply the following.
Letf − ,f + : Due to (16), there is a one-to-one correspondence between stationary probability measures for the system {f − , f + } on Λ with probabilities p − , p + and for the system {f − ,f + } with probabilities p − , p + , both considered on the σ-algebra of Borel sets. Since there is a unique stationary probability measure µ for {f − , f + } on Λ, there is also a unique stationary probability measurẽ µ for {f − ,f + }. Moreover, µ = π * μ . Now we determine the structure of the measureμ.
Using this together with (19), we check that (20) for even n ∈ N (split into four cases: j < −1, j > 1, j = −1, j = 1, respectively) is equivalent to the following system of equations: (where we write r instead of r 1 ). Now we solve the system (21) together with (18). The first two equations of (21) agree with the definitions of η − , η + . Substituting them, respectively, into the third and fourth ones, we obtain Summing this over r ∈ {1, . . . , k} and using the second and third equation of (18), we have and substituting the second and first equation of (21) respectively, we arrive at a single equation which together with the first equation of (18) gives Using (22), we finally obtain The numbers c ± , β ± 1 , . . . , β ± k satisfy (21) and (18). In this way we showed that the system of equations (21) and (18) has a unique solution for which the measure ν is stationary. By the uniqueness of such a measure, we have ν =μ.
Finally, we determine the Hausdorff dimension of the measure µ. Since µ| I j = π * (η j δ j ⊗ ν j ) for j ∈ Z * by Proposition 5.14, we have Note that the measure π * (η j δ j ⊗ν j ), supported on the Cantor set Λ j , is bi-Lipschitz isomorphic (after normalization) to the measure π * (η −1 δ −1 ⊗ ν −1 ), which (after normalization) is the selfsimilar measure for the iterated function system {φ r • φ s } k r,s=1 with probabilities (β − r β + s ) k r,s=1 . It is well-known (see e.g. [Edg98, Theorem 5.2.5]) that the Hausdorff dimension of such a measure is equal to the ratio of the entropy of the measure and its Lyapunov exponent, i.e.
6. Proof of Theorem 2.10. Case l > 1 Preliminaries. In Theorem 2.10 we consider a symmetric AM-system {f − , f + } of disjoint type with probabilities p − , p + , positive Lyapunov exponents and a (k : l)-resonance for some relatively prime k, l ∈ N, k > l. In this section we deal with the case l > 1. Our approach is similar to the case l = 1, however the combinatorics of the obtained system of intervals is more complicated and produces Cantor sets which are attractors for infinite iterated function systems.
In particular, this shows that the condition ρ < η implies that the system is of disjoint type, which proves Remark 2.11. Finally, notice that the positivity of the Lyapunov exponents of the system is equivalent to Construction of the set Λ. Let us define the basic intervals I j ∈ Z * in the same manner as in the case l = 1, i.e.
Let us now explain briefly the differences compared to the case l = 1. Unlike previously, the union j∈Z * I j is no longer forward-invariant under {f − , f + }. More precisely, f + (I −k ∪ . . . ∪ I −l ) ⊂ I l , but f + (I −l+1 ∪ . . . ∪ I −1 ) is situated between I l and I l+1 , inside a larger interval J l (see Lemma 6.3 and Figure 5). Therefore, our first step is extending the family {I j } j∈Z * to a larger family {I j } j∈J consisting of similar copies of intervals f + (I −l+1 ), . . . , f + (I −1 ) and their further iterates which are not contained in the intervals obtained in previous steps of the construction (see Figures 4 and 5). As a result, we obtain a forward-invariant family of intervals, which has infinitely many elements inside each of the (disjoint) intervals J j . As before, we iterate the intervals from this family to produce a fully invariant and minimal union of disjoint Cantor sets. The corresponding iterated function system {Φ r } r∈R on I −1 is generated by the action of f + on the interval [x + , f − (x − )], which maps some of the intervals I j into I l . This infinite IFS has a Cantor attractor Λ −1 ⊂ I −1 , which is copied inside each of the intervals I j to form a suitable invariant minimal set Λ ⊂ (0, 1).
Note that this notation is compatible with our previous definition of I j for j ∈ Z * . Furthermore, for j ∈ Z * let The following lemmas describe the combinatorics of the intervals I j , j ∈ J .
Lemma 6.2. The following statements hold.
(b) The segments J j , j ∈ Z * , are pairwise disjoint.
(c) For j ∈ Z * , the segments I j , j ∈ J j , are pairwise disjoint subsets of J j .

See Figures 4 and 5.
I j,1 J j I j,l−1 I j,1,l−1 I j,1,1 I j,l−1,l−1 I j,l−1,1 I j Figure 4. A schematic view of the location of the intervals I j,j 1 ,...,jn within J j for j < 0.
Proof. The assertion (a) is straightforward. To show (b), it is enough to use (26) and check sup J j−1 < inf J j for j < 0 (and use the symmetry of the system). By a direct computation, the latter inequality is equivalent to (24). By symmetry and the definition of I j and J j , showing (c)-(f) we can assume j = −1. First, we prove (c). Since I −1 ⊂ J −1 , Lemma 6.1 implies I j ⊂ J −1 for j ∈ J −1 . To show the disjointness of I j , suppose that I −1,j 1 ,...,jn ∩ I −1,j 1 ,...,j n = ∅ for some distinct (−1, j 1 , . . . , j n ), (−1, j 1 , . . . , j n ) ∈ J −1 . We can assume n ≥ n. Applying suitable sequence of inverses of maps φ r to both segments, we can suppose j 1 = j 1 or I −1,j 1 ,...,j n = I −1 . In the first case we have a contradiction with the last assertion of Lemma 6.1, while the second case contradicts with the first assertion of it. This proves (c). The first part of (d) is straightforward. Together with (c), it shows the second part. The assertion (e) follows from (c) and the fact φ 1 < · · · < φ l−1 . The first part of (f) holds by a direct checking. In turn, together with the fact that the maps φ r reverse the orientation and φ 1 < · · · < φ l−1 , it proves the second part by induction. The assertion (g) is straightforward.
The following lemma is a direct consequence of the definition of the maps f ± and Lemma 6.2. See Figure 5.
so the family {Φ r } r∈R is a countable infinite iterated function system of contractions in I −1 satisfying lim s→∞ |φ rs (I −1 )| = 0 for any sequence (r s ) ∞ s=1 of mutually distinct elements of R. Moreover, the definition of Φ r implies This together with Lemma 6.1 implies that Φ r (I −1 ), r ∈ R, are pairwise disjoint. Similarly as before, we are interested in the limit set of this system. As the family {Φ r } r∈R is infinite, there are two limit sets one can consider: and its closure It is easy to see that they satisfy (see e.g. [MU96, Section 2]). As our goal is to find the minimal attractor of the system {f − , f + } (which equals also the support of µ), we will focus on the Λ −1 . However, we will use the set L in the proof of Proposition 6.14, as it is better suited for calculating the Hausdorff dimension.
Description of trajectories. Lemma 6.3 implies the following.
Lemma 6.5. The following statements hold.
Proof. If x ∈ J j for j < 0 (resp. j > 0), then it is enough to notice that by Lemma 6.3, Enumerate the components of (0, 1) \ J by U j , j ∈ Z, such that U j is the gap between J j−1 and J j for j < 0, U 0 is the gap between J −1 and J 1 , and U j is the gap between J j and J j+1 for j > 0.
Lemma 6.7. A trajectory {f i N • · · · • f i 1 (x)} ∞ N =0 of a point x ∈ I j , j ∈ J , does not jump over the central interval at any time 0 ≤ s < N , for some N ≥ 0, if and only if where sgn(j) = sgn(j ) and n(j) − n(j ) is even, or sgn(j) = sgn(j ) and n(j) − n(j ) is odd.
Proof. If a trajectory {f i N • · · · • f i 1 (x)} ∞ N =0 of x ∈ I j does not jump over the central interval at any time 0 ≤ s < N , then by Lemmas 6.3 and 6.5, where j ∈ J such that sgn(j) = sgn(j ) and n(j) − n(j ) is even, or sgn(j) = sgn(j ) and n(j) − n(j ) is odd. Consequently, F j,j is defined on I j and (F j,j ) −1 • f i N • · · · • f i 1 | I j is an increasing affine homeomorphism from I j onto itself, so it is identity. Therefore, f i N • · · · • f i 1 | I j = F j,j . The other implication follows from Lemmas 6.3 and 6.5 and the definitions of the maps F j,j , G ± j , H ± .
holds for some r ∈ R.
Proof. Follows directly from Lemma 6.5.
Proof. The definitions of σ s , t, t and the conditions for j imply that all the considered maps are well-defined. The assertions of the lemma follow directly from Lemmas 6.7 and 6.8, and (34), (35), (36), (37).
Define r m+1 ∈ R by r m+1 = l if t = t , p is odd, or t = t , p is even (l, 1) if t = t , p is even, or t = t , p is odd .
Take now y ∈ {0, 1}. Then, by Lemma 6.10, we see for s > 0, the trajectory defined by jumps over the central interval infinitely many times and has y as its limit point. Hence, Proposition 6.12. We have Proof. The first assertion follows directly from Lemma 6.4, while Proposition 6.11 implies minimality.
Singularity of µ. Proof. Similarly as for the case l = 1, it is enough to use Proposition 6.12.
We will prove now that dim H Λ −1 = dim H L, i.e. taking the closure does not increase the Hausdorff dimension of L. To that end, let L(∞) be the "asymptotic boundary" of the system {Φ r } r∈R , i.e. the set of all limit points of sequences (x s ) ∞ s=1 , where x s ∈ Φ rs (I −1 ) and {r s } ∞ s=1 consists of mutually distinct elements of R. It follows from [MU96, Lemma 2.1] that As the above sum is countable and the transformations Φ r are bi-Lipschitz, we obtain Using Lemmas 6.1 and 6.2, it is easy to see that where K is the limit set of the iterated function system {φ r } l−1 r=1 on J −1 . By Lemma 6.1, this system satisfies the Strong Separation Condition, so its box and Hausdorff dimension are both equal to the unique solution t 0 ∈ (0, 1) of the equation ρ lt 0 − 2ρ t 0 + 1 = 0. As noted above, we have t 0 < d, hence dim H Λ −1 = d. By Lemma 6.2, the sets Λ j , j ∈ J , are disjoint similar copies of Λ −1 , so dim H j∈J Λ j = dim H Λ −1 . To end the proof, note that Λ \ j∈J Λ j = j>0 ρ j K ∪ I(ρ j K) , hence Finally, this implies dim H Λ = d.
The following proposition gives some information about the structure of the measure µ in the case of equal probabilities p − , p + .
Like in the proof of Lemma 5.2, let and for j ∈ Z * define U j = ρ −j U 0 for j < 0 I(ρ j U 0 ) for j > 0 .
By Lemma 5.1, the following statements hold. (a) U −j = I(U j ) for j ∈ Z.

Proof of Theorem 2.16
We consider a symmetric AM-system with probabilities p − = p + = 1/2 and positive Lyapunov exponents, which exhibits (5 : 2)-resonance and satisfies ρ = η. The latter condition is equivalent to (45) ρ 7 − 2ρ 6 + 2ρ − 1 = 0 and to ρx − = 1/2. Note that this implies f − (x − ) = ρ 2 x − < 1/2, so the system is of disjoint type (see the beginning of the proof of Theorem 2.10 in the case l > 1). Define segments J j , j ∈ Z * as in the case ρ < η. We have J j = [ρ −j /2, ρ −j+1 /2] for j < 0 [I(ρ j /2), I(ρ j−1 /2)] for j > 0 , so the segments J j have pairwise disjoint interiors, each two consecutive intervals (according to the order in Z * ) have a common endpoint and j∈Z * J j = (0, 1). Similarly, defining maps φ r and intervals I j , j ∈ J as in the case ρ < η and proceeding as in the proofs of Lemmas 6.1 and 6.2, we check that for each j ∈ Z * , the intervals I j , j ∈ J j are contained in J j , have disjoint interiors and satisfy j∈J j |I j | = |J j |. Analogously, we can define maps Φ r , r ∈ R, intervals I j;r 1 ,...,rm and sets Λ j , Λ in the same way as in the case ρ < η. The maps Φ r form an iterated function system in I −1 , such that the intervals Φ r (I −1 ) have disjoint interiors and r∈R |Φ r (I −1 )| = r∈R |I −1;r | = |I −1 |. Hence, Λ −1 = I −1 and the pressure (39) satisfies P (1) = 0. The combinatorics of the intervals I j;r 1 ,...,rm is the same as in the case ρ < η, so Lemmas 6.3 and 6.4 and Propositions 6.11 and 6.12 still hold. We have Λ j = I j for j ∈ J and Λ = (0, 1).
By Theorem 3.6, there exists a unique stationary measure µ, and Proposition 6.12 implies supp µ = Λ ∪ {0, 1} = [0, 1]. By Proposition 3.11, the measure µ is non-atomic. Hence, the measure of the endpoints of the intervals I j;r 1 ,...,rm is zero. In particular, Proposition 6.15 holds in this case with the same proof.
The above facts show that {Φ r } r∈R is a countable iterated function system of contracting similarities on I −1 satisfying the Open Set Condition, with the attractor Λ −1 = I −1 . By Proposition 6.15, the probability measure is the self-similar measure for this system with probabilities β r , r ∈ R.
To prove Theorem 2.16, we show dim H µ < 1. Since by Proposition 6.15, the measure µ is a countable linear combination of µ −1 and its similar copies µ| I j , j ∈ J , it is sufficient to show dim H µ −1 < 1. Let h(µ −1 ) = − r∈R β r log β r be the entropy of µ −1 . The proof splits into two cases depending whether h(µ −1 ) is finite or infinite. To shorten the proof, we do not determine which case actually takes place, but we consider both possibilities. Suppose first that h(µ −1 ) is infinite. Then we have dim H µ −1 ≤ t 0 < 1, where t 0 = inf{t > 0 : P (t) < ∞} is the unique solution of the equation ρ lt 0 − 2ρ t 0 + 1 = 0 (see the proof of Proposition 6.14). This fact follows from [BJ18, Proposition 3.1], which is based on [KPW01, Theorem 4.1]. Actually, the mentioned results in [BJ18,KPW01] are formulated for a more specific class of iterated function systems, but the proofs are valid in the general case of self-similar systems on the interval.
Suppose now that h(µ −1 ) is finite. Recall that the self-similar iterated function system {Φ r } r∈R on I −1 is regular with the attractor Λ −1 = I −1 . In particular, the normalized Lebesgue measure L = Leb | I −1 /|I −1 | is the Gibbs and equilibrium state for the geometrical potential in dimension 1 and also the 1-conformal measure for this system on I −1 (see [MU03,Section 4.4]). Moreover, the Lyapunov exponent χ(L) = r∈R Φ r log Φ r of the measure L is finite, since (similarly as in (39)) by the definition of the set R in the considered case, ρ r+n log(ρ r+n ) + 5 r=2 ρ r log(ρ r ) > −∞.
In such a situation [MU03,Theorem 4.4.7] (see also [HMU02,Theorem 4.6]) asserts that either the self-similar measure µ −1 is equal to L or dim H µ −1 < dim H Λ −1 = 1. Therefore, to end the proof of the theorem, it is sufficient to show µ −1 = L.