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Synchronization in minimal iterated function systems on compact manifolds

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds.


Introduction
We consider iterated functions systems generated by a collection of diffeomorphisms on a compact manifold. Our focus will lie on the combination of two properties. First, the iterated functions systems are minimal, meaning that orbits of the semigroup are dense in the manifold. Second, we study the occurrence of synchronization for minimal iterated function systems, meaning that typically orbits converge to each other when iterated by the same sequence of diffeomorphisms. Iterated function systems are described by a skew product system over a shift operator. This puts the study of synchronization for iterated function systems into the larger perspective of generalized synchronization, which we introduce next.
1.1. Generalized synchronization. Generalized synchronization refers to synchronization caused by external forcing rather than coupling. It is a phenomenon in dynamical systems that are driven by other dynamical systems. In the context of iterated maps, this presumes dynamics 1 for a state variable x(k) ∈ M and a driving system y(k + 1) = g(y(k)) on a space N . The entire dynamics (y(k + 1), x(k + 1)) = F (y(k), x(k)) with F (y, x) = (g(y), f (x, y)) thus is a skew product system on N × M with base space N and fibers {y} × M . Generalized synchronization is the effect that typical orbits x 1 (k) and x 2 (k) of (1) converge to each other as k → ∞ under the same driving dynamics, i.e. identical orbits of (2). It is explained by a single attracting invariant graph for the skew product system [25,26]. In specific cases such an invariant graph may be continuous, but in general one should consider measurable graphs. The basin of attraction of the invariant graph will then also not be an open neighborhood. There exists substantial literature exploring generalized synchronization, but little mathematical theory to back up the observations. The book [22] and the extensive review paper [6] both contain excellent overviews of different aspects of synchronization, and include chapters on synchronization by external forces.
Rigorous mathematical results related to generalized synchronization exist mostly for driven systems acting on Euclidean spaces, allowing for stronger contraction properties than available on compact manifolds, and for circle diffeomorphisms driven in various ways. We give a brief account of contexts where rigorous results have been derived. An iterated functions system generated by k maps f 0 , . . . , f k−1 yields a skew product system driven by a Bernoulli shift on k symbols. Synchronization obviously arises in the classical case where f 0 , . . . , f k−1 are affine contractions on a Euclidean space. Extensions have been considered where the maps contract only on average [4,11,25,27]. Rigorous results for driven dynamical systems on compact manifolds have been derived in cases where the compact manifold is the circle. A large body of work is available on quasiperiodically forced circle diffeomorphisms; the circle diffeomorphisms x → f (x, y) are forced by g(y) = y + α mod 1 with α irrational. This research relates to the existence of strange non-chaotic attractors, see [15] and references therein. Reference [13] treats circle diffeomorphisms driven by an expanding circle map; in this case the driving system is mixing and not just transitive. Randomly perturbed circle diffeomorphisms have been considered in the contexts of iterated function systems [1,10,17] and absolutely continuous noise [16,18,30]. Other contexts with results in this direction include random logistic maps [28] and stochastic parabolic partial differential equations [20].
We contribute to the theory of generalized synchronization for classes of iterated function systems generated by a collection of diffeomorphisms, on compact manifolds M of any dimension. Here, an iterated function system consists of a collection of maps with a fixed distribution from which a map is chosen, independently at each iterate. More specifically, we consider two cases: (1) diffeomorphisms depending on a random parameter from an absolutely continuous distribution, and (2) iterated function systems generated by a finite collection of diffeomorphisms.
1.2. Random diffeomorphisms. We will treat random diffeomorphisms f a with a random parameter a chosen independent and identically distributed. We choose the setup where the random parameter has the effect that a point is mapped into a set according to an absolutely continuous invariant measure, as in [2,29]. More formally, let D be the unit ball of the same dimension as M . We consider random diffeomorphisms f a : M → M , x → f a (x), depending on a random parameter a ∈ D drawn from a measure η on D with density function g : D → R. A stationary measure m on M is a measure that equals its average push-forward under the random diffeomorphisms: Here f a m with f a m(A) = m(f −1 a (A)) denotes the push-forward measure of m by f a . The random diffeomorphisms in this setup admit a finite number of absolutely continuous stationary measures with disjoint supports.
The following result treats random diffeomorphisms with a stationary measure of full support on M . It gives conditions for synchronization in terms of Lyapunov exponents (see further the discussion below) and a global dynamical property for one of the diffeomorphisms. Take the product measure, denoted by ν + , on Σ + = D N . For ω ∈ Σ + , write f n ω (x) = f ω(n−1) • · · · • f ω(0) (x). Theorem 1.1. Let {f a }, a ∈ D, be random diffeomorphisms so that (1) one of the diffeomorphisms fâ,â ∈ D, is a Morse-Smale diffeomorphism with a single attracting fixed point, (2) {f a } admits an absolutely continuous stationary measure m of full support. If {f a } has negative Lyapunov exponents, then for almost all ω ∈ Σ + , x, y ∈ M .
The same result holds if some finite composition of the random diffeomorphisms has an attracting fixed point with open, dense basin. The reasoning to prove the above result allows to obtain variants for cases where the stationary measure does not have full support. We include a possible variant here. A stationary measure m is isolated if there is an open neighborhood U of its support so that the closure of f a (U ) is contained in U for all a ∈ D, and m is the unique stationary measure with support in U . The neighborhood U is called an isolating neighborhood, see [29,Definition 1.5].
Theorem 1.2. Let {f a }, a ∈ D, be random diffeomorphisms so that (1) {f a } admits an absolutely continuous isolated stationary measure m with support in an isolating neighborhood U . (2) one of the diffeomorphisms fâ,â ∈ D, is a diffeomorphism with an attracting fixed point in U , containing U in its basin.
If {f a } on U has negative Lyapunov exponents, then for almost all ω ∈ Σ + , x, y ∈ U .
1.3. Iterated function systems. The occurrence of synchronization for iterated function systems that are generated by finitely many diffeomorphisms can be treated in the same framework. Consider finitely many diffeomorphisms {f 0 , . . . , f k−1 } on a compact manifold M . The diffeomorphisms f i are picked at random with probability p i > 0. For these given probabilities on {0, . . . , k − 1}, let ν + be the product measure on Σ + = {0, . . . , k − 1} N . The following result establishes the robust occurrence of synchronization for minimal iterated function systems on compact manifolds. Minimal here signifies that orbits under the semigroup action of the iterated function system lie dense in the manifold for each initial point. Further explanations and precise conditions are formulated in Section 3. There also special cases of iterated function systems on spheres and two-dimensional surfaces are treated, where stronger results are obtained for iterated function systems generated by two diffeomorphisms. (1) the iterated function system is minimal, so that there is a stationary measure m of full support. A main difference with the context of random diffeomorphisms is that we do not know absolute continuity of the stationary measure. The statement is therefore weaker. Theorem 1.4 below provides a different perspective, focusing on invariant measures with certain properties, of the same result.
1.4. Skew product systems. We return to a setup with skew product systems. The random diffeomorphisms and iterated function systems are expressed as skew product systems, where the driving system is the shift on a symbol space. When treating this for one sided time N, this results in the skew product system F + on Σ + × M : with the shift operator σ(ω)(i) = ω(i + 1) on Σ + = B N for B = D in case of random diffeomorphisms or B = {0, . . . , k − 1} for iterated function systems generated by k diffeomorphisms. The skew product system F + admits an ergodic invariant product measure µ = ν + × m, where ν + is the product invariant measure for the driving system (we also call m an ergodic stationary measure if ν + × m is ergodic for F + ). The natural extension of F + is obtained when the shift acts on two sided time Z; this yields a skew product system F : Σ × M → Σ × M with Σ = B Z and given by the same expression We will adopt notation F n (ω, x) = (σ n ω, f n ω (x)) = (σ n ω, f ω(n−1) • · · · • f ω(0) (x)) for iterates of F .
Invariant measures for F + with marginal ν + and invariant measures for F with marginal ν are in one to one relationship, as detailed in [3] and Proposition 2.1 below. A stationary measure m thus, through the invariant measure ν + × m for F + , gives rise to an invariant measure µ for F . The measure µ has marginal ν (the product invariant measure for the shift on Σ) and conditional measures µ ω , for ν-almost all ω given as the weak star limit µ ω = lim n→∞ f n σ −n ω m of push forwards of the stationary measure. Meaning For an ergodic stationary measure m, one has that ν + × m almost everywhere there are fiber Lyapunov exponents λ 1 ≥ . . . ≥ λ dim M . The top fiber Lyapunov exponent is computed as Crauel proves that µ ω is a finite union of delta measures, with equal weight, in case λ 1 < 0 [9] (compare also [18,23]). That is, assuming that the fiber Lyapunov exponents with respect to the stationary measure are negative, then by [9] F admits an invariant measure µ with marginal ν and atomic conditional measures µ ω along the fibers {ω} × M . So for functions X i : Σ → M . Our aim is to identify conditions under which the conditional measures are supported on a single atom where K = 1, and subsequently to derive the synchronization theorems from it. Recent work by Bochi, Bonatti and Díaz [7] establishes for each compact manifold M a C 2 open set of iterated function systems, generated by finitely many diffeomorphisms on M , for which the corresponding skew product system on Σ × M admits an invariant measure of full support for which all fiber Lyapunov exponents are zero. The following result phrases Theorem 1.3 in similar terms. The corresponding skew product system F on Σ × M admits an invariant measure µ, so that (1) F has negative fiber Lyapunov exponents with respect to µ, (2) µ has full support, (3) the marginal of µ on Σ is the Bernoulli measure ν, (4) the conditional measures µ ω on almost all fibers are delta measures.
The open class of iterated function systems in [7] is given in terms of conditions which they term minimality (of an induced iterated function system on a flag bundle) and maneuverability. The construction in this paper makes clear that these conditions can occur simultaneously with the conditions defining the set of iterated function systems in Theorem 1.4. So Theorem 1.4 combined with [7] yields the following result. I am grateful to Masoumeh Gharaei for many discussions on the paper.

Random diffeomorphisms
In this section we prove Theorem 1.1. Recall that we are given a family of diffeomorphisms f a : M → M on a compact manifold M , depending on a parameter a from the unit ball of the same dimension as M . The parameter a is picked at random from a probability measure with density function g. The map (a, x) → f a (x) is assumed to be smooth. The skew product systems F + and F are defined in (4) and (5).
A Riemannian structure on M defines a Riemannian measure on M . The ergodic stationary measure m is absolutely continuous and equivalent to Riemannian measure.
We quote the following result that makes precise the correspondence between stationary measures m (and thus invariant measures ν + × m for F + ) and invariant measures for the skew product system F . Write M for the space of probability measures on M , endowed with the weak star topology. Proposition 2.1 gives an invariant measure µ for F with Lebesgue measure λ as marginal and with conditional measures µ ω , ω ∈ Σ. Assume F + has negative fiber Lyapunov exponents (with respect to the ergodic measure ν + × m), so that µ ω is atomic for ν all ω; for functions X i : Σ → M .
The following lemma follows from the theory of nonuniform hyperbolicity [5] and can be found in [9] or [21,Lemma 10.5]. Write W s (X i ) for the stable set of X i inside the fiber {ω} × X i . Let d denote a distance function from a Riemannian structure on M .
. We can in fact, for all ε > 0, get δ and A in the lemma with ν(A) > 1 − ε, see [5,Section 8.1]. With these preparations we can give the proof of our main result.
Proof of Theorem 1.1. Let µ be as in Proposition 2.1. We will establish that the corresponding disintegrations µ ω are delta measures, i.e. K = 1 in (6). The theorem will follow from this.
Let Q 0 be the attracting fixed point of fâ from the conditions of the theorem. Write Λ 0 = M \ W s (Q 0 ) for the complement of its basin of attraction. Since m is absolutely continuous, m(Λ 0 ) = 0. It is therefore possible to take a closed subset D ⊂ W s (Q 0 ) with m(D) close to 1. There is a small ball of parameter values a aroundâ for which f a has an attracting fixed point near Q 0 and D is in its basin of attraction. Such a small ball constitutes a set of parameters with positive measure for η.
Recall that M is metrizable, see e.g. [19]. Let ∆ ⊂ M be the subset of delta measures on M . Fix ε > 0. Define By Poincaré recurrence, σ −n (ω) lies in A for infinitely many n ∈ N, for almost all ω ∈ Σ. We make the following claim: given ε > 0, there exists L ∈ N so that for each ω ∈ A, there exists B ω + ⊂ Σ − and l < L so that f l σ −l ω , ω ∈ B ω + × A ω + , maps a set of stationary measure more than 1 − ε into one of the balls Namely, for any r > 0, there is a sufficiently large iterate of fâ that maps D into a neighborhood of radius r of the attracting fixed point Q 0 of fâ. Likewise for f a with a close toâ. Since the stationary measure has full support, there is a finite composition of the random diffeomorphisms that map B r (Q 0 ) into B δ (X i ). Combining the two statements proves the claim. Note that by Lusin's theorem we may take A slightly smaller so that B δ (X i ) depends continuously on ω ∈ A. The uniform bound on the number of iterates L in the above claim implies that ν − (B ω + ) is uniformly bounded away from zero. Consequently the union B = ∪ ω + B ω + × A ω + has positive measure. As a consequence, for almost all ω, its orbit under σ −1 intersects B. Thus F has zero measure. This proves that for almost all ω ∈ Σ, the sequence f n σ −n ω m accumulates on the set of delta measures (consider a countable set of ε's converging to zero). Hence, by Proposition 2.1, f n σ −n ω m converges to a delta measure δ X(ω) as n → ∞.
It remains to explain the synchronization (3) from the appearance of delta measures as disintegrations. We have that f n σ −n ω m converges to δ X(ω) almost surely. This implies convergence in measure, and since σ leaves ν invariant, also that f n ω m converges to δ X(σ n ω) in measure. That is, for any ε > 0, Here d M is a metric on M generating the weak star topology. This in turn implies that for some subsequence n k → ∞, (see e.g. [24, Theorem II.10.5]). We combine this with the existence of stable sets around X(σ n ω). Lemma 2.1 and the comment following it, yields sets S(δ) ⊂ Σ + with ν + (∪ δ S(δ)) = 1 and So for all ε > 0 there is δ > 0 with ν + (S(δ)) > 1 − ε. Once orbits are in a δ-ball around X(ω) inside the stable manifold of X(ω), distances to the orbit of X(ω) decrease to zero. This implies that f n ω m converges to δ X(σ n ω) almost surely. In more detail, let (7) implies that for any given ε > 0 there is N > 0 with ν + (Γ(δ, N )) > 1 − ε. A measure is close to a delta measure if most of the measure is in a small ball: for any δ there isδ > 0 so that d M (µ, δ x ) <δ implies µ(B δ (x)) > 1 − δ. Combining the above we get that for any ε > 0 there is N > 0 so that the set of ω ∈ Σ + where both B δ (X(σ n ω)) ⊂ W s (X(σ n ω)) and f n ω m(B δ (X(σ n ω))) > 1 − δ for n ≥ N has ν + -measure more than 1 − ε. This implies the synchronization property (3).
Essentially the same reasoning proves Theorem 1.2.

Iterated function systems
This section develops results for iterated functions systems resulting in the proof of Theorem 1.3 and of Theorem 1.4. In addition it contains a study of special cases of iterated function systems on spheres and two-dimensional surfaces.
We start by developing necessary terminology and notation. Let F, G be two families of diffeomorphisms on a compact manifold M . Denote and for k ∈ N, Write gen F + for the semi-group generated by F, that is, gen F + = ∪ k∈N F k . The action of the semi-group gen F + is called the iterated function system associated to F and we denote it by IFS (F). For x ∈ M , we write the orbits of the action of this semi-group as A sequence {x n : n ∈ N} is called a branch of an orbit of IFS (F) if for each n ∈ N there is f n ∈ F such that x n+1 = f n (x n ). We say that IFS (F) is minimal (or minimal on M ) if any orbit has a branch which is dense in M . Likewise IFS (F) can be minimal on an invariant set F(B) = B. We say that a property holds C r -robustly for IFS (F) if it holds for any familyF whose elements are C r perturbations of elements of F.
where (σω)(j) = ω(j + 1) is the left shift operator. The symbol space Σ + is as usual endowed with the product topology. The natural extension of Here Σ is endowed with the product topology.
A stationary measure m on M is a measure that satisfies The product measure ν + × m is then invariant for F + .
Lemma 3.1. For a minimal iterated function system, a stationary measure m has full support.
Proof. See [17,Proposition 5]. The support A of m is closed and nonempty.
In the following sections we treat special cases of iterated function systems generated by diffeomorphisms on spheres and surfaces, before treating the general case and proving Theorems 1.3 and 1.4. For iterated function systems on spheres and surfaces we have improved results, involving iterated function systems generated by only two diffeomorphisms and clearer conditions.

Iterated function systems on spheres.
The following result discusses synchronization for minimal iterated function systems on spheres. Let S n denote the n-dimensional sphere. (1) the iterated function system is minimal, so that there is a stationary measure m of full support. (2) F + has negative fiber Lyapunov exponents (with respect to ν + × m), (3) the corresponding disintegrations µ ω are delta measures (here µ is the invariant measure of F as in Proposition 2.1).
A north-south diffeomorphism on S n is a diffeomorphism with a hyperbolic repelling and a hyperbolic attracting fixed point, where the basin of attraction of the attracting fixed point is the sphere minus the repelling fixed point. In agreement with the construction of C 1 robustly minimal iterated function systems in [14], we take f 0 to be a north-south diffeomorphism. The attracting fixed point Q 0 of f 0 has a basin of attraction W s (Q 0 ) that lies open and dense in S n ; the complement S n \ W s (Q 0 ) is a repelling fixed point P 0 . An essential ingredient for the proof of Theorem 3.2 is the fact that the basin of attraction W s (Q 0 ) has full stationary measure, expressed by the following lemma. As it may be of independent interest we provide a variant of the argument, in which we replace the shift map acting in the base by a generalized Baker map. Namely, the shift map is topologically semi-conjugate to a generalized Baker map that preserves Lebesgue measure, such that the Bernoulli measure is pushed forward to Lebesgue measure. The Bernoulli shift and the generalized Baker maps are isomorphic measure preserving systems. We give the formulas below. We take a setup suitable for iterated function systems generated by k > 1 diffeomorphisms f i , 0 ≤ i < k, picked with probabilities p i . Working with a generalized Baker map will allow us to employ arguments based on the Lebesgue density theorem.
Consider the generalized Baker map B : [0, 1] → [0, 1] that expands vertical strips I i , 0 ≤ i < k, by a factor 1 p i in the horizontal direction, while contracting by a factor p i in the vertical direction. Thus, writing l i = i−1 j=0 p j (with l 0 = 0), we get I i = [l i , l i + p i ) and Write also B + (y) = y p i + l i for the expanding map B + : [0, 1] → [0, 1]. Note that B + preserves one-dimensional Lebesgue measure and B preserves two-dimensional Lebesgue measure. We denote both one-dimensional and two-dimensional Lebesgue measure by λ. An itinerary ω ∈ Σ + corresponds to a point y ∈ When working with the Baker map as base map, we use shorthand notation . . • f ⌊k(ky mod 1)⌋ • f ⌊ky⌋ . Consider F given by (5). Let m be an ergodic stationary measure. Proposition 2.1 gives an invariant measure µ for F with Lebesgue measure λ as marginal and with conditional measures µ (y,z) , (y, z) ∈ [0, 1] 2 . Assume F + has negative fiber Lyapunov exponents (with respect to the ergodic measure λ × m), so that µ (y,z) is atomic for Lebesgue almost all (y, z): µ (y,z) = 1 K K i=1 δ X i . Write W s (X i ) for the stable set of X i inside the fiber {(y, z)} × X i . Let d denote a distance function from a Riemannian structure on S n . Lemma 2.1 in this case reads as follows. Lemma 3.3. There is δ > 0 and a set A ⊂ [0, 1] 2 of positive Lebesgue measure so that W s (X i ) contains a ball B δ (X i ) of radius δ around X i , i = 1, . . . , K, for fibers over A. And there are C > 0, 0 < λ < 1 so that for all . Second proof of Theorem 3.2. Consider F as a skew product over the Baker map in the base [0, 1] 2 . Since the fiber Lyapunov exponents are negative, by [9] µ ω = 1 z). We must prove that K = 1 under the assumptions of the theorem.
Let A be as in Lemma 3.3. Since the fiber coordinates of iterates of (y, z, x) → F B (y, z, x) do not depend on z, we may consider A as a product set A y × [0, 1]. We recall again that the space of probability measures M endowed with the weak star topology is metrizable.
Since m(P 0 ) = 0 by Lemma 3.2, it is possible to take a closed subset D ⊂ W s (Q 0 ) with m(D) close to 1. Consider a fixed y 0 ∈ [0, 1]. Fix ε > 0. Let ∆ be the subset of delta measures µ. Define m is at least ε away from ∆ for all i} is the complement of all Q n 's. We claim that R has no density points in [0, 1] and hence has zero Lebesgue measure.
For almost all (y 0 , z 0 ), B −n (y 0 , z 0 ) lies in A for infinitely many n ∈ N. We make the following claim: there exists an integer L > 0, so that the following holds for each n with (y −n , z −n ) = B −n (y 0 , z 0 ) ∈ A: there is 0 ≤ l ≤ L and y −n−l × [0, 1] ⊂ B −l (y −n × [0, 1]), for which f n+l y −n−l m lies within ε of ∆. For any r > 0, there is a sufficiently large iterate of f 0 that maps D into a neighborhood of radius r of the attracting fixed point Q 0 of f 0 . Since the iterated function system IFS (F) is minimal, for any s > 0 there is t > 0 so that F t is s-dense in S n . There is thus a finite composition of the diffeomorphisms f i that map B r (Q 0 ) into B δ (X i (y −n , z −n )). Combining the two statements proves the claim.
Following [10, Proposition 5.5], we note that the stationary measure, under these conditions and negative top Lyapunov exponent, λ 1 < 0, is unique. Letting n → ∞ we get That is, Now suppose m 1 and m 2 are stationary measures and write m = 1 2 m 1 + 1 2 m 2 . Then we get an equality for almost all ω, so that X m 1 = X m 2 for almost all ω. Hence 3.2. Iterated function systems on surfaces. (1) the iterated function system is minimal, so that there is a stationary measure m of full support. (2) F + has negative fiber Lyapunov exponents (with respect to ν + × m), (3) the corresponding disintegrations µ ω are delta measures (here µ is the invariant measure of F as in Proposition 2.1).
The construction of minimal iterated function systems in [14] involves the choice of a Morse-Smale diffeomorphism with a unique attracting fixed point as one of the diffeomorphisms, say f 0 . The attracting fixed point, Q 0 , of f 0 has a basin of attraction W s (Q 0 ) that lies open and dense in M . The complement Λ 0 = M \ W s (Q 0 ) is a stratification consisting of the stable manifolds of finitely many hyperbolic fixed or periodic points. Recall that a stratification is a compact set consisting of finitely many manifolds W i with if x n ∈ W j converges to y ∈ W i , then there is a sequence of d-planes E n ⊂ T xn W j of dimension d = dim W i that converge to T y W i . Two stratifications N 1 , N 2 inside M are transverse if at intersection points the tangent spaces span the tangent space of M . We refer to [12], see in particular [12,Exercise 3.15], for the transversality theorem for stratifications. It implies that for an open and dense set of diffeomorphisms g, g(Λ 0 ) is transverse to Λ 0 .
Instead of proving that the basin of attraction of Q 0 has full stationary measure, in analogy of Lemma 3.2, we bound its stationary measure from below by 1/2. This suffices to conclude Theorem 3.3.
Proof of Theorem 3.3. The result follows from the proof of Theorem 3.2, taking a subset of D ⊂ W s (Q 0 ) of stationary measure more than 1/2 instead of measure close to one (this still implies K = 1). The argument providing iterated function systems with negative Lyapunov exponents, by perturbing minimal iterated function systems, is postponed to Section 3.3 below.
As in the previous section, the stationary measure under the conditions of the theorem is unique. Start with a minimal iterated function system generated by two diffeomorphisms f, g. For k > 0 to be determined later, let f 0 , . . . , f k−1 be small perturbations of f and write g = f k . The iterated function system generated by f 0 , . . . , f k is minimal and has a stationary measure m of full support. Each f i , 0 ≤ i < k, has an attracting fixed point Q i with open and dense basin of attraction W s (Q i ). Write Λ i for its complement.
For a C 1 open and dense set of k diffeomorphisms f i , the collection of stratifications Λ i is transverse (as noted above, recall [12,Exercise 3.15]). The following lemma bounds the stationary measure on the stratifications. The same argument allows the conclusion that for any c < 1/2, m(Λ i ) < c for some 0 ≤ i < k if k is big enough.
Proof. Assume m(Λ i ) ≥ 1/2 for all i. The smallest possible total stationary measure on a union of stratifications Λ i 1 ∪ . . . ∪ Λ i l occurs if the measure is supported on the common intersection, if this is nonempty. Note that for l = dim(M ) − 1 this intersection, if nonempty, is zero dimensional. Also, the intersection of dim(M ) different stratifications is always empty.
To calculate the smallest possible total measure on Λ 0 ∪. . .∪Λ k−1 , suppose there is measure 1/2 on each Λ i . Consider sets S j that occur as maximal intersections of sets Λ i 1 ∩ . . . ∩ Λ i l ; meaning such that each intersection with a further stratification Λ i is empty.
Think of a distribution of measure, i.e. an assignment of measures n j to the S j 's. We seek the minimal total measure, among variation of distributions. The argument will be combinatorial. For the purpose of bounding the minimal total measure, we may assume that each collection of l stratifications, l = dim(M ) − 1, has a nonempty intersection by possibly adding imaginary intersections. This indeed only adds possible distributions of measure (previous distributions assign zero measure to the new imaginary intersections), hence does not increase total measure.
The minimal possible total measure j n j on Λ 0 ∪ . . . ∪ Λ k−1 occurs at an equidistribution among the different disjoint sets S j . At equidistribution each S j carries the same measure, say n j = n. To see that this gives the minimal possible total measure, first observe that a linear combination of distributions, preserving the total measure, is again a distribution. Then by symmetry, permuting the sets S j and averaging distributions, one obtains the equidistribution. This therefore has minimal total measure.
There are k l intersections S j and k−1 l−1 intersections in a fixed stratification Λ i . At equidistribution, the measure of S j equals 1/2 k−1 l−1 as the measure of Λ i is 1/2. The total measure j n j is 1 2 k l / k−1 l−1 = k/2l. This number is bigger than 1 if k > 2l. The lemma follows.
Proof of Theorems 1.3 and 1.4. The reasoning in the proofs of Theorems 3.2 and 3.3 can be followed to find that the conditional measures µ ω are delta measures.
Next we prove the synchronization property as stated in Theorem 1.3. For almost all ω ∈ Σ, W s (X(ω)) is open. Indeed, take y ∈ W s (X(ω)). Then some iterate f n ω (y) is contained in a small open ball around X(σ n (ω)) and this bal is a subset of W s (X(σ n ω)). And therefore a small neighborhood of y lies in W s (X(ω)). Since f n ω m converges to δ X(ω) for almost all ω, the stationary measure of M \W s (X(ω)) is zero for almost all ω (note that W s (X(ω)) depends only on the present and future of ω, i.e. on ω(i), i ≥ 0). Now because m assigns positive measure to open sets and W s (X(ω)) is open, it follows that W s (X(ω)) is open and dense.
To prove that iterated function systems as postulated exist, we must show that we can construct the iterated function systems as above so that they have negative Lyapunov exponents. We start with a minimal iterated function system and perform perturbations to force negative Lyapunov exponents. Denote by P : M → M the map Stationary measures are fixed points of P. By Lemma 3.6 below, P is continuous and changes continuously under variation of the maps generating the iterated function system in the C 0 topology. The set of fixed points of P therefore varies upper semi-continuous when varying the iterate function system in the C 0 topology. That is, for any neighborhood O of the closed set of stationary measures of IFS ({f i }), there is a C 0 neighborhood of IFS ({f i }) so that each iterated function system from it has its stationary measures contained in O.
Takef 0 near f 0 , equal to f 0 outside a neighborhood of Q 0 , making it contracting stronger near Q 0 with even Df 0 ≡ 0 in a small neighborhood N 0 of Q 0 . Letf i = f i for i = 0. It follows from the construction in [14] that this can be done with IFS ({f i }) minimal. After having concluded this proof we return to this claim, and indicate how it follows from [14]. Take a sequencef 0,j converging tof 0 in the C 1 topology. We don't change the diffeomorphisms f 1 , . . . , f k ; for convenience we writef i,j = f i for 1 ≤ i ≤ k. Letm j be a ergodic stationary measure of IFS ({f i,j }). By passing to a subsequence we have thatm j converges to a measurem. Lemma 3.6 shows thatm is a stationary measure of IFS ({f i }). By Lemma 3.1m has full support.
For an ergodic measure ν ×m j , the top Lyapunov exponent of IFS ({f i,j }) satisfies, for ν ×m j almost all (ω, x), If ln Df 0,j ≤ B j on N 0 and ln Df 0,j ≤ C on M (uniformly in j), then M ln Df 0,j dm j ≤ C + B jmj (N 0 ).
As B j → −∞ andm j →m for j → ∞,m j (N 0 ) converges to a positive numberm(N 0 ) and is upper-semicontinuous in a at a = 0 (R a goes to zero as a → 0 by the above noted continuity of the set of stationary measures atm j ). So the fiber Lyapunov exponents are stably negative, C 1 close to IFS ({f i,j }). The observation that Lemma 3.5 remains valid for IFS ({f i,j }) and its perturbations finishes the proof of Theorem 1.3.
As before, the stationary measure under the conditions of the theorem is unique.
Consider P acting on the space M of probability measures endowed with the weak star topology, induced by the iterated function system as in (9). The following lemma was used above. since any x ∈ T −1 (A) is contained in T −1 n (A) for all large enough n. So T n µ converges to T µ. This argument also shows that P depends continuously on f 0 , . . . , f k−1 .
Finally we indicate how it follows from the constructions in [14] that minimal iterated function systems with very strong contraction on some small set N 0 , as required in the above proof of Theorems 1.3 and 1.4 exist. For details on the construction we refer to [14]. The construction in [14] involves two Morse-Smale diffeomorphisms f and g, where g is a perturbation of f −1 . The diffeomorphism f possesses a unique attracting fixed point and nearby there is a unique repelling fixed point of g. In a chart containing the attracting fixed point of f , f and g appear (almost) linear; f and g −1 as well as f • g are contractions in the chart. There is a box B in the chart (containing the repelling fixed point of g) so that B ⊂ f (B) ∪ f • g(B).
Since f and f • g are contractions it follows easily that the iterated function system generated by f an f • g acts minimally on a set that contains B. Key here is the observation that the image of B under a large number n of compositions of f and f • g has diameter going to zero as n → ∞, while the union of the images under all 2 n such compositions covers B. Minimality on the entire manifold is ensured by generic assumptions such as no common fixed points for f and g. Now perturb f , making it a strong contraction on a small ball N 0 in B, while not altering f on ∂B and g(∂B). The above key observation and the proof of minimality remain valid.