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 function systems generated by finitely many diffeomorphisms on a compact manifold. We thus consider compositions of diffeomorphisms f 0 , . . . , f k acting on a compact manifold M. For each iterate a diffeomorphism f i is picked Fig. 1 The left frame shows numerically computed time series for the iterated function system generated by f 0 (x) = x + 1 4π sin(2π x) (chosen with probability p 0 = 0.5) and f 1 (x) = x + 1 π mod 1 (chosen with probability p 1 = 0.5). Time series with two different initial conditions but identical compositions are shown, where the points are connected by lines. The orbits appear to converge to each other. The right frame shows the histogram of a numerically computed orbit, that appears to lie dense in [0,1] independently with a given probability p i . Our focus will lie on the combination of two properties: (i) the iterated function systems are minimal, meaning that for each point there is a sequence of diffeomorphisms that gives a dense orbit in the manifold; (ii) the iterated function systems displays synchronization, meaning that typically orbits converge to each other when iterated by the same sequence of diffeomorphisms.
We moreover demand robustness where the properties persist under small perturbations of the generating diffeomorphisms Minimal iterated function systems on compact manifolds have been constructed before in Homburg and Nassiri (2014), Ghane et al. (2010) and Bochi et al. (2014). Synchronization for minimal iterated function systems on compact manifolds has been established before only for iterated function systems on the circle (Antonov 1984), see also Deroin et al. (2007), Kleptsyn and Nalskii (2004), Kaijser (1993), Zmarrou and Homburg (2008) and Malicet (2017).
For the purpose of illustration we describe an example. Let f 0 be a Morse-Smale diffeomorphism on the circle with a unique attracting and a unique repelling fixed point. Let f 1 be a smooth diffeomorphism on the circle with irrational rotation number, so that its orbits are dense. Consider the iterated function system generated by f 0 , f 1 , were f 0 , f 1 are picked independently with positive probabilities p 0 , p 1 = 1− p 0 . This iterated function system is clearly minimal. It is not difficult to demonstrate that iterated functions systems generated by diffeomorphisms that are C 1 -small perturbations of f 0 , f 1 are also minimal. It has been established that such iterated function systems display synchronization, as illustrated in Fig. 1.
Aim of this paper is to provide constructions of minimal iterated function systems that display synchronization, in a robust way, on any compact manifold. Figure 2 illustrates a two dimensional example: it shows synchronization of a minimal iterated function system on the two dimensional torus. At the end of this paper in Proposition 3.2 we establish minimality and synchronization of this iterated func- Fig. 2 Iterates of 1000 randomly chosen initial points under random compositions of the torus diffeomorphisms f 0 (x, y) = (x + 1 4π sin(2π x), y + 1 4π sin(2π y)) and f 1 (x, y) = (x + 1 π , y + 1 √ 2π ) mod 1, picked randomly with probability 1/2 for both. The randomly chosen initial points in the left frame are iterated 50, 100, 150 iterates in the following frames from left to right tion system and of C 1 -small perturbations of it. Analogous examples can be given for iterated function systems on higher dimensional tori. Synchronization phenomena as illustrated in Figs. 1 and 2 and as considered in this paper fall into the larger concept of master-slave synchronization or synchronization by noise. The classical concept of synchronization is the phenomenon that different oscillations in coupled systems will converge to oscillations that move with identical frequency. It has been realized that external forcing or noise, rather than coupling, can also synchronize dynamics. An example of synchronization by noise that applies to iterated function systems on the circle is by Antonov (1984). An illustrative example for linear differential equations forced by the Lorenz equations, is given by Pecora and Carroll (1990). Other examples are from random interval diffeomorphisms (Gharaei and Homburg 2017), random logistic maps Steinsaltz (2001) and stochastic differential equations (Masmoudi and Young 2002;Flandoli et al. 2017a, b). The book (Pikovsky et al. 2001) contains an excellent overview of different aspects of synchronization and includes discussions of synchronization by external forces or noise.
In the context of iterated maps, master-slave synchronization involves dynamics, for a state variable x(k) ∈ M and a driving system, on a space N . The entire dynamics (y(k + 1), thus is a skew product system on N × M with base space N and fibers {y} × M.
Master-slave synchronization is the effect that typical orbits of (1) converge to each other under the same driving dynamics, i.e. identical orbits of (2): where (y, x) = x and d denotes the distance on M. The effect can be explained by the existence of an attracting invariant graph for the skew product system Stark (1997Stark ( , 1999. Iterated function systems fit this description: they are studied using a formulation as a skew product system over a shift operator (see below).

Minimal Iterated Function Systems
One ingredient of this study is the robust occurrence of minimal iterated function systems. Consider a collection F = { f 0 , . . . , f k } of diffeomorphisms on a compact manifold M.
Definition 1.1 The iterated function system IFS (F) generated by F is the set of all possible finite compositions f ω n • · · · • f ω 0 of maps in F, that is, the semi-group generated by these maps.
By Homburg and Nassiri (2014), any compact manifold admits a pair of diffeomorphisms f 0 , f 1 that generates a C 1 -robustly minimal iterated function system.

Synchronization in Iterated Function Systems
Consider finitely many diffeomorphisms F = { f 0 , . . . , f k } on a compact manifold M and fix positive probabilities p i , 0 ≤ i ≤ k, with k i=0 p i = 1. The diffeomorphisms f i are picked at random, independently at each iterate, with probability p i . In this context of Markov processes a central notion is that of stationary measure. Definition 1.3 A stationary measure m for the iterated function system IFS (F) is a probability measure that is equal to its average pushforward under the diffeomorphisms: where It is well known that IFS (F) always admits at least one stationary measure, because M is a compact space.

Lemma 1.1 For an iterated function system that is minimal on M, a stationary measure m has full support M.
Proof See Kleptsyn and Nalskii (2004, Proposition 5). The support supp m of m is closed and nonempty. From (3) we get so that supp m = M by minimality.
For the given probabilities p i on {0, . . . , k}, let ν + be the product measure (or Bernoulli measure) on The following result provides conditions for an iterated functions system by diffeomorphisms to display synchronization. We first recall the notion of fiber Lyapunov exponent. With the stationary measure m as in the statement of Theorem 1.1, and in light of Lemma 1.2 below, one has that for (ν + × m)-almost all (ω, x) ∈ + k+1 × M, exists. The number of limit values, counting multiplicity, equals the dimension of M.
The possible limit values are the fiber Lyapunov exponents. If ν + × m is ergodic (the stationary measure m is then also called ergodic), the fiber Lyapunov exponents are independent of (ω, x). We refer to e.g. Viana (2014) for more information.   To obtain the synchronization property (1) we make use of the theory of nonuniform hyperbolicity; for this reason we require f 0 , . . . , f k to be C 2 diffeomorphisms and not just C 1 .
For diffeomorphisms on the circle Theorem 1.1 is known to hold for iterated function systems with two generators, i.e. with k = 1 (see Kleptsyn and Nalskii 2004). We will show in Proposition 3.1 that also on compact surfaces it holds with k = 1.
The arguments that have been applied to obtain synchronization for iterated function systems on the circle used specific properties of one dimensional systems, such as the property that connected sets of small measure have small diameter. This is not true in higher dimensions, so that a partly different approach was needed. We use the theory of nonuniformly hyperbolic systems, in particular the existence of local stable manifolds in cases of negative fiber Lyapunov exponents. There are three main conditions that we use to establish synchronization: a minimal iterated function system, negative fiber Lyapunov exponents yielding local contraction, and global contraction in the form of the existence of an open set of sufficiently large (stationary) measure that is mapped into an arbitrary small ball by suitable iterates.

Invariant Measures for Skew Product Systems
Compositions of diffeomorphisms f 0 , . . . , f k on M can be studied in a single framework given by a skew product system F + : Indeed, the coordinate in M iterates as, We will write, with the shift operator (σ ω) i = ω i+1 on + k+1 . As the dependence of the fiber maps f ω on ω is on ω 0 alone, the skew product maps are of a restricted kind called step skew product maps.
An explicit computation on sets that generate the Borel sigma-algebra shows the following connection, which is standard (see e.g. Gharaei and Homburg 2017).

Lemma 1.2 A probability measure m is a stationary measure if and only if
. We 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 : and given by the same expression

Recall the notation
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. We quote the following result that precises this correspondence.
Write M for the space of probability measures on M endowed with the weak star topology.

Proposition 1.1 Let m be a stationary measure for IFS (F). Then there exists a measurable map
as n → ∞, ν-almost surely. The measure μ on k+1 × M with marginal ν and conditional measures μ ω = L(ω) is an F-invariant measure.
Proof See Arnold (1998, Theorem 1.7.2) for a general result that implies the proposition, or Gharaei and Homburg (2017, Appendix A) or Viana (2014, Chapter 5) for this setting.
A stationary measure m thus, through the invariant measure ν + × m for F + , gives rise to an invariant measure μ for F, with marginal ν. The measure μ has conditional measures μ ω , meaning for Borel sets A.

Proposition 1.2 Let m be an ergodic stationary measure for IFS (F).
Assume F + has negative fiber Lyapunov exponents with respect to the ergodic measure ν + × m. Then the conditional measures μ ω for the F-invariant measure μ are a finite sum of K delta measures of equal mass 1/K , for ν-almost all ω.
Proof See Crauel (1990) and Le Jan (1987). Bochi et al. (2014) 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 k+1 × M admits an invariant measure of full support for which all fiber Lyapunov exponents are zero. The following result phrases Theorem 1.1 in similar terms. Proposition 1.3 Let M be a compact manifold. There is k ≥ 1 and a C 1 open set of minimal iterated function systems generated by C 2 diffeomorphisms f 0 , 0 ≤ i ≤ k, on M and picked with positive probabilities p i , with the following properties. The iterated function system admits a unique stationary measure m. The corresponding invariant measure μ for the skew product system F on k+1 × M has the following properties:

Recent work by
(i) F has only negative fiber Lyapunov exponents with respect to μ; (ii) μ has full support; (iii) the marginal of μ on k+1 is the Bernoulli measure ν; (iv) the conditional measures μ ω are delta measures: The open class of iterated function systems in Bochi et al. (2014) 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 Bochi et al. (2014)  The corresponding skew product system F on k+1 × M admits simultaneously (i) an invariant measure μ that has full support, Bernoulli measure as marginal, delta measures as conditional measures on fibers, and negative fiber Lyapunov exponents; (ii) an invariant measure ν that has full support and zero fiber Lyapunov exponents.
The techniques in this paper may possibly be extended from the step skew product systems given by iterated function systems to more general skew product systems. An example is in Homburg (2017), where the ideas of synchronization have been used to clarify properties of the disintegrations of Lebesgue measure along center manifolds in certain conservative partially hyperbolic dynamical systems.
2.1. the existence of minimal iterated function systems with negative fiber Lyapunov exponents (proving items (i), (iv) in Theorem 1.1); 2.2. properties of the stationary measure that are needed in the proof; 2.3. the existence of atomic conditional measures of the F-invariant measure μ along fibers (finishing the proof of Proposition 1.3); 2.4. the occurrence of synchronization (item 1 in Theorem 1.1); 2.5. the uniqueness of the stationary measure (item (ii) in Theorem 1.1, finishing the proof of Theorem 1.1).

Minimality and Negative Fiber Lyapunov Exponents
This section constructs a C 1 open set of minimal iterated function systems with negative fiber Lyapunov exponents. We start with a number of results on iterated function systems generated by two maps. We collect statements from Homburg and Nassiri (2014) that we need in the sequel.
Recall that a Morse-Smale diffeomorphism on M is a diffeomorphism whose recurrent set consists of finitely many fixed and periodic points, all hyperbolic. Moreover, their stable and unstable manifolds are transverse.
Lemma 2.1 Let M be a compact manifold. Then there is a pair of diffeomorphismŝ g 0 ,ĝ 1 on M that generates a C 1 -robustly minimal iterated function system. The diffeomorphisms satisfy the following properties. The diffeomorphismĝ 0 is a Morse-Smale diffeomorphism with a unique attracting fixed point Q 0 , whose basin of attraction is open and dense in M. There is a small neighborhood U 0 of Q 0 and a compact ball B 0 ⊂ U 0 with the following properties: Denote h 0 =ĝ 0 , h 1 =ĝ 0 •ĝ 1 . By classical theory of iterated function systems (Hutchinson 1981), there is a unique compact set S with B 0 ⊂ S ⊂ U 0 and, Further, for all ω ∈ + 2 and uniformly in n. By (4), for each n ∈ N, With (5) this implies that IFS ({ĝ 0 ,ĝ 0 •ĝ 1 }) is minimal on S.
To obtain minimal iterated function systems with negative fiber Lyapunov exponents, we start with an iterated function system as in Lemma 2.1 and modify to bring strong contraction on a region. This is elaborated in the proof of Lemma 2.3 below. In the proof we need a statement on the dependence of stationary measures on the iterated function system, which we provide first. Recall that M denotes the space of probability measures on M endowed with the weak star topology. Take a metric d M that generates the weak star topology, see e.g. Mañé (1987). Denote by P : M → M the map, Note that stationary measures are fixed points of P.

Lemma 2.2
The map P is continuous. It also depends continuously on f 0 , . . . , f k if these vary in the C 0 topology.
For the proof we refer to Gharaei and Homburg (2017). Because of the vanishing derivative on D 0 ,g 0 is not a diffeomorphism. Such maps can be constructed by modifying the constructions in Homburg and Nassiri (2014) as follows. By working in a chart containing U 0 we may assumeĝ 0 ,ĝ 1 are diffeomorphisms on Euclidean space R n , with B 0 containing the origin. Let φ : R → R be a smooth odd function with φ ≡ 0 on a small neighborhood of 0, and otherwise increasing plus equal to the identity outside a neighborhood of 0. For s > 0 small, let r : R n → R n be given by r (x) = sφ( x/s ) and letg 0 =ĝ 0 • r .
Althoughg 0 is not a diffeomorphism, there are diffeomorphisms in any C 1neighborhood of it: to perturb to a diffeomorphism it suffices to perturb φ to a smooth increasing function. Keepĝ 1 unchanged and writeg 1 =ĝ 1 .
By Lemma 2.2, the set of fixed points of P is a compact subset of M that varies upper semi-continuously when varyingg 0 ,g 1 in the C 1 topology. That is, for any neighborhood O ⊂ M of the closed set of stationary measures of IFS ({g 0 ,g 1 }), there is a C 1 neighborhood of IFS ({g 0 ,g 1 }) so that each iterated function system from it has its stationary measures contained in O.
We claim the existence of an open neighborhood ofg 0 ,g 1 in the C 1 topology, so that for all pairs of diffeomorphisms g 0 , g 1 from it, and for any ergodic stationary measure m of IFS {g 0 , g 1 }, IFS {g 0 , g 1 } has negative fiber Lyapunov exponents.
Suppose otherwise. Then there is a sequenceg 0, j converging tog 0 andg 1, j converging tog 1 in the C 1 topology, with a nonnegative fiber Lyapunov exponent for some ergodic stationary measurem j . By passing to a subsequence we may assume thatm j converges to a measurem. Lemma 2.2 shows thatm is a stationary measure of IFS ({g 0 ,g 1 }). By Lemma 1.1,m has full support. (For these statements it plays no role thatg 0 is not a diffeomorphism.) The top fiber Lyapunov exponent λ 1, j of IFS ({g 0, j ,g 1, j }) satisfies, for (ν + ×m j )almost all (ω, x), We can bound ln Dg 0, j ≤ C on M uniformly in j. There is also a bound ln Dg 0, j ≤ B j on C 0 with B j → −∞ as j → ∞. So we get, M ln Dg 0, j dm j ≤ C + B jm j (C 0 ).

Regularity of Stationary Measures
Let g 0 , g 1 be Morse-Smale diffeomorphisms as in Lemma 2.3. Recall that the attracting fixed point Q 0 of g 0 has a basin of attraction W s (Q 0 ) that is open and dense in M. The reasoning in the following sections would work for IFS ({ f 0 , f 1 }) with f 0 = g 0 and f 1 = g 1 (i.e. k = 1 in Theorem 1.1 and Proposition 1.3) if the basin W s (Q 0 ) has sufficiently large stationary measure; m(W s (Q 0 )) > 1/2. As we do not know whether this is the case, we will add sufficiently many diffeomorphisms, all small perturbations of g 0 , as generators of an iterated function system and we bound the stationary measures of the basins of the attracting points of these extra generators. It turns out that at least one of these basins has stationary measure more than 1/2, which will suffice for the reasoning in the following sections.
The complement 0 = M\W s (Q 0 ) is a stratification consisting of the stable manifolds of finitely many hyperbolic fixed or periodic points.

Definition 2.1 A stratification is a compact set consisting of finitely many manifolds
Definition 2.2 Two stratifications N 1 , N 2 inside M are transverse if at intersection points the tangent spaces span the tangent space of M. A collection of stratifications N 1 , . . . , N l is said to be transverse at a common intersection point if any N k is transverse to any intersection ∩ j N i j of a subcollection not containing N k . A collection of stratifications N 1 , . . . , N l is transverse if any subcollection is transverse at a common intersection points of the subcollection.
Starting point for the following is a robustly minimal iterated function system IFS ({g 0 , . . . , g 0 , g 1 }), with k copies of g 0 . Given are probabilities p 0 , . . . , p k to pick the diffeomorphisms from. We will assume that IFS ({g 0 , . . . , g 0 , g 1 }) has robustly negative fiber Lyapunov exponents. It follows from the discussion in Sect. 2.1 that for given positive probabilities p i , such an iterated function system exists. Proof We already have the first two items. We must check the remaining items (iii) and (iv). Item (iii) is fulfilled for any f i sufficiently close to g 0 since g 0 is a Morse-Smale diffeomorphism. So it remains to find an open set of diffeomorphisms for which item (iv) holds. We refer to Hirsch (1976), see in particular Hirsch (1976, Exercise 3.15), for the transversality theorem for stratifications. It implies that for a C 1 open and dense set of k diffeomorphisms f i , 0 ≤ i ≤ k − 1, the collection of stratifications i is transverse.

Lemma 2.4 Let k be a positive integer. In any neighborhood of IFS
The following lemma bounds the stationary measure on the stratifications. Proof We will show that if k is large enough, any probability measure on M satisfies m( i ) < 1/2 for some 0 ≤ i < k. Assume m is a probability measure on M with, for all i. The smallest possible total measure on a union of stratifications i 1 ∪· · ·∪ i l , varying over the probability measures on M, occurs if the measure is supported on the common intersection, if this is nonempty. By transversality we have that for l = dim(M) this intersection, if nonempty, is zero-dimensional. Also, the intersection of dim(M) + 1 different stratifications is always empty.
To calculate the smallest possible total measure on 0 ∪ · · · ∪ k−1 , suppose there is measure 1/2 on each i . Consider sets S j that occur as maximal intersections of sets i 1 ∩ · · · ∩ i l ; meaning such that each intersection with a further stratification i is empty. Think of an assignment of mass n j = m(S j ) to the S j 's. We seek the minimal total measure, among variation of such assignments. 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), has a nonempty intersection by possibly adding imaginary intersections. This indeed only adds possible assignments of mass (previous assignments assign zero measure to the new imaginary intersections), hence does not increase total measure. The minimal possible total measure j n j on 0 ∪. . .∪ 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 convex combination of assignments of mass, preserving the total measure, is again an assignment of mass. Then by symmetry, permuting the sets S j and averaging assignments, 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 i . At equidistribution, the measure of S j equals 1 2 / k−1 l−1 as the measure of 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.
We have now constructed iterated function systems IFS (F), F = { f 0 , . . . , f k }, that are C 1 -robustly minimal, so that moreover, for any ergodic stationary measure m, the fiber Lyapunov exponents are all negative and (after relabeling the diffeomorphisms if necessary) f 0 admits a unique attracting fixed point Q 0 with m(W s (Q 0 )) > 1/2.

Delta Conditional Measures
Given an ergodic stationary measure m, let μ be the associated invariant measure for F given by Proposition 1.1. We will establish that the corresponding conditional measures μ ω are delta measures, i.e. K = 1 in Proposition 1.2. For this it suffices to establish that μ ω contains a point measure of mass larger than 1/2, for ν-almost all ω. Indeed, for each 0 ≤ c ≤ 1, the set of points (ω, x) for which μ ω = c is an invariant set. By ergodicity this set has μ-measure equal to 0 or 1. This observation implies the following lemma.
Proof The existence of a set A ⊂ k+1 so that (7) holds, follows from the theory of nonuniform hyperbolicity (Pesin 2004, Lemma 10.5).
Write π : k+1 → + k+1 for the natural projection. Note that the fiber coordinates of F(ω, x) do not depend on ω − . Hence, if A + = π A, when replacing A by π −1 A + , estimate (7) still applies. That is, we may consider A as a product set − × A + . By Lemma 2.5, it is possible to take a closed subset D ⊂ W s (Q 0 ) so that, m(D) > 1/2, (after relabeling the diffeomorphisms if necessary). Recall that d M is a metric on M that generates the weak star topology. Let ⊂ M be the subset of probability measures on M that assign at least mass m(D) to some point, Note that is a closed subset of M. Let A be a subset of k+1 as provided by Lemma 2.7.
Lemma 2.8 There exists L ∈ N so that for each ω ∈ A, there exists B ω + ⊂ − so that for ζ ∈ B ω + × {ω + }, Proof For any r > 0, a sufficiently large iterate of f 0 maps D into a neighborhood of radius r of the attracting fixed point Q 0 of f 0 . By minimality of IFS { f 0 , . . . , f k }, the set ∪ ω∈D N f n ω (Q 0 ) intersects each open set. Hence there is, for any e > 0, an integer L 1 so that for any ball B ⊂ M of diameter e, there are symbols a 1 , . . . , a L 1 with f a L 1 • · · · • f a 1 (Q 0 ) ∈ B. Combining the above statements, there is a composition . We let B ω + consist of the sequences in − that end with these symbols. Given L 1 we may choose L 2 so that L = L 1 + L 2 does not depend on ω. This proves the lemma with L = L 1 + L 2 .
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, This concludes the proof of Proposition 1.3.

Synchronization
We continue with the statement of Theorem 1.1 that describes synchronization (item 1 in its statement). For ν-almost all ω ∈ k+1 , the fiber Lyapunov exponents at (ω, X (ω)) exist and are strictly negative. Write W s (X (ω)) for the stable set of X (ω) inside the fiber {ω} × M; The theory of nonuniform hyperbolicity, as in Lemma 2.7, yields the following. Write D δ (X (ω)) for the δ-ball around X (ω). Then for all ε > 0 there is δ > 0 so that, Once orbits are in a δ-ball D δ (X (ω)) for ω ∈ S(δ), distances to the orbit of X (ω) decrease to zero, which we may assume to happen at a uniform rate as in (7).
It remains to show that W s (X (ω)) is dense in M for ν-almost all ω ∈ k+1 . We have that ( f n σ −n ω ) * m converges to δ X (ω) , ν-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, as before, 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. Shiryayev 1984, Theorem II.10.5). We combine this with the existence of stable sets around X (σ n ω) to prove that d M (( f n ω ) * m, δ X (σ n ω) ) → 0 almost surely. In more detail, let, Now (9) implies that for any given ε > 0,δ > 0, there is N > 0 with, A measure is close to a delta measure if most of the measure is in a small ball: for any ε, δ there isδ > 0 so that d M (μ, δ x ) <δ implies μ(D δ (x)) > 1 − ε. So (10) gives that for any ε > 0, δ > 0 there exists N > 0 so that, With (8) we get that for all ε > 0, there exists δ > 0 and N > 0 so that the set Let U ⊂ k+1 be the set of ω ∈ k+1 with σ i ω ∈ T ε for infinitely many integers i for each ε. Note that ν(U ) = 1. Suppose ω ∈ U . Take y ∈ M and a small ball B around it. To prove that W s (X (ω)) is dense in M, we must show that B contains a point in W s (X (ω)). Note that m(B) > 0 since m has full support. For ε > 0 small enough we have m(B) > ε. Therefore, for ω ∈ U , there is z ∈ B with σ i ω ∈ T ε and f n+i (z) ∈ B δ (X (σ n+i ω)) ⊂ W s (X (σ n+i ω)), n ≥ N .

Uniqueness of the Stationary Measure
This section addresses the uniqueness of the stationary measure stated in item (ii) of Theorem 1.1. Lemma 2.10 Assume the conditions of Theorem 1.1. Then the stationary measure m is the unique stationary measure.
Proof Let m be a stationary measure with only negative fiber Lyapunov exponents and assume there exists a different stationary measurem. We may takem to be an ergodic stationary measure. By Proposition 1.1 there is a F-invariant measureμ with marginal ν and conditional measuresμ ω satisfying, for ν-almost all ω ∈ k+1 .
Recall that for ν-almost all ω ∈ k+1 , W s (X (ω)) is open and dense (Lemma 2.9). We can therefore take e 1 and a subset T ⊂ k+1 with ν(T ) > 0 so that for ω ∈ T , W s (X (ω)) contains a closed ball B(ω) of diameter e 1 . By Lemma 1.1,m has full support and thus assigns positive measure to any open set B ⊂ M. We can therefore take e 2 > 0 and decrease T if necessary to find, for ω ∈ T . By taking T still smaller if needed, we may moreover assume that there are e 3 > 0 and N > 0 so that f n σ −n ω (B(σ −n ω)) is contained in a ball of diameter e 3 around X (ω), if n ≥ N and σ −n ω ∈ T . In particular f n σ −n ω (B(σ −n (ω))) converges to X (ω) if σ −n ω ∈ T and n → ∞. For ν-almost all ω ∈ k+1 , σ −n ω ∈ T for infinitely many values of n. For such ω we find by (12) that ( f n σ −n ω ) * m ( f n σ −n ω (B(σ −n (ω))) > e 2 . Since f n σ −n ω (B(σ −n (ω))) converges to X (ω) we get that there is an accumulation point of ( f n σ −n ω ) * m that assign positive measure to X (ω). By (11), However, by ergodicity,μ = μ impliesμ ω (X (ω)) = 0 for ν-almost all ω ∈ k+1 . A contradiction has been derived and the lemma is proved.

Iterated Function Systems on Compact Surfaces
On compact two-dimensional surfaces one obtains Theorem 1.1 with iterated function systems generated by two diffeomorphisms. The proof of Theorem 1.1 can be followed, with Lemma 2.5 being replaced by Lemma 3.2 below. This lemma uses that IFS Proof Write α = m( 0 ). Since m has full support, α < 1. Since M is twodimensional, 0 intersects f −1 1 ( 0 ) in a set of dimension zero, if it intersects, so in a set of stationary measure zero by Lemma 3.1. Therefore m( 0 ∪ f −1 1 ( 0 )) = m( 0 ) + m( f −1 1 ( 0 )) = α + f 1 m( 0 ), so that α + f 1 m( 0 ) < 1. The measure m being stationary implies α = p 0 α + p 1 ( f 1 ) * m( 0 ). So, and α + ( f 1 ) * m( 0 ) < 1 implies α < 1/2. Figure 2 gives a numerical demonstration of synchronization in a specific iterated function system on the two dimensional torus. Here we provide a robust synchronization result for small perturbations of this specific iterated function system, partly to illustrate the results of this paper.
Assume the diffeomorphisms f 0 , f 1 are picked independently with positive probabilities p 0 , p 1 = 1 − p 0 . Then there is a C 1 neighborhood of IFS { f 0 , f 1 } so that any iterated function system IFS {g 0 , g 1 } with C 2 diffeomorphisms from it, is minimal and displays synchronization.
Let B be a small ball around the attracting fixed point (1/2, 1/2) of f 0 . Then f 0 (B) is strictly contained inside B. Take vectors (u 1 , v 1 ), . . . (u l , v l With the above observation it is easily seen that there are numbers n 1 , . . . , n l so that, Note further that there is a finite number N so that Since (13) and (14) are robust under small C 1 perturbations of f 0 , f 1 , it follows that IFS { f 0 , f 1 } is robustly minimal on T 2 . Now IFS { f 0 , f 1 } has a stationary measure m = m 1 × m 2 with negative fiber Lyapunov exponents. By Lemma 2.10, m is the unique stationary measure for IFS { f 0 , f 1 }. Since m 1 and m 2 are nonatomic (Kleptsyn and Nalskii 2004), we find m(W s ((1/2, 1/2)) = 1 for the basin of attraction W s ((1/2, 1/2)) of the stable fixed point (1/2, 1/2) of f 0 .
Consider now C 2 diffeomorphisms g 0 , g 1 that are C 1 close to f 0 , f 1 . Then by Lemma 2.2, any stationary measure for IFS {g 0 , g 1 } is close to m in the weak star topology. So g 0 has an attracting fixed point whose basin has stationary measure at least 1/2. It now follows from Theorem 1.1 that IFS {g 0 , g 1 } has a unique stationary measure with negative fiber Lyapunov exponents, and displays synchronization.