Skew product systems with one-dimensional fibres

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1 Skew product systems with one-dimensional fibres T. Jäger Abstract These lecture notes grew (and continue to grow) out of several courses given on various occasions in different places, including in particular the European Summer School on Topological Methods in Surface Dynamics held in Grenoble in June 2006, the Workshop on Skew Product Dynamics and Multifractal Analysis held in Luisenthal in October 2012 and the Spring School in Dynamical Systems held in Banská Bystrica in April The aim is to give a comprehensive description, from a measure-theoretical and topological viewpoint, of the dynamics of skew product systems that have either monotone interval maps or circle homeomorphisms on the fibres. The first part concentrates on forced monotone interval maps. The main results which are included are a characterization of ergodic invariant measures due to Furstenberg, a description of minimal sets due to Stark and a classification of forced concave maps going back to Keller. The latter also provides the basis for a description of non-autonomous saddle-node bifurcations in forced interval maps. Finally, as a particular example, we also give a proof for the existence of strange non-chaotic attractors in so-called pinched skew products, which were introduced by Grebogi et al and later treated rigorously by Keller. In the second part we focus on forced circle homeomorphisms. Since forced monotone interval maps embed naturally in this setting, we can rely on many of the results of the previous chapter. However, the additional freedom given by the circular fibres leads to a greater diversity of the dynamical behaviour. The chapter starts with a dichotomy concerning the structure of ergodic invariant measures due to Furstenberg and continues with Herman s definition of the fibred rotation number, which provides the basis for the rotation theory of forced circle maps. Restricting to irrational rotations as forcing transformations, we then give an analogue to Poincaré s celebrated classification of circle homeomorphisms in the quasiperiodically forced case. This classification entails a number of further consequences concerning topological transitivity, the uniqueness of minimal sets and the invariant measures and the occurrence of mode-locking. 1 Forced monotone interval maps 1.1 Notation and terminology We consider skew product maps (1.1) f : X X, f(ω, x) = (γ(ω),f ω(x)) on a product space X and refer to these as γ-forced maps. is called the base or driving space and is equipped with a σ-algebra F, such that (, F) becomes a measurable space. In most situations we will assume that is a compact metric space, in which case F is always the (completed) Borel σ-algebra. In this case, we always assume that f is a continuous map. The space X is called the phase space, usually it is a smooth (Riemannian) manifold. In these notes, X will mostly be either a real interval or the circle. The Borel-σ-algebra on X will be denoted by B. The product space X is also called the extended phase space. The base transformation γ (also driving system or forcing process) is always measurable, and in case is a topological space we automatically assume it to be continuous. We often write γω instead of γ(ω). Department of Mathematics, TU Dresden, Tobias.Oertel-Jaeger@tu-dresden.de 1

2 2 Skew products with one-dimensional fibres The maps f ω : X X are called fibre maps. We will always assume them to be continuous. By f n ω = (f n ) ω we denote the fibre maps of the iterates of f, that is, f n ω(x) = π 2 f n (ω, x) = f γ n 1 ω... f ω(x). Here π i denotes the projection to the i-th coordinate (i = 1, 2). When X is a smooth manifold and all fibre maps f ω are C r, we call f a γ-forced C r -map. If X = [a, b] R, a, b R {± } and all fibre maps are monotonically increasing (not necessarily strictly), we call f a γ-forced monotone interval map. By M(γ), we denote the set of all γ-invariant probability measures on. When m M(γ) the tupel (, F, m, γ) is a measure-preserving dynamical system, abbreviated mpds (compare [1]). In this case, f is sometimes called a continuous random map with base (, F, m, γ). 1.2 Invariant graphs and invariant measures In the context of forced systems, a substitute for fixed points of unperturbed maps which is natural from the measure-theoretical point of view is given by invariant graphs. A measurable function ϕ : X is called an f-invariant graph if it satisfies (1.2) f ω(ϕ(ω)) = ϕ(γ(ω)) for all ω. In this case Φ = {(ω, ϕ(ω)) ω } is f-invariant. Slightly abusing terminology, the term f-invariant graph will be used both for ϕ and Φ. If (1.2) only holds m-a.s. for some m M(γ), we call f an (f, m)-invariant graph. In this case, we identify invariant graphs which are equal m-a.s. and implicitely speak of equivalence classes. To any (f, m)-invariant graph ϕ, an f-invariant measure m ϕ can be assigned by defining (1.3) m ϕ(a) = m(π 1(A Φ)) for any measurable set A X. The measure m ϕ is ergodic if and only if m is ergodic. A converse of this holds for ergodically forced monotone interval maps. Given m M(γ), we denote by M m(f) the set of all f-invariant measures µ which project to m in the first coordinate, that is, m = µ π 1 1 = π 1µ. Theorem 1.1 (Furstenberg [2], see also Theorem in [1]). Suppose f is a γ-forced monotone interval map, m M(γ) and µ M m(f) is ergodic. Then there exists an (f, m)-invariant graph ϕ such that µ = m ϕ. A more general version providing a dichotomy for the structure of invariant ergodic measures of forced circle homeomorphisms will be given in Section 2.2. For the proof of the above statement, we need some information on the decomposition of measures (see [1, Chapter 1.4]). Suppose µ is a probability measure on a product space X, where X is polish 1, and µ projects to a measure m on. Then there exists a family of probability measures (µ ω) ω such that for all integrable functions g : X R ω R g(ω,x) dµω(x) is measurable; X R g dµ = R R g(ω,x) dµω(x)dm(ω). X X The measures µ ω on X will be called the fiber measures of µ. If A X is measurable and we use the notation (1.4) A ω := {x X (ω,x) A}, then we have (1.5) µ(a) = µ ω(a ω) dm(ω). Further, the fiber measures are mapped to each other by the action of f, that is, (1.6) µ γω = µ ω f 1 γω for m-a.e. ω. This can easily be seen by using the fact that since X is polish, its Borel-σ-algebra B has a countable -stable generator. This observation will be crucial in a number of arguments. It immediately implies that the topological support of µ ω is mapped to that of µ γω, that is, f ω(supp(µ ω)) = supp(µ γω) for m-a.e. ω. 2 1 Metrizable in a complete and second countable way. 2 Recall here that the support of a probability measure on a topological space X is defined as supp(ν) = {x X ν(u) > 0 for all neighbourhoods U of x}, and if X is second countable then ν(supp(ν)) = 1.

3 3 Proof of Theorem 1.1. Suppose f is a γ-forced map, m M(γ) and µ M m(γ). Let ϕ(ω) = sup{x R µ ω(, x] 1/2} and define A,B R by A ω = (,ϕ(ω)] and B ω = [ϕ(ω),+ ). Then µ ω(a ω) 1 and 2 µ ω(b ω) 1 for all ω and hence µ(a) 1 and µ(b) At the same time, A and B are f-invariant. This follows from the fact that µ ω fω 1 = µ γω, which implies x ϕ(γω) µ γω(, x) 1 2 µ ω(,f 1 γω (x)) 1 2 f 1 γω (x) ϕ(ω) x f ω(ϕ(ω)) and hence f ω(ϕ(ω)) = ϕ(γ(ω)). Ergodicity of µ implies µ(a) = µ(b) = 1 and hence µ(a B) = µ(φ) = 1. Since µ projects to m, we obtain µ = m ϕ. 1.3 Minimal strips and minimal sets of minimally forced monotone interval maps As we have seen above, invariant graphs replace fixed points of unperturbed systems in a measurable setting. From the topological point of view, however, suitable substitutes of fixed points should have some topological structure as well. It turns out that the right concept are so-called invariant strips, which we will introduce in the following. Throughout the section, we assume that is a compact metric space and f is a γ-forced monotone interval map with a minimal homeomorphism γ : on the base. A compact set A R is called a strip, if A ω is a non-empty compact interval for all ω. We allow A ω to be reduced to a single point, such that particular examples are continuous curves of the form Φ = {(ω, ϕ(ω)) ω }. A strip A is called minimal, if it does not strictly contain any smaller strip. Note that thus minimality of strips is different from the minimality of a set in the dynamical sense. In order to see the connection to invariant graphs, suppose A T 1 is compact and f- invariant. Then π 1(A) is non-empty, compact and γ-invariant and thus equals by minimality. Therefore, we can define functions ϕ ± A : R, called the bounding graphs of A by (1.7) ϕ + A(ω) := sup A ω and ϕ A(ω) := inf A ω. As a simple consequence of monotonicity, ϕ ± A are invariant graphs. Further, as (1.8) {ω ϕ + A(ω) b} = π 1(A [b,+ )) is closed for all b R, ϕ + A is upper semicontinuous (usc) and (1.9) lim sup ϕ + A(ω ) ϕ + A(ω) for all ω. ω ω Similarly, ϕ A is lower semicontinuous (lsc). Consequently (1.10) [ϕ A, ϕ + A] := {(ω, x) X ϕ A(ω) x ϕ + A(ω)} is compact and therefore defines an f-invariant strip that contains A. Hence, every compact invariant set is contained in an invariant strip, and both are bounded by semicontinuous invariant graphs. We note that the above construction of bounding graphs does not depend on the dynamics and can therefore also be applied to non-invariant sets. It is easy to check that a nested intersection of strips is again a strip. Hence, the Lemma of orn easily implies that ever strip contains a minimal strip. However, this does not allow to conclude that every invariant strip contains an invariant minimal strip, since in this case orn s Lemma only yields the existence of a minimal element amongst all invariant strips. For this reason, we need a more explicit construction to obtain minimal strips. To that end, suppose that ϕ + is an usc graph with corresponding point set Φ + := {(ω, ϕ + (ω)) ω }. Then we let ϕ + = ϕ Φ +. Similarly, if ϕ is lsc we let ϕ + = ϕ + and then define Φ ϕ+ +, ϕ + by the corresponding concatenations of these procedures. We say ϕ + and ϕ reflexive if ϕ + = ϕ + + and ϕ = ϕ +, respectively. ϕ + and ϕ are called a pair of reflexive graphs if ϕ = ϕ + and ϕ + = ϕ +. In this case both graphs are reflexive. Note that every strip A R can be written as A = [ϕ A, ϕ+ A ] as in (1.10). Minimality of the strip is then characterized by the reflexivity of its bounding graphs.

4 4 Skew products with one-dimensional fibres Lemma 1.2. A strip A R is minimal if and only if ϕ A and ϕ+ A graphs. form a reflexive pair of Proof. Suppose A is a minimal strip. Since Φ + A A, we have that ϕ+ A ϕ. Hence, if the two graphs were not equal then [ϕ + A, ϕ+ A ] would be a strict subset of A, contradicting the minimality of the strip A. Conversely, suppose that ϕ A and ϕ+ A form a reflexive pair and A A is a strip as well. Then either ϕ + A ϕ + A or ϕ A ϕ A. In the first case this means that Φ A is contained in the compact set [ϕ A, ϕ+ ], such that ϕ + A A ϕ+ and thus ϕ + A A ϕ+ A, contradicting reflexivity. The second case is symmetric. Lemma 1.3. Suppose that f is a minimally forced monotone map and M is a minimal set. Then ϕ M and ϕ+ M form a reflexive pair of graphs. Conversely, if ϕ+ is usc and reflexive, then Φ + is a minimal set. Proof. By minimality of M, we have Φ M = Φ M, which immediately implies the reflexivity of the pair ϕ + M, ϕ+ M. Conversely, suppose ϕ+ is usc and reflexive and let M = ϕ +. If M contains a strict compact invariant subset M, then ϕ + ϕ M and hence ϕ + = ϕ M ϕ, but M M also M implies ϕ M ϕ M, so that we obtain ϕ M = ϕ M. However, this further means that ϕ + = ϕ + + = ϕ + M and we obtain Φ + Φ M M, contradicting M M. Since this means that every minimal set M is contained in a minimal strip [ϕ M, ϕ+ M ], this raises the obvious question whether M = [ϕ M, ϕ+ M ]. However, the general answer to this question is no, see [3] for examples. We can now turn to a general procedure to obtain reflexive graphs. Lemma 1.4. Suppose ϕ : T 1 R is upper semi-continuous. Then ϕ + and ϕ + + form a reflexive pair of graphs. We have to show that ϕ + + = ϕ +. First of all, as ϕ + ϕ and due to the semi-continuity of the two graphs, the set [ϕ +, ϕ + ] is compact and contains Φ +. Therefore it also contains Φ + +, so that ϕ + ϕ + + ϕ +. However, as ϕ + + is lower semi-continuous, the set [ϕ + +, ϕ + ] is compact as well, consequently it contains Φ + and thus also Φ +. This proves the reverse inequality ϕ + + ϕ +. Corollary 1.5. Given any compact set A T 1 with π 1(A) =, the set S + (A) = [ϕ + A is a minimal strip., ϕ+ + A ] We note that if either the graph ϕ + in Lemma or the set A in Corollary are f-invariant, then so are the resulting reflexive graphs and the set S + (A). If A, B are minimal strips which are invariant for a minimally forced monotone map f, then the γ-invariant set π 1(A B) is either empty of all of. In the second case, the intersection A ω B ω is a non-empty compact interval for all ω, such that A B is an invariant strip as well. By minimality this implies A = B. Hence, we conclude that the family of invariant minimal strips is strictly ordered in R. We write A B : ϕ A ϕ B and ϕ + A ϕ + B A B : ϕ + A < ϕ B. Using this ordering, the invariant minimal strip S + (A) associated to an invariant compact set A can be characterized as the highest invariant minimal strip contained in [ϕ A, ϕ+ A ]. In a completely analogous way, one can define the lowest invariant minimal strip contained in S (A). Given a compact set A T 1 with π 1(A) =, we define the set of pinched points as and say A is pinched if P(A) is non-empty. P(A) = {ω #A ω = 1} Lemma 1.6. If ϕ is lsc, ϕ + is usc and Φ Φ + or vice versa, then A = [ϕ, ϕ + ] is pinched and P(A) is a residual set. Since {(ω, x) x ϕ + (ω)} is closed, we must have ϕ ϕ +. The set P(A) is a residual set, since it is the intersection of the open sets P n(a) = {ω diam(a ω) 1/n}. Thus, it remains to show that for each n N the set P n(a) is dense. To that end, fix ω and choose a sequence (ω n) n N converging to ω such that lim n ϕ + (ω n) = ϕ (ω). By semi-continuity and since ϕ ϕ +, we also have that lim n ϕ (ω n) = ϕ (ω).

5 5 This shows that P n(a) intersects every neighbourhood of ω. Corollary 1.7. Every minimal strip and every minimal set of a minimally forced monotone map is pinched. Corollary 1.8. If a strip A has empty interior, then it contains a unique minimal strip. Proof. If A 1, A 2 A are different minimal strips, there exists a residual set P(A 1) P(A 2). Since C i = A i R has compact closure contained in A i and there are no semicontinuous graphs in A i except the bounding graphs, we have ϕ + A i = ϕ + C i. Consequently, if A 1 A 2, then we must have some ω such that A 1,ω A 2,ω, say ϕ A 1 (ω) = ϕ + A 1 (ω) < ϕ A 2 (ω) = ϕ + A 2 (ω). However, due to the semicontinuity of the graphs this implies that for some ε, δ > 0 the set A contains an open box of the form B δ (ω) [ϕ + A 1 (ω) + ε, ϕ A 2 (ω)]. Remark 1.9. Based on these notions, a classification of transitive sets was given by Stark [4]. A decomposition of the phase space into (1) transitive annuli, (2) annuli with wandering dynamics and (3) regions densely filled by minimal strips is described in [5]. The concept of minimal strips plays a crucial role in the theory of forced circle homeomorphisms [6, 7]. An analogous concept can be used in a non-skew product setting to describe the dynamics of periodic-point free maps [8, 9, 5]. 1.4 Lyapunov exponents Suppose f is a γ-forced C 1 -interval map and m M(γ). The the (vertical) Lyapunov exponent of an (f, m)-invariant graph is defined as (1.11) λ m(ϕ) := log f ω(ϕ(ω)) dm(ω) = log f ω(ϕ(ω)) dm ϕ(ω,x). Lemma 1.10 (e.g. Proposition 3.3 in [10]). In the above situation, if λ m(ϕ) < 0, then there exists a measurable function η : R + which is m-a.s. strictly positive and satisfies X (1.12) f n ω(x) ϕ(γ n ω) n 0 for all x B η(ω) (ϕ(ω)). Corollary If (1.13) ψ(ω) = sup n o x > ϕ(ω) fω(x) n ϕ(γ n (ω)) n 0 is m-a.s. finite, then ψ is an (f, m)-invariant graph with λ m(ψ) 0. Strict negativity of the Lyapunov exponents on a compact invariant set typically implies that this set is a continuous curve, or a union of continuous curves. We provide two results in this direction, one for forced monotone C 1 -interval maps and one for minimally forced C 1 -maps with higher dimensional phase space. We start with the version for forced interval maps. Theorem Suppose γ is a homeomorphism of a compact metric space, f is a γ-forced C 1 -interval map and K is a compact f-invariant set such that K ω is an interval for all ω. Further, assume that for all m M(γ) and all (f, m)-invariant graphs ϕ contained in K we have λ m(ϕ) < 0. Then K is a continuous f-invariant curve. For the proof, we need the following semi-uniform ergodic theorem from [11]. Given a measurepreserving transformation T of a probability space (Y, B, µ) and a subadditive sequence of integrable functions g n : Y R (that is, g n+m(y) g n(y) + g m(t n y)), the limit ḡ(y) = lim n gn(y)/n exists µ-a.s. by the Subadditive Ergodic Theorem (e.g. [1, 12]). Furthermore ḡ is T-invariant. Consequently, when T is ergodic then ḡ is µ-a.s. equal to the constant µ(ḡ) = R Y ḡ dµ. Theorem 1.13 (Theorem 1.12 in [11]). Suppose that T : Y Y is a continuous map on a compact metrizable space Y and g n : Y R (n N 0) is a subadditive sequence of continuous functions. Let τ be a constant such that µ(ḡ) < τ for every T-invariant ergodic measure µ. Then there exist δ > 0 and N N, such that g n(y)/n τ δ for all y Y, n N.

6 6 Skew products with one-dimensional fibres Proof of Lemma Due to Theorem 1.1, any f-invariant ergodic measure µ is of the form µ = m ϕ for some γ-invariant ergodic measure m and an (f, m)-invariant graph ϕ. Consequently, we have (1.14) log f ω(x) dµ(ω,x) = log f ω(ϕ(ω)) dm(ω) = λ m(ϕ) < 0. X Hence, Theorem 1.13 with Y = X, T = f, τ = 0 and g n(ω,x) = log(f n ω) (x) implies that for some N N and α (0,1) we have (1.15) fω N (x) α (ω, x) K. If we let C := `sup ω ϕ + (ω) ϕ (ω), then this implies (1.16) diam (K ω) = diam f N γ N (ω) `Kγ 1 (ω) α diam `K γ N (ω) α C ω, which yields C α C. This means that C = 0, such that K is the graph of the continuous function ϕ ϕ +. When the underlying homeomorphism γ is minimal, then a similar statement holds in much greater generality, namely for arbitrary compact invariant sets of γ-forced C 1 -maps on any Riemannian manifold. For the case of quasiperiodic forcing by an irrational rotation of the circle, this was shown by Sturman and Stark [11, Theorem 1.14]. The proof presented below is taken from [13, Section 5]. In the remainder of this section we let X be a Riemannian manifold, endowed with the canonical distance function d induced by the Riemannian metric. We suppose f is an γ-forced C 1 -map on X. The upper Lyapunov exponent of (ω, x) X is (1.17) λ max(ω, x) = lim sup n 1 n log Dfn ω (x), where Df ω(x) is the derivative matrix of f ω in x and denotes the usual matrix norm. Given any f-invariant probability measure µ, we define the upper Lyapunov exponent of µ by (1.18) λ max(µ) = λ max(ω, x) dµ(ω,x). Note that by Kingmans Subadditive Ergodic Theorem, we have 1 (1.19) lim n n log Dfn ω (x) = λ max(µ) µ-a.s. Further, we let X k = {x X k x i x j if i j} and endow X k with the Hausdorff topology. Theorem Suppose γ : is a minimal homeomorphism, X is a Riemannian manifold, f is an γ-forced C 1 -map on X and K is a compact invariant set of f. Further, assume that λ max(µ) < 0 for all f-invariant ergodic measures µ supported on K. Then there exist k N and a continuous map ψ : X k such that K is the graph of ψ, that is, K = (ω, ψ i(ω)) ω, i = 1,..., k. Remark (a) Note that since we do not assume any specific structure on, it does not make sense to speak of the smoothness of the curve ψ in this setting (in contrast to [11]). However, when is a torus and γ and irrational rotation, then the smoothness of ψ follows from its continuity [14]. In fact, ψ is as smooth as f itself in this situation. In general, smoothness can only be expected when γ is an isometry. (b) If f is invertible, as in the case of forced monotone interval maps, the conclusion of Theorem 1.14 also holds if λ max(µ) > 0 for all ergodic measures µ. Proof. Applying Theorem 1.13 to Y = X, T = f, τ = 0 and g n(ω,x) = log Df n ω (x), we obtain that for some N N and α (0, 1) (1.20) Df N ω (x) α (ω,x) K.

7 7 Replacing f by f N if necessary, we may assume without loss of generality N = 1. By compacity, there exist some ε > 0 and α (α,1) such that (1.21) Df ω(x) α (ω,x) B ε(k). If we let ˆB ε(k) = {(ω,x) X x B ε(k ω)}, then together with the invariance of K (1.21) implies that (1.22) f( ˆB ε(k)) ˆB ε(k). It follows that for any (ω, x) ˆB ε(k) (1.23) Df n ω (x) α n n N. Consequently, we have (1.24) x,x K ω and d(x,x ) < 2ε d(f n ω(x),f n ω(x )) α n d(x,x ) n N. We now proceed in 4 steps. Step 1: K intersects every fibre in a finite number of points. Let K ω := {x X : (ω,x) K}. As K is compact, there exist (ω 1, x 1),..., (ω m, x m) such that (1.25) K m[ B ε(ω k, x k ). k=1 We will show that for any ω the cardinality of K ω, denoted by #K ω, is at most m. Suppose for a contradiction that there exists ω 0 with #K ω0 > m. We choose m+1 distinct points ξ 1,..., ξ m+1 K theta0 and let a = min d(ξi, ξj). i j Further, we fix n N such that 2ε α n < a and choose, for each for i = 1,..., m + 1, some ξ i 1 f n γ n (ω 0 ) {ξi} K (note that such ξ i exist since f(k) = K and therefore f n γ n (ω 0 ) (K γ n (ω 0 )) = K ω0 ). Due to (1.25), there exist l {1,..., m} and i, j {1,..., m + 1} such that ξ i and ξ j both belong to ˆB ε(x l ). Hence, the distance between the two points is less than 2ε. Using (1.24) we conclude that (1.26) d(ξ i, ξ j) = d contradicting the definition of a. Step 2: #K ω is constant on. We let f n γ n (ω 0 ) (ξ i), f n γ n (ω 0 ) (ξ j) k := min ω #Kω α n 2ε < a, and fix ω 0 with #K ω0 = k. Suppose there exists ω with #K ω > k. Similar as in Step 1, we choose points ξ 1,..., ξ k+1 K ω, let a = min i j d(ξ i, ξ j) and fix n 0 N such that α n 2ε < a n n 0. Due to the compacity of K, there exists δ > 0 such that (1.27) K ω B ε(k ω0 ) ω B δ (ω 0). By the minimality of γ on, there exists n n 0 with γ n (ω) B δ (ω 0), such that K γ n (ω) B ε(k ω0 ). However, as K ω0 only consists of m points, at least two of the points ξ 1,..., ξ m+1, say ξ i and ξ j, must have preimages ξ i and ξ j under f n γ n (ω) such that d(ξ i, ξ j) < 2ε. Using (1.24) again we obtain (1.28) d(ξ i, ξ j) = d f n γ n (ω) (ξ i), f n γ n (ω) (ξ j) α n 2ε < a, contradicting the definition of a. Step 3: The distance between distinct points in K ω is at least 2ε.

8 8 Skew products with one-dimensional fibres The proof of this step is almost completely identical to that of Step 2. If there exists ω 0 such that two points in K ω0 have distance less than 2ε, then for any n with γ n (ω) sufficiently close to ω 0 at least two of the k points in K ω will have preimages that are 2ε-close. Choosing n sufficiently large and using (1.24) once more, this leads to a contradiction in the same way as in (1.26) and (1.28). Step 4: The mapping ω K ω is continuous. Fix ω 0. We have to show that given any γ > 0 there exists δ > 0 such that d(ω,ω 0) < δ implies d H(K ω, K ω0 ) < γ, where d H denotes the Hausdorff distance on the space of subsets of X. We may assume without loss of generality that γ < ε. Due to the compacity of K, there exists δ > 0 such that d(ω,ω 0) < δ implies K ω B γ(k ω0 ). However, since K ω and K ω0 consist of exactly k points which are at least 2ε apart, there must be exactly one point of K ω in the γ-neighbourhood of any point in K ω0. Thus, we obtain d H(K ω, K ω0 ) < γ as required. 1.5 Forced monotone interval maps with concave fibres The aim of this section is to give a classification, with respect to the number and stability of invariant graphs, of forced monotone interval maps with concave fibres. In contrast to the preceeding sections, the topological structure of does not play a role and it suffices to assume that (, B) is a measurable space. Theorem Let f be a γ-forced monotone C 1 -interval map and m M(γ). Suppose ψ ψ + : R are measurable functions such that for m-a.e. ω the fibre maps f ω are strictly concave on A ω := [ψ (ω),ψ + (ω)]. Further, assume that η(ω) := inf x Aω log f ω(x) has an integrable minorant. Then there exist at most two (f, m)-invariant graphs in A, and if there are two such graphs ϕ 1 ϕ 2 then λ(ϕ 1) > 0 and λ(ϕ 2) < 0. For the proof, we need the following measure-theoretic lemma. Lemma 1.17 (Lemma 2 in [15]). Suppose (, B, m,γ) is a mpds and g is measurable. If g γ g has an integrable minorant, then g γ g dm = 0. Note that the statement is trivial when g is integrable, since then R g γ g dm = R g γ dm R g dm = 0 by γ-invariance of m. In the general case, the proof relies on an application of Fatou s Lemma. Proof of Theorem Suppose there exist three (f, m)-invariant graphs ϕ 1 ϕ 2 ϕ 3 in A. By concavity, we have f ω(ϕ 3(ω)) f ω(ϕ 2(ω)) ϕ 3(ω) ϕ 2(ω) = ϕ3(γω) ϕ2(γω) ϕ 3(ω) ϕ 2(ω) < ϕ2(γω) ϕ1(γω) ϕ 2(ω) ϕ 1(ω) for m-a.e. ω and hence «ϕ3(γω) ϕ 2(γω) (1.29) log ϕ 3(ω) ϕ 2(ω) dm(ω) < log «ϕ2(γω) ϕ 1(γω) ϕ 2(ω) ϕ 1(ω) dm(ω). However, if we let g(ω) = log(ϕ i+1(ω) ϕ i(ω)) for i = 1, 2 then Lemma 1.17 implies (1.30) log «ϕi+1(γω) ϕ i(γω) ϕ i+1(ω) ϕ i(ω) dm(ω) = g γ(ω) g(ω) dm(ω) = 0, contradicting the strictness of the inequality in (1.29). For the Lyapunov exponents of two (f, m)-invariant graphs ϕ ϕ +, note that again due to the strict concavity of the fibre maps we have log f ω(ϕ 1(ω)) < log «ϕ2(γω) ϕ 1(γω) ϕ 2(ω) ϕ 1(ω) < log f ω(ϕ 2(ω)) m-a.s. After integration with m, the statement therefore follows from (1.30) again.

9 9 1.6 Approximating curves Theorem Let f be a γ-forced monotone C 1 -interval map, m M(γ) and suppose ϕ 0 : R satisfies f ω(ϕ 0(ω)) < ϕ 0(γω) m-a.s. Further, assume that ω inf x ϕ0 (ω)(x) has an integrable minorant. Let ϕ n(ω) := f n γ n ω (ϕ0(γ n ω)) and ϕ + (ω) := lim n ϕn(ω). Then, if ϕ + is m-a.s. finite, it is an invariant graph and we have λ m(ϕ + ) 0. Proof. By induction, it is easy to see that the sequence ϕ n is decreasing. If ϕ n(ω) = ϕ + (ω) on a set of positive measure for some n N, we must have f ω(ϕ + (ω)) = 0 on and hence λ m(ϕ + ) =. Hence, we may assume that the sequences ϕ n(ω) are m-a.s. strictly decreasing. The graph ϕ + is f-invariant since f ω(ϕ + (ω)) = f ω( lim n ϕn(ω)) = lim n fω(fn γ n ω(ϕ 0(γ n ω)) = lim n ϕn+1(γω) = ϕ+ (γω). For the Lyapunov exponent, we have λ m(ϕ + ) = log f ω(ϕ + (ω)) dm(ω) «Fatou fω(ϕ n(ω) f ω(ϕ + (ω)) lim inf log dm(ω) n ϕ n(ω) ϕ + (ω) «ϕn+1(γω) ϕ + (γω) = lim inf log dm(ω) n ϕ n(ω) ϕ + (ω) «ϕn(γω) ϕ + (γω) Lemma 1.17 lim inf log dm(ω) = 0 n ϕ n(ω) ϕ + (ω) 1.7 Strange nonchaotic attractors in pinched skew products We consider the system (1.31) f α : T 1 [0, 1] T 1 [0, 1], f α(ω,x) = (ω + ρ mod 1,tanh(αx) sin(πω)), which is a slight modification of a paradigm example introduced by Grebogi et al in [16]. The rigorous description was given by Keller in [15]. f α is a γ-forced monotone C 1 -interval map with the irrational rotation of angle ρ R \ Q on the base. The global attractor of the system is defined as (1.32) A α = \ n N f n α(t 1 [0, 1]) = [0, ϕ + ], where ϕ + is the so-called upper bounding graph and given by with ϕ + (ω) = sup A α ω = lim n ϕn(ω) ϕ n(ω) = f n α,γ n ω (1). By Theorem 1.18, ϕ + is an invariant graph with λ m(ϕ + ) 0, where m now denotes the Lebesgue measure on T 1. As f 0 = 0 and due to invariance, we have ϕ + (nρ mod 1) = 0 for all n N. However, at the same time we have λ m(0) = log f ω(0) dω = log α + log sin πω dω = log α log 2. This means that for α > 0 the graph ϕ + cannot coincide with the 0-line, and must therefore be m-a.s. strictly positive and consequently noncontinuous. Further, Theorem 1.16 yields λ m(ϕ + ) < 0. Due to the combination of its properties complicated topological structure, attractivity and the non-chaotic dynamics of the system ϕ + is called a strange nonchaotic attractor.

10 10 Skew products with one-dimensional fibres 2 Forced circle homeomorphisms Henri Poincarés celebrated classification of orientation-preserving circle homeomorphisms [17] presents one of the cornerstones of Dynamical Systems Theory. Motivated by celestial mechanics and the study of ordinary differential equations on the two-torus, Poincaré observed that toral flows generated by non-vanishing differentiable vector fields always admit an essential closed curve transverse to the direction of the flow which is crossed infinitely often by every orbit. The return map to this so-called Poincaré section is an orientation-preserving circle homeomorphism f, and the dynamics of the original flow can be understood by analyzing the long-term behaviour of the orbits of f. Given a lift F : R R of f to the real line, Poincaré introduced the rotation number of F as (2.1) ρ(f) = lim n (F n (x) x)/n and proved that the limit in (2.1) always exists and does not depend on the starting point x R. Since different lifts of f coincide modulo an integer constant, so do their rotation numbers, and consequently the rotation number ρ(f) = ρ(f) mod 1 of f is well-defined. The surprising discovery was that this simple topological invariant almost completely determines the dynamics of f. This is the content of the famous Poincaré Classification Theorem. Theorem 2.1. Let f be an orientation-preserving circle homeomorphism. Then (a) ρ(f) is rational if and only f has a periodic orbit. (b) ρ(f) is irrational if and only if there exists an order-preserving continuous onto map h : T 1 T 1 such that h f = r ρ(f) h, where r ρ(f) : x x + ρ(f) is the irrational rotation with angle ρ(f) on T 1 = R/. Up to date, this is still one of the simplest and at the same time most elegant classification results in Dynamical Systems. Since both circle homeomorphisms with periodic orbits, which basically reduce to monotone interval maps, and irrational rotations are well-understood, a wealth of further information can be deduced from this fundamental result. In particular, when ρ(f) is irrational then f has a unique minimal set and a unique invariant probability measure. Furthermore, the rotation number of f is sensitive to perturbations in this case, meaning that ρ : ε ρ(r ε f) is strictly monotone in ε = 0. Conversely, the existence of an attracting or repelling periodic orbit leads to mode-locking, which means that ρ is constant in a neighbourhood of ε = 0. These facts, together with some basic results from complex analysis, easily allow to verify the surprising observation that the rotation number of the Arnold circle map f α,τ(x) = x + τ + α sin(2πx) as a function of τ is a 2π devils staircase, meaning that it is locally constant on a dense and open subset of T 1, see Figure 2.1. The aim of this section is to develop, as far as this is possible, an analogous theory for forced circle homeomorphisms. As we will see, even in the simplest case of an irrational rotation the influence of the external forcing leads to a variety of additional dynamical phenomena. 2.1 Multivalued invariant graphs Throughout this and the next section, we will assume that f is a γ-forced map with ergodic base (, F, m, γ). When going from monotone interval maps to circle homeomorphisms, one major change is the possibility of periodic orbits with period > 1. In the forced situation, this corresponds to the existence of periodic graphs. If f is a γ-forced map and ϕ 1 : X is measurable, we say ϕ 1 is p-periodic if for m-a.e. ω we have fω(ϕ j 1(ω)) ϕ 1(γ j ω) if j = 1,..., p 1 and fω(ϕ p 1(ω)) = ϕ 1(γ p ω). In order to express this in another way, we let T p = {x T p x i x j for all i j} and define ϕ j(ω) = f j 1 (ϕ(γ (j 1) ω)) for j = 2,..., p and ϕ : T p γ (j 1) by ϕ = (ϕ 1,..., ϕ p). Then we have that (2.2) f ω({ϕ 1(ω),...,ϕ p(ω)}) = {ϕ 1(γω),..., ϕ p(γω)}. In general, we call a measurable function ϕ : T p satisfying (2.2) a multivalued invariant graph. Now, it is a simple but important observation in the context of forced systems that there is another possible reason, besides periodicity, for the multivaluedness of an invariant graph. In particular, not all multivalued invariant graphs correspond to the orbit of a periodic graph. In order to understand this, it suffices to look at the simple examples given by = T 1, γ(ω) = ω + α for some α T 1 \ Q and f(ω, x) = (γω,x + α/p) with arbitrary p N. For these maps, p-valued

11 11 Figure 2.1: The devils staircase τ ρ(f α,τ) of the Arnold circle map with α = invariant graphs are given by ϕ = (ϕ 1,..., ϕ p) with ϕ j(ω) = ω + j 1. Note that the corresponding p p point set Φ = {(ω, ϕ j(ω)) ω T 1, j = 1,..., p} can be identified with T 1 by the homeomorphism h(ω, ϕ j(ω)) = ϕ j(ω) with inverse h 1 (ξ) = (pξ mod 1, ξ), and h conjugates f Φ to the irrational rotation with angle α/p. Any p-periodic one-valued subgraph ψ of ϕ would define a γ-invariant subset h(ψ) on T 1 of measure 1/p, contradicting the ergodicity of γ. This shows that ϕ cannot be decomposed in any measurable way into smaller invariant or periodic subgraphs. Accordingly, we will use the following general definition of multivalued invariant graphs. Suppose f is a γ-forced map and p, q N. Then a p, q-invariant graph is a measurable function ϕ : T 1 T pq, ω (ϕ i j(θ)) 1 i p 1 j q which satisfies (2.3) f ω({ϕ 1 1(ω),..., ϕ p q(ω)}) = {ϕ 1 1(γω),...,ϕ p q(γω)} for m-a.e. ω T 1 and has the following additional properties: (MIG1) ϕ cannot be (measurably) decomposed into smaller disjoint invariant subgraphs. (MIG2) The graphs ϕ 1,..., ϕ p are p-periodic, but cannot further be decomposed into smaller invariant or periodic subgraphs. Note that unlike in Section 1.1, we now supress the dependence for the sake of a simpler notation. As before, we denote the point set Φ := {(ω, ϕ i j(ω)) ω T 1, 1 i p, 1 j q} by the corresponding capital letter. Since p, q-invariant graphs are only defined m-a.s., we again implicitly consider them as equivalence classes and thus identify two such point sets Φ and Ψ if the set {ω Φ ω = Ψ ω} has full measure. If p = q = 1, we will sometimes say that ϕ is a simple invariant graph. An invariant graph is called continuous if it is continuous as a function T 1 T pq. Note that by this convention an invariant graph is a minimal object in the sense that it is not decomposable into smaller invariant objects. Therefore the union of two or more invariant graphs will not be called an invariant graph again. On the other hand, it is always possible to decompose an invariant set which is the graph of a n-valued function into the disjoint union of invariant graphs. Lemma 2.2. Let f be a γ-forced circle homeomorphism with ergodic base (, F, m, γ) and suppose ψ : T n satisfies f ω({ψ 1(ω),..., ψ n(ω)}) = {ψ 1(γω),...,ψ n(γω)} m-a.s. Then Ψ := graph(ψ) can be decomposed in exactly one way (modulo permutation) into k disjoint p, q-invariant graphs Φ i, i.e. Ψ = S k i=1 Φk, with kpq = n. 3 3 For the sake of completeness, we should also mention the following: Suppose a measurable set consists

12 12 Skew products with one-dimensional fibres Proof. In order to show the uniqueness of the decomposition, note that any two invariant graphs are either disjoint or equal on m-a.e. fiber. Otherwise their intersection would define an invariant subgraph. Thus if Ψ 1,..., Ψ l is another decomposition of graph(ψ) into invariant graphs, then every Ψ j must be equal to some Φ i (as it cannot be disjoint to all of them). This immediately implies the uniqueness of the decomposition. For the existence of a decomposition, observe that graph(ψ) is either an invariant graph itself or can be decomposed into subgraphs which are invariant as point sets. The same is true for any such subset, and after at most n steps this yields an invariant graph ˆϕ. This is either a 1, q-periodic invariant graph, or it contains a periodic subgraph. Again, after a finite number of steps this yields some p-periodic graph ϕ = ϕ 1 1,..., ϕ 1 q which is not further decomposable into invariant or periodic subgraphs. Now, by ergodicity of m, the intervals [ϕ 1 i (ω), ϕ 1 i+1(ω)] contain the same number of points ψ j(ω) for m-a.e. ω. Otherwise it would be possible to define an f p -invariant subgraph of ϕ 1. Thus, by setting ϕ l i(ω) := lth point of {ψ 1(ω),...,ψ n(ω)} in [ϕ 1 i (ω), ϕ 1 i+1(ω)], the required decomposition of graph(ψ) into p,q-invariant graphs can be defined. Example 2.3. Slighly extending the above example, γ-forced maps with p, q-invariant graphs for arbitrary p and q are given by torus translations: Consider f : T 2 T 2, (ω,x) (ω+α, x+ k α+ l ) q pq with α irrational, k relatively prime to q and l relatively prime to p. Then ϕ i j(ω) := k q ω + j q + i pq defines a continuous p, q-invariant graph. Note that the motion on Φ i induced by the action of f p is equivalent to a rotation by pα/q on T 1, such that again it is not possible to decompose these graphs into smaller invariant components. Similar to Section 1.2, p, q-invariant graphs can be identified with invariant ergodic measures. Suppose ϕ is a p, q-invariant graph. Then (2.4) µ ϕ(a) := 1 pq px i=1 j=1 qx m({ω T 1 : (ω, ϕ i j(ω)) A}) A F B defines an invariant ergodic measure. The aim of the next section will be to prove a partial converse to this, similar to Theorem 1.1. However, the situation for circle maps is slightly more complicated. As it can be seen from irrational translations of the torus, which have the Lebesgue measure as the unique invariant measure, not all ergodic measures of forced circle maps correspond to invariant graphs. 2.2 Furstenbergs classification of ergodic invariant measures For an orientation-preserving circle homeomorphism f, either all ergodic invariant measures are equidistributions on periodic orbits or f is uniquely ergodic and the unique invariant measure is continuous. Here, we call ν non-atomic or continuous if ν({x}) = 0 for all x X. We say x X is an atom of a measure ν on X if ν({x}) > 0. Even without invoking rotation numbers, this statement has a direct analogue in the context of forced circle homeomorphisms. This is the main result of this section. In order to state it, we say a γ-forced map with base (, F, m, γ) is m-uniquely ergodic, if M m(f) contains a unique invariant measure. Theorem 2.4 (Furstenberg, Theorem 4.1 in [2]). Let f be a γ-forced circle homeomorphism with ergodic base (, F, m,γ). Then the following holds. (i) If there exists a p, q-invariant graph ϕ, then every µ M m(f) is of the form µ = m ϕ for some some p, q-invariant graph ϕ. (ii) If there exists no invariant graph, then f is m-uniquely ergodic and the fiber measures of the unique invariant measure µ M m(f) are continuous. of n points on each fiber. One might ask whether such a set is always the graph of a n-valued measurable function. Proposition in [1] provides a positive answer to this, namely (a) If A T 1 is a measurable set with #A ω = n for m-a.e. ω, then A is the graph of a n-valued measurable function ψ. (b) Let A T 1 be a measurable set which is f-invariant and satisfies #A θ < m-a.s. Then #A ω is m-a.s. constant and there exists a multi-valued function ψ such that A ω = (graph(ψ)) ω m-a.s.

13 13 In case γ is a uniquely ergodic transfomation of a compact metric space and m is the unique invariant measure, we have that M m(f) = M(f). Hence, m-unique ergodicity is equivalent to unique ergodicity. Thus, we obtain Corollary 2.5. Suppose is a compact metric space and γ : is a uniquely ergodic homeomorphism. Then either every f-invariant ergodic measure is associated to a mulitvalued invariant graph or f is uniquely ergodic. For the proof of Theorem 2.2, we need some preliminary notions and results. Suppose f is a γ-forced map with base (, F, m,γ) and µ M m(f). Then we define the fiberwise support of µ as (2.5) S(µ) := [ ω T 1 ({ω} supp(µ ω)) and let S ω(µ) := supp(µ ω) = (S(µ)) ω. 4 Note that if is a topological space, so that supp(µ) is defined, the fiberwise support is generally not equal to supp(µ) and we only have S θ (µ) supp(µ) ω m-a.s. If µ is ergodic, we have the following simple observation. Lemma 2.6. Let f be a γ-foced circle homeomorphism with ergodic base (, F, m,γ) and suppose µ M m(f) is ergodic. Then either the fiber measures µ ω are m-a.s. continuous or µ is associated to some p, q-invariant graph ϕ. Proof. Define Λ δ := {(ω, x) µ ω{x} δ}. 5 These sets are invariant by (1.6), and so is Λ 0 := S 1 n N Λ n. Suppose µω is not m-a.s. continuous. Then µ(λ 0 ) > 0, and by ergodicity µ(λ 0 ) = 1. This means that δ := sup{δ > 0 : µ(λ δ ) = 1} > 0, as Λ δ ր Λ 0 if δ ց 0. Now let Λ δ := T n N Λ δ 1 n \ Λ δ+ 1 n. Then Λ δ is again an invariant set of measure 1 and therefore µ ω(λ δ ω ) = δ #Λ δ ω = 1 for m-a.e. ω. Consequently #Λ δ ω = n and δ = 1 n for some n N. As µ is ergodic Λ δ cannot be decomposed into invariant subsets and must therefore be the point set of some p, q-invariant graph with n = pq. 6 A skew translation over the base (, F, m, γ) is a γ-forced map T of the form T : T 1 T 1, (ω, x) (γω, x + τ(ω)) for some measurable function τ : T 1. The following result about skew translations is wellknown. The proof is again taken from [2], in slightly modified form (compare also [18]). Theorem 2.7. An skew translation T over an ergodic base (, F, m, γ) is m-uniquely ergodic if and only if m Leb T 1 is ergodic with respect to T. Proof. If T is m-uniquely ergodic, then since m Leb T 1 is obviously invariant it is the unique measure in M m(f). Suppose m Leb T 1 is not ergodic, such that there exists an f-invariant set A T 1 with m Leb T 1(A) (0, 1). By ergodicity and invariance Leb T 1(A ω) is m-a.s. constant. But then (m Leb T 1) A /m Leb T 1(A) defines another f-invariant measure in M m(f), contradicting m-unique ergodicity. Hence m Leb T 1 must be ergodic. Conversely, assume that m Leb T 1 is ergodic and suppose for a contradiction that ν M m(f) is different from m Leb T 1. Then we can choose a function g : T 1 R such that the family (x g(ω,x)) ω is equicontinuous and R T 1 g dm Leb T 1 < R T 1 g dν. In order to see that this is possible, note that since the σ-algebra B on T 1 is countably generated, there exists a set A with m(a) > 0 and an interval I T 1 such that Leb T 1(I) < ν ω(i) for all ω A. By uniformly approximating the indicator function of I with a function ḡ : T 1 R, we obtain a function g(ω,x) = 1 A(ω) ḡ(x) with the above properties. Given α T 1, let g α(ω,x) := g(ω,x+α). Due to Birkhoff s Ergodic Theorem, given a countable dense set {α n n N} T 1, there exists a set B T 1 with m Leb T 1(B) = 1 and n 1 X (2.6) lim g αn f i (ω,x) = n i=0 T 1 g dm Leb T 1 4 S(µ) is a measurable set, as its complement is S r,s Q {ω K(ω,[r, s]) = 0} [r, s], compare [1, Corollary 1.6.5]. 5 These sets are measurable as Λ δ = T n N S r Q ω µω([r, r + 1 n ]) δ [r, r + 1 n ]. 6 Compare footnotes 3 and 8.

14 14 Skew products with one-dimensional fibres for all (ω, x) B and n N. As f is a skew rotation (2.6) just means that n 1 X lim g f i (ω, x + α n n) = g dm Leb T 1, T 1 i=0 and due to the equicontinuity of (x g(ω,x)) ω, the density of {α n n N} and the fact that R is a skew rotation, this implies n 1 X lim g f i (ω,x) = g dm Leb T 1 n T 1 i=0 for all ω π 1(B) and all x T 1. However, since ν(π 1(B) T 1 ) = 1, we obtain n 1 X lim g f i (ω,x + α n n) = g dm Leb T 1 ν-a.s., T 1 i=0 contradicting the Birkhoff Ergodic Theorem for the ergodic measure ν. Now we can turn to the Proof of Theorem 2.4. (i) Suppose there exists a p, q-invariant graph ϕ = (ϕ 1,..., ϕ pq), and ν is an invariant ergodic measure. Then similar as in the proof of Lemma 2.6 it can be seen that ν ω([ϕ i(ω),ϕ i+1(ω))) = 1 m-a.s. i = 1... pq. In analogy to the proof of Theorem 1.1, pq define j r i(ω) := sup r [0, ϕ i+1(ω) ϕ i(ω)) ν ω([ϕ iω, ϕ i(ω) + r) 1 ff 2pq and ψ i(ω) := ϕ i(ω) + r i(ω). This gives a graph ψ, and (1.6) implies that the corresponding point set Ψ is invariant ν-a.s. If ψ could be decomposed into invariant subgraphs, then so could ϕ, thus ψ is a p,q-invariant graph as well. Now and thus ν ω([ϕ i(ω),ψ i(ω))) 1 2pq ν! n[ [ϕ i, ψ i) 1 2 i=1 and νω([φi(ω),ψi(ω)]) 1 2pq and ν! n[ [ϕ i, ψ i] 1 2. Using the ergodicity of ν and the invariance of these sets, one can see that they must have measure 0 and 1, respectively. Hence, their intersecction Ψ has measure ν(ψ) = 1, meaning that ν = m ψ. (ii) There always exists at least one invariant measure µ, and by the Ergodic Decomposition Theorem µ can be chosen ergodic. As there is no invariant graph, µ ω is m-a.s. continuous by Lemma 2.5. Suppose there exists another invariant ergodic measure ν. We will show that this leads to a contradiction, proceeding in two steps: First we show ν(s(µ)) = 0. Then we prove that ν(s(µ) c ) = 1 implies the existence of an invariant graph, in contrast to the assumption that there is no such graph. Note that ergodicity of ν and invariance of S(µ) imply that either ν(s(µ)) = 0 or ν(s(µ)) = 1. Step 1: ν(s(µ)) = 0. Since S ω(µ) is closed, its complement S ω(µ) c consists of at most countably many open intervals. Let A ω be the union of all the right endpoints of these intervals and set A := S ω T {ω} A ω. 7 Now let Λ be a γ-invariant set of full measure, 1 such that (1.6) holds and µ ω is continuous for all ω Λ. Let S := (S(µ) \ A) π 1 1 (Λ). As µ ω is continuous m-a.s. and ν projects to m, ν(s(µ) \ S) = 0. We will now construct an isomorphism h between f and a skew translation T which is invertible on S and maps µ to the measure m Leb T 1, that is, µ h 1 = m Leb T 1. As µ is ergodic with respect to f, so will be m Leb T 1 with respect to T. By Theorem 2.7 m Leb T 1 then is the unique T-invariant measure on T 1, and thus µ will be the only f-invariant measure on S. Define h : T 1 T 1 by i=1 (2.7) h(ω, x) := (ω, µ ω([0, x])). Then S 7 T This set is measurable, as A = S(µ) n N m N Sr Q {ω T1 µ ω((r, r +. m 1 m 1 n )) = 0} (r, r + 1 n )

15 15 h is injective on S and onto π 1 1 (Λ): If ω ω, then h(ω, x) h(ω, x ). Suppose x < x and (ω, x),(ω, x ) S. Then x, x S ω(µ) and x cannot be the right endpoint of a connected component in S ω(µ) c. Thus (x, x ) S ω(µ) and therefore π 2(h(ω,x )) π 2(h(ω, x)) = µ ω((x, x ]) > 0. The fact that h is onto π 1 1 (Λ) follows from the continuity of the fiber measures µ ω on Λ. T := h f h 1 is a skew translation: As h : S π 1 1 (Λ) is invertible, T is m Leb T 1-a.s. well-defined. Set τ(ω) := T ω(0) and let x T 1 be arbitrary. If f ω(h 1 ω (x)) [f ω(0),0] then T ω(x) = µ γω([0, f ω(h 1 ω (x))]) = Analogously, if f ω(h 1 ω (x)) [0, f ω(0)] then = µ γω([0, f ω(0)]) + µ γω([f ω(0), f ω(h 1 ω (x))]) = T ω(0) + µ ω([0, h 1 ω (x)]) = τ(ω) + x. T ω(x) = µ γω([0, f ω(h 1 ω (x))]) = = µ γω([0, f ω(0)]) µ γω([f ω(h 1 ω (x)),f ω(0)]) = T ω(0) + µ γω([f ω(0), f ω(h 1 ω (x))]) mod 1 = τ(ω) + x mod 1. µ h 1 = m Leb T 1: This follows directly from (2.7), as µ ω(h 1 ω ([0, x])) = x by definition. Thus T is an skew translation with ergodic measure m Leb T 1. Step 2: Existence of an invariant graph Suppose ν(s(µ) c ) = 1. Then define Λ >α := [ ω T 1 ν ω((r,s)) > α, µ ω((r, s)) = 0 (r, s). r,s Q (Λ >α) ω contains exactly those connected components in S ω(µ) c with measure ν ω bigger than α. Similar to the proof of Lemma 2.6, there is a unique n (0,1) such that Λ 1 n := T ε>0 (Λ 1 n ε \ Λ 1 n ε) has measure 1. Thus, (Λ 1 n ) ω contains exactly n intervals in m-a.e. fiber, and the left (or right) endpoints of these intervals constitute an invariant graph ψ. 8 This contradicts the assumption that there is no such graph in case (ii) and thus completes the proof. The isomorphism h constructed in Step 1 of (ii) deserves some more attention. To that end, given two γ-forced circle homeomorphisms f and g we say f is fiberwise semi-conjugate to g if there exists a measurable map h : T 1 T 1 with the following properties. (i) π 1 h(ω, x) = ω (ω,x) T 1 (ii) For every ω the map h ω : T 1 T 1, x π 2 h(ω, x) is continuous and order-preserving. (iii) h f = g h If the fiber maps h ω are all homeomorphisms, then f and g are called fiberwise conjugate. h is called a fiberwise semiconjugacy or fiberwise conjugacy, respectively. The proof of Theorem 2.4 now already contains Lemma 2.8. Let f be a γ-forced circle homeomorphisms with ergodic base (, F, m, γ) and suppose there exists no invariant graph for f. Then f is fiberwise semiconjugate to an m-uniquely ergodic skew translation, and the semiconjugacy h maps the unique f-invariant measure µ M m(f) to the measure m Leb T The fibred rotation number Given a γ-forced circle homeomorphism f, we call F : R R a lift of f if it satisfies π F = f π, where π(ω,x) = (ω,π(x)). Note that in this case F is a γ-forced monotone map on the real line and each fibre map F ω is a lift of the circle homeomorphism f ω. In particular, for two different lifts F and G of f and all ω, F ω and G ω differ by an integer constant n(ω). However, unless the base space is a connected topological space and f is continuous, this integer may depend on ω. In this situation, the rotation numbers of different lifts of f, which we show to exist below, are not necessarily integer translates of each other. Consequently, it does not make sense to assign a rotation number ρ(f) T 1 to f itself, even if ρ(f) it is well-defined for every fixed lift F of f. 8 Measurability of ψ follows again from standard arguments and we omit the details.

16 16 Skew products with one-dimensional fibres Theorem 2.9 (Herman). Suppose f is a γ-forced circle homeomorphism with ergodic base (, F, m, γ) and F is a lift of f. Then there exists a set with m( ) = 1 and a number ρ m(f) R, called the fibred rotation number of F, such that for all (ω,x) R we have (2.8) ρ m(f) = lim n (F n ω (x) x)/n. Further, if is a compact metric space and γ is a uniquely ergodic transformation, then (2.8) holds for all (ω,x) T 1 and the convergence is uniform. Proof. Fix an ergodic measure µ M m(f) and let ϕ(ω, x) = F ω(ˆx) ˆx, where ˆx π 1 ({x}) is arbitrary. Note that since F ω is a lift of the circle homeomorphism f ω, ϕ(ω, x) does not depend on the choice of the particular lift ˆx of x and is therefore well-defined. By Birkhoff s Ergodic Theorem, we have n 1 (2.9) lim (F ω n 1 X (x) x)/n = lim ϕ f i (ω, x) = n n n i=0 T 1 ϕ dµ =: ρ(f) µ-a.s. on T 1. In particular, as µ projects to m, there exists a set of full measure such that for each ω there is some x ω R which satisfies (2.9). Moreover, this is also true for all integer translates of x ω. Therefore, given any x R we may assume without loss of generality that x [x ω, x ω + 1). However, this implies that F n ω (x) [F n ω (x ω), F n ω (x ω) + 1) for all n N, and consequently x satisfies (2.9) as well. Hence, (2.9) holds for all (ω,x) T 1. Furthermore, we see that (2.9) also holds ν-a.s. for any other measure ν M m(f). In case γ is uniquely ergodic, this means that R T 1 ϕ dν = ρ m(f) for all f-invariant measures ν. The uniform convergence therefore follows from Theorem We now turn to the case where γ is a uniquely ergodic transformation on a compact and connected space. In this case, the above theorem asserts the existence of a unique rotation number ρ(f) for any lift F of f, and since it is easy to check that different lifts of f coincide up to an additive integer constant, the rotation number ρ(f) = ρ(f) mod 1 is well-defined. Nevertheless, there is still one decisive difference to the rotation theory of circle homeomorphisms. When f Homeo +(T 1 ) and F : R R is a lift, then F n (x) x nρ(f) 1 for all x R and n N. Hence, the convergence of the rotation number is even faster than merely uniform, with an a priori error estimate of 1 n after n iterations. In the forced situation, such a uniform bound need not exist. Given a γ-forced circle homeomorphism f with uniquely ergodic base transformation γ and lift F, we define the deviations from the constant rotation by (2.10) D(n, ω,x) := F n ω (x) x nρ(f). Providing examples where these quantities are unbounded is not an easy task. Hence, we will first restrict ourself to some general observations. Lemma 2.10 ([19]). Suppose is a compact connected metric space, γ is strictly ergodic 9 and f is a γ-forced circle homeomorphism with lift F. Then either C = sup{ D(n, ω, x) (ω,x) R, n N} <, or sup n N D(n, ω, x) = for all (ω, x) R. In the latter case, there still exist ω, ω + T 1 such that for all x R we have sup n N D(n, ω, x) < and inf n N D(n, ω +, x) >. Proof. First of all, as all maps F n ω are lifts of circle homeomorphisms, we have that for all n N and ω T 1 (2.11) D(n, ω, x 1) D(n, ω, x 2) 1 x 1, x 2 R. Therefore, boundedness of the deviations of an orbit does not depend on the x-value. Suppose sup n D(n, ω 0, x 0) c for some c > 0 and (ω 0, x 0) R. Then D(n, γ k (ω),x) 2c + 1 k N, x R. Now assume that there exist orbits with unbounded deviations and choose (ω, x) T 1 R and n N such that D(n, ω,x) > 2c + 2. By (2.11) this implies D(n, ω,y) > 2c + 1 y R. However, 9 Recall that strictly ergodic means uniquely ergodic and minimal.