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The Denjoy type-of argument for quasiperiodically forced circle diffeomorphisms

The Denjoy type-of argument for quasiperiodically forced circle di?eomorphisms
Tobias H. J¨ger and Gerhard Keller a
Mathematisches Institut, Friedrich-Alexander-Universit¨t Erlangen-N¨ rnberg, Germany a u

arXiv:math/0312023v1 [math.DS] 1 Dec 2003

MSC Classi?cation Numbers: 37C70, 37C60, 37H15

1st February 2008
Abstract We carry the argument used in the proof of the Theorem of Denjoy over to the quasiperiodically forced case. Thus we derive that if a system of quasiperiodically forced circle di?eomorphisms with bounded variation of the derivative has no invariant graphs with a certain kind of topological regularity, then the system is topologically transitive.

1

Introduction

When studying quasiperiodically forced circe maps, it is an obvious question to ask to what extent the results for unperturbed circle homeomorphisms and di?eomorphisms carry over. Invariant graphs serve as a natural analogue for ?xed or periodic points. Unfortunately, a system without invariant graphs is not necessarily conjugated to an irrational torus translation (at least not if the conjugacy is required to respect the skew product structure; a simple counterexample will be given in Section 2). However, there is a result by Furstenberg ([Fur61]), which may be considered as an anologue to the Poincar? classi?cation for circle homeomorphisms in a measuree theoretic sense: It states that for a system of quasiperiodically forced circle maps either there exists an invariant graph, then all invariant ergodic measures are associated to invariant graphs, or the system is uniquely ergodic with respect to an invariant measure which is continuous on the ?bres. In the latter case the system will be isomorphic to a skew translation of the torus by an isomorphism with some additional nice properties. We will brie?y discuss Furstenbergs results after we have introduced the concept of invariant graphs in Section 2. The (?brewise) rotation number can be de?ned in the same way as the rotation number of a circle homeomorphism, but Herman already gave examples where invariant graphs exist in combination with arbitrary rotation numbers (see [Her83]). Thus the connection between the existence of ?xed or periodic points and rational rotation numbers seems to break down in the forced case. However, this is di?erent if we require a certain amount of topological regularity of the invariant graphs, in particular when the invariant graphs are continuous. Then the structure of an invariant graph already determines the ?brewise rotation number, and this is described in Section 3. It might be considered well-known folklore, but as we knew of no apropriate reference and need a slight generalization afterwards it seemed apropriate to include some details. Section 4 contains our main result, namely Theorem 4.4. TBV denotes the class of systems where the variation of the logarithm of the derivative of the ?bre maps is integrable. The concept of a regular p, q-invariant graph and its implications for the rotation numbers are introduced in Sections 2 and 3, as mentioned. The statement we derive then reads as follows: : If T ∈ TBV is not topologically transitive, then there exists a regular p, q-invariant l graph for T . In particular, ρT depends rationally on ω, i.e. ρT = k ω + pq mod 1 for q suitable integers k and l.

1

The question whether such systems are minimal (as in the unforced case) has to be left open here. However, if there exists a minimal strict subset of the circle then some further conclusions about its structure can be drawn.

2

Invariant graphs and invariant measures

We study quasiperiodically forced circle homeomorphisms and di?eomorphisms, i.e. continuous maps of the form T : T2 → T2 , (θ, x) → (θ + ω, Tθ (x)) , (2.1) where the ?bre maps Tθ are either orientation-preserving circle homeomorphisms or orientationpreserving circle di?eomorphisms with the derivative DTθ depending continuously on (θ, x). To ensure all required lifting properties we additionally assume that T is homotopic to the identity on T2 . The classes of such systems will be denoted by Thom and Tdi? respectively. In all of the following, m will be the Lebesgue-measure on T1 , λ the Lebesgue-measure on T2 and πi : T2 → T1 (i = 1, 2) the projection to the respective coordinate. When considering ?bre n maps of iterates of T or their inverses, we use the convention Tθ := (T n )θ ?n ∈ Z. Finally, by n n T? we denote the set of all points (x1 , . . . , xn ) ∈ T with xi = xj ?i, j : 1 ≤ i < j ≤ n. Invariant graphs. Due to the aperiodicity of the forcing rotation, there cannot be any ?xed or periodic points for a system of the form (2.1). The simplest invariant objects will therefore be invariant graphs. In contrast to quasiperiodically forced monotone interval maps, where such invariant graphs are always one-valued functions, these might be multi-valued when circle maps are considered and the de?nition therefore needs a little bit more care. We will usually not distinguish between invariant graphs as functions and as point sets. This might seem a bit confusing at some times, but it is very convenient at others. De?nition 2.1 (p, q-invariant graphs) Let T ∈ Thom , p, q ∈ N. A p, q-invariant graph is a measurable function ? : T1 → Tpq , θ → ? 1≤i≤p (?i (θ))1≤j≤q with j Tθ ({?1 (θ), . . . , ?p (θ)}) = {?1 (θ + ω), . . . , ?p (θ + ω)} 1 q 1 q for m-a.e. θ ∈ T1 , which satis?es the following: (i) ? cannot be decomposed (in a measurable way) into disjoint subgraphs ?1 , . . . , ?m (m ∈ N) which also satisfy (2.2). (ii) ? can be decomposed into p p-periodic q-valued subgraphs ?1 , . . . , ?p . (iii) The subgraphs ?1 , . . . , ?p cannot further be decomposed into invariant or periodic subgraphs. If p = q = 1, ? is called a simple invariant graph. The point set Φ := {(θ, ?i (θ)) | θ ∈ T1 , 1 ≤ j i ≤ p, 1 ≤ j ≤ q} will also be called an invariant graph, but labeled with the corresponding capital letter. An invariant graph is called continuous if it is continuous as a function T1 → Tpq . ? Note that by this convention an invariant graph is a minimal object in the sense that it is not decomposable into smaller invariant parts. 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 T ∈ Thom and suppose F : T1 → Tn satis?es Tθ ({F1 (θ), . . . , Fn (θ)}) = {F1 (θ + ω), . . . , Fn (θ + ω)} m-a.s. . 2 (2.2)

Then graph(F ) can be decomposed in exactly one way (modulo permutation) into k disjoint k p, q-invariant graphs Φi , i.e. graph(F ) = i=1 Φk , with kpq = n. 1 Proof: Uniqueness: Any two invariant graphs are either disjoint or equal on m-a.e. ?bre, otherwise their intersection would de?ne an invariant subgraph. Thus if Ψ1 , . . . , Ψl is another decomposition of graph(F ) into invariant graphs, then every Ψj must be equal to some Φi (as it cannot be disjoint to all of them). This immediately implies the uniqueness of the decomposition. Existence: graph(F ) 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 subgraph and after at most n steps this yields an invariant graph ?. This is either a 1, q-periodic invariant graph, or it contains ? a periodic subgraph. Again, after a ?nite number of steps this yields some p-periodic graph ?1 = ?1 , . . . , ?1 which is not further decomposable into invariant or periodic subgraphs. W.l.o.g. 1 q we can assume that the points ?1 are ordered in T1 , i.e. ?1 (θ) < ?1 (θ) < . . . < ?1 (θ) < ?1 (θ). 1 2 q 1 j Now for m-a.e. θ ∈ T1 all the intervals [?1 (θ), ?1 (θ)] contain the same number of points i i+1 Fj (θ), otherwise it would be possible to de?ne an T p -invariant subgraph of ?1 . Thus, by setting ?l (θ) := lth point of {F1 (θ), . . . , Fn (θ)} in [?1 (θ), ?1 (θ)], the required decomposition i+1 i i of graph(F ) into p, q-invariant graphs can be de?ned. P Of course, an example of a p, 1-invariant graph can always be given by choosing a circle homeomorphism f with a periodic point of period p and taking Tθ = f ?θ ∈ T1 . Then the periodic orbit of f de?nes a constant p, 1-invariant graph. For q = 1 the situation is slightly more complicated, and there is no direct analogue for 1, q-invariant graphs in the unperturbed case. The appropriate picture in this case is rather an invariant line for a ?ow on the torus, as the following example shows. The fact that invariant graphs may correspond to these two di?erent types of dynamics, or both at once, will come up again later, namely in the proof of Thm. 4.4 . Simple examples for p, q-invariant graphs with arbitrary p and q are given by torus translal tions: Consider T : T2 → T2 , (θ, x) → (θ + ω, x + k ω + pq ) with k relatively prime to q and l q relatively prime to p. Then ?i (θ) := j i ? 1 + (j ? 1)p k θ+ q pq (1 ≤ i ≤ p, 1 ≤ j ≤ q)

de?nes a continuous p, q-invariant graph. The meaning of the numbers k and l will be explained in Section 3. Note that the motion on Φi induced by the action of T p is equivalent to an irrational ? l lk rotation on T1 (by p ω + q where l = ? mod q), such that it is not possible to decompose these q graphs into smaller invariant components. Furstenberg’s results. As mentioned before, even in the absence of invariant graphs conjugacy to an irrational torus translation cannot be expected, at least not if we want the conjugacy to respect the skew product structure: Example 2.3 A skew translation of the torus is a map T : T2 → T2 , (θ, x) → (θ + ω, x + g(θ) mod 1) with some measurable function g : T1 → R. When g is continuous, a ?brewise rotation number ρT can be assigned to T (see Def. 3.1), and ρT will be equal to τ := T1 g(θ) dθ. As conjugacy
1 For the sake of completeness, we should also mention the following: Suppose a measurable set consists of n points on each ?bre. One might ask whether such a set is always the graph of a n-valued measurable function. Prop. 1.6.3 in [Arn98] provides a positive answer to this, namely

(a) If A ? T2 is a measurable set with #Aθ = n ?θ ∈ T1 , then A is the graph of a n-valued measurable function F . (b) Let A ? T2 be a measurable set which is T -invariant and satis?es #Aθ < ∞ m-a.s. . Then #Aθ is m-a.s. constant and there exists a multi-valued function F such that Aθ = (graph(F ))θ m-a.s. .

3

preserves the rotation numbers, T can only be conjugated to the irrational torus translation R : (θ, x) → (θ + ω, x + τ ). l Now suppose τ ∈ Q := { k ω + p mod 1 | k, q, l, p ∈ Z, p, q = 0}. Then R is uniquely ergodic, / q and if T and R are conjugated (via a conjugacy h) so will be T . Thus h must preserve the Lebesgue measure, which is invariant under both transformations. If we require the conjugacy to respect the skew product structure of the system, i.e. π1 ? h(θ, x) = θ ?(θ, x) ∈ T2 , then all ?bre maps hθ must also preserve Lebesgue measure and therefore be rotations. Hence h will be a skew translation as well, i.e. h(θ, x) = (θ, x + ?(θ) mod 1) for some continuous function ? : T1 → R. As R = h?1 ? T ? h this function ? is a continuous solution of the cohomology equation g(θ) ? τ = ?(θ + ω) ? ?(θ) . (2.3) Conversely, if a function g has measurable, but no continuous solutions of (2.3), then on the one hand the resulting system T cannot be conjugated to an irrational torus translation. On the other hand T will still not have any invariant graphs, because a (non-continuous) solution ? of (2.3) still allows to de?ne an isomorphism h between T and R as above. Thus T will still be uniquely ergodic with respect to the Lebesgue measure on T2 . An example of such a function g is explicitly constructed in [KH97] (Section 12.6(b)). This means that a possible analogue to the Poincar? classi?cation for circle homeomorphisms e must be somewhat weaker in nature. It turned out that the right perspective is to look at the invariant ergodic measures (see [Fur61]). First note that to any invariant graph an invariant ergodic measure can be assigned: De?nition 2.4 (Associated measure) Let ? be a p, q-invariant graph. Then 1 ?? (A) := pq
p q

m({θ ∈ T1 : (θ, ?i (θ)) ∈ A}) ?A ∈ B(T2 ) j
i=1 j=1

(2.4)

de?nes an invariant ergodic measure. If ? = ?? for some p, q-invariant graph ?, then ? and ? are called associated to each other. In order to study general invariant ergodic measures, it is useful to look at the so-called ?bre measures. Any invariant measure ? can be disintegrated in the way ? = m × K where K is a stochastic kernel from T1 to T1 , such that ?(A) = T1 K(θ, Aθ ) dθ ?A ∈ B(T2 ). The measures ?θ := K(θ, .) on T1 are the ?bre measures of ?. The ?bre measures are mapped to each other by the action of T , i.e. ?1 ?θ+ω = ?θ ? Tθ+ω for m-a.e. θ ∈ T1 . (2.5) All this is throughouly discussed for general random dynamical systems in Chapter 1.4 in [Arn98]. Note that an invariant ergodic measure ? is associated to some invariant graph if and only if its ?bre measures are m-a.s. point measures. Otherwise the ?bre measures are continuous. The following result of Furstenberg ([Fur61], Thm. 4.2) now provides a partial converse of Def. 2.4 and a substitute for the Poincar? classi?cation in the unforced case. Furstenberg used e quite di?erent terminology, but his result can be stated in the following way: Theorem 2.5 Let T ∈ Thom . Then one of the following is true: (i) There exists a p, q-invariant graph ?. Then every invariant ergodic measure is associated to some p, q-invariant graph. (ii) There exists no invariant graph. Then T is uniquely ergodic and the ?bre measures of the unique invariant measure ? are continuous.

4

Indeed the original result is even slightly more general: Furstenberg only assumes that the base of the skew product is uniquely ergodic, not necessarily an irrational rotation. The main ingredient in the proof is the construction of an isomorphism h between the original system and a skew translation of the torus. This isomorphism has some additional properties, which we describe by the following De?nition 2.6 (Fibrewise conjugacy) Let T, S ∈ Thom . T is said to be ?brewise semi-conjugated to S, if there exists a measurable map h : T2 → T2 with the following properties: (i) π1 ? h(θ, x) = θ ?(θ, x) ∈ T2 (ii) For every θ ∈ T1 the map hθ : T1 → T1 , x → π2 ? h(θ, x) is continuous and orderpreserving. (iii) h ? T = S ? h If the ?bre maps hθ are all homeomorphisms, then T and S are called ?brewise conjugated. h is called a ?brewise semi-conjugacy or ?brewise conjugacy, respectively. Now Furstenbergs proof of Thm. 2.5 already contains Lemma 2.7 Let T ∈ Thom and suppose there exists no invariant graph for T . Then T is ?brewise semiconjugated to an uniquely ergodic skew translation of the torus, and the semi-conjugacy h maps the unique T -invariant measure ? to the Lebesque-measure on T2 . There is an interesting consequence of Thm. 2.5 concerning Lyapunov exponents, which are n de?ned pointwise by λ(θ, x) := limn→∞ log DTθ (x). If the system is uniquely ergodic all these 2 limits exist and the convergence is uniform on T . As no iterate of T can be uniformly expanding or contracting, all Lyapunov exponents must be zero in the uniquely ergodic case. Conversely, the existence of points with non-existent or non-zero Lyapunov exponents implies that the system is not uniquely ergodic and therefore has invariant graphs by Thm. 2.5 .

3

Rotation numbers and regular invariant graphs

Throughout this and the next section, we will repeatedly work with lifts of di?erent objects on T2 to either T1 × R or R2 (both of which are covering spaces of T2 ). Therefore, we have to make some conventions regarding notation and terminology: ? Projections: π will denote the natural projection either from R2 → T2 , T1 × R → T2 or R → T1 . If we want to project to one coordinate and in addition project this coordinate onto the circle (if it is not in T1 anyway) we will denote this by π1 and π2 . In those cases where we want to project to an R-valued coordinate we will use π1 or π2 , respectively. R? ? ? valued variables will be denoted by θ, x, y , . . . . The only exception is the rotation number ω on the base, which we will always identify with its unique lift in [0, 1). ? A lift of a map T ∈ Thom to either T1 × R or R2 will usually be denoted by T (i.e. T ? π = π ? T ), for a second lift of the same map we will use T . It is easy to see that T = T + (l, m) (3.1)

for some l, m ∈ Z whenever T , T are two lifts of the same map T , and that a lift is uniquely ? ? determined by choosing its value T (θ0 , x0 ) = (θ1 , x1 ) at one point (θ0 , x0 ). Fibrewise rotation numbers. The ?brewise rotation number of a quasiperiodically forced circle homeomophism can be de?ned exactly in the same way as the rotation number in the unforced case: 5

Theorem and De?nition 3.1 (Fibrewise rotation number) Let T ∈ Thom and T : T1 × R → T1 × R be a lift of T . Then the limit ρT := lim
n→∞

1 n (T (x) ? x) mod 1 n θ

(3.2)

is independent of θ, x and the choice of the lift T : T1 × R → T1 × R. It is called the ?brewise rotation number of T . Further more, ρT = lim ?
n→∞

1 n

T1

n Tθ (0) dθ

(3.3)

and the convergence in (3.2) is uniform on T1 × R. This result is due to Herman ([Her83]), a very nice and more elementary proof can be found in [SFGP02]. As we will see, the existence of a continuous invariant graph implies that the ?brewise rotation number depends rationally on the rotation number on the base, i.e. ρT = l k ω+ mod 1 q n

holds for some k, q, l, n ∈ Z. ρT is called rationally independent of ω, if it is not rationally dependent. Dynamics of continuous invariant graphs. In order to describe the structure of continuous invariant graphs, we need the following concept: Suppose a continuous function γ : R → R satis?es γ(θ + q) ? γ(θ) = k and γ(θ + l) ? γ(θ) ∈ Z ?θ ∈ R, 1 ≤ l < q / (3.4) for some q ∈ N and k ∈ Z. We can then project it down to T2 to obtain a q-valued graph γ(θ) = (γ1 (θ), . . . , γq (θ)), where γi (θ) = π(γ(θ + i ? 1)) (θ ∈ [0, 1)),

which wraps around the circle q times in the θ-direction and never intersects itself. In this situation we will call both γ and the corresponding point set Γ := {(θ, γi (θ)) | θ ∈ T1 , 1 ≤ i ≤ q} a q-curve, a lift of γ will always be a function γ : R → R as in (3.4). If γ is one-valued, it will be called a simple curve. Such a q-curve γ has the property that it is a continuous function T1 → Tq and consists of ? one connected component only (as a point set). Conversely, it follows from the usual arguments for the existence of lifts that any graph γ : T1 → Tq with these properties is a q-curve, as it ? can be lifted to a function γ : R → R which satis?es (3.4) and projects down to γ. Such a lift is uniquely determined by choosing the value at one point, and by γx we will denote the lift of γ which satis?es γx (0) = x (with x ∈ π ?1 (γ(0))). Suppose ? is a continuous 1, q-invariant graph. Then Φ consists of one connected component only (otherwise the connected components would be permuted, and Φ could therefore be decomposed into periodic subgraphs). It follows from the comments made above that ? is a q-curve. Similarly, any continuous p, q-invariant graph ? is a collection of p disjoint q-curves ?1 , . . . , ?p . We now want to describe, how the structure and dynamics of an invariant graph determine the rotation number. To that end, we need to de?ne two characteristic numbers which can be assigned to an invariant graph, the winding and the jumping number. De?nition 3.2 (Winding number) Let γ be a q-curve and k ∈ Z, such that γ(θ + q) ? γ(θ) = k (see (3.4)). Then k is called the winding number of γ. In other words, the winding number is the number of times γ winds around the circle in the x-direction before it closes. Some simple observations about the winding number are collected in the following: 6

Lemma 3.3 Let γ be a q-curve with winding number k. (a) If q > 1 then k = 0. (b) q and k are relatively prime. (c) If ζ is another q-curve which does not intersect γ, then ζ has the same winding number as γ. Proof: If γ is a lift of γ, then γ(θ + q) ? γ(θ) = k ?θ ∈ R. Thus, for any j ∈ N we have 1 q
q

γ(θ + j) ? γ(θ) dθ =
0

=

j q

q

γ(θ + 1) ? γ(θ) dθ =
0

j q2

q

γ(θ + q) ? γ(θ) dθ =
0

jk q

(3.5)

(a) Suppose q > 1 and k = 0. Taking j = 1 in (3.5) the Mean Value Theorem yields the existence of at least one θ ∈ [0, q) with γ(θ + 1) = γ(θ), a contradiction to the fact that γ does not intersect itself (see (3.4)). (b) Suppose q and k are not relatively prime and take j < q, such that
jk q

∈ Z. Then, as
jk q

above, there is at least one θ ∈ [0, q) which satis?es γ(θ + j) ? γ(θ) = contradicting (3.4).

∈ Z, again

? (c) Suppose ζ has winding number k < k and take lifts γ of γ and ζ of ζ with ζ(0) ∈ ? [γ(0), γ(0) + 1). But then ζ(q) = ζ(0) + k < γ(0) + k = γ(q), and the two curves have to ? intersect somewhere in between. The case k > k is treated analogously. P In order to de?ne the jumping number, suppose T ∈ Thom has a continuous p, q-invariant graph ? = (?1 , . . . , ?p ), consisting of p p-periodic q-curves ?1 , . . . , ?p . Let x1 , . . . , xp ∈ [0, 1) be lifts 1 q 1 q of ?1 (0), . . . , ?p (0). W.l.o.g. we can assume that these points are ordered in the way 1 q x1 < x2 < . . . < xp < x1 < . . . < xp < . . . < xp . 1 1 2 q 1 2 (Note that any of the intervals [?1 (0), ?1 (0)] must contain the same number of points from j j+1 the other graphs ?i .) To each of the points xi we can assign a lift ?i := ?i i of the q-curve ?i , j j x
j

and there is exactly one lift T of T which maps ?1 to one of the other lifts ?i (see Fig. 1 for an 1 j example). De?nition 3.4 (Jumping number) Let 0 ≤ m ≤ p ? 1 and 0 ≤ n ≤ q ? 1 be such, that there is a lift T of T for which T (θ, ?1 (θ)) = (θ + ω, ?1+m (θ + ω)) . 1 1+n Then l := m + np is called the jumping number of ? (with respect to T ). Remark 3.5 If l = m + np is the jumping number of a p, q-invariant graph ?, then m and p are relatively prime, which is equivalent to l and p being relatively prime. Otherwise p | mp′ holds for some ′ p′ ∈ {1, . . . , p ? 1}. But then the union of Φ1 , Φ1+m , . . . , Φ1+(p ?1)m is invariant, contradicting the minimality of Φ.

7

Lift T → R

2

2

T ¨ ¨¨ ¨¨ ¨¨ ¨ q ? ¨ ¨¨ ¨¨ ¨¨ ¨ ¨¨ ¨¨ ¨¨ l=1 ¨ ¨¨ ¨¨ ¨¨ ¨ E
l=3 θ θ+ω

?2 2

q P ¨¨ ¨¨ ¨¨ ¨¨ ¨ ?2 ¨¨ ¨¨ ?1 ¨ ¨ ¨¨ ¨

q ?

?1 2 ?2 1 ?1 1

(i, j = 1, 2). 1 This is a 2, 2-invariant graph for both (θ, x) → (θ + ω, x + + and (θ, x) → (θ + ω, x + 2 ω + 3 ), 4 but the jumping number is di?erent in the two cases (l = 1 and l = 3 respectively). The di?erent lifts mentioned before Def. 3.4 are shown on the right.
1 ω 2 1 ) 4

Figure 1: Depicted on the left is the graph ? = (?1 , ?2 ) with ?i (θ) = 1 θ + j 2

i?1+2(j?1) 4

Proposition 3.6 Suppose T ∈ Thom and ? is a continuous p, q-invariant graph with winding number k and jumping number l. Then l k mod 1 . (3.6) ρT = ω + q pq Conversely, the numbers p, q, k and l associated to a continuous invariant graph in the above way are uniquely determined by the rotation numbers ω and ρT . Proof: Obviously, instead of working with a lift to T1 × R we can also work with a lift to R2 . By Def./ Thm. 3.1 we can use any such lift, any initial point (θ, x) and any sequence of iterates to determine the ?brewise rotation number. Let therefore ?1 , . . . , ?p be as in Def. 3.4 and choose 1 q the lift T : R2 → R2 , such that T (0, ?1 (0)) = (ω, ?1+m (ω)). After pq iterates, we will return to 1 1+n a copy of ?1 which is vertically translated exactly by l, i.e. 1
pq Tθ (?1 (0)) = ?1 (pqω) + l . 1 1

Thus we get ρT = lim
n→∞

1 l ?1 (npqω) ? ?1 (0) + . 1 npq 1 pq

By using the continuity of ?1 and (3.4) we can replace the irrational translation with integer 1 translation, i.e. lim ω 1 k 1 1 1 ?1 (npqω) ? ?1 (0) = lim ?1 (xω) ? ?1 (0) = lim (?1 (jq) ? ?1 (0)) = ω . 1 1 1 1 j→∞ jq x→∞ x npq q

n→∞

l Putting all this together yields ρT = k ω + pq . q It remains to show, that p, q, k and l are uniqely determined by ω and ρT . To that end, suppose

l′ l k′ k ω+ = ′ω+ ′ ′ , q pq q pq and the numbers on both sides belong to continuous invariant graphs. Then by Lemma 3.3 k and q as well as k ′ and q ′ must be relatively prime and therefore k = k ′ and q = q ′ . The same argument then applies to l, p and l′ , p′ (see Rem. 3.5). P 8

Regular invariant graphs. So far we have seen that the existence of continuous invariant graphs allows to draw conclusions about the ?brewise rotation number. This stays true if the assumption of continuity is weakened to some extent, namely for the class of regular invariant graphs which we will introduce in this section. This concept will then turn out to be vitally important for both the formulation and the proof of Thm. 4.4 . The regularity we require is, that the invariant graphs are boundary lines of certain compact invariant sets. This is hardly surprising, considering that such boundary lines, which are necessarily semi-continuous functions, also play a prominent role in the study of quasiperiodically forced interval maps (see e.g. [Kel96, Sta03]). However, instead of working with the compact invariant sets it will be more convenient for our purposes to use their complements. De?nition 3.7 (p, q-invariant strips and p, q-invariant open tubes) Let T ∈ Thom . (a) A compact invariant set A is called a 1, q-invariant strip if it consists of q disjoint closed and nonempty intervals on every ?bre and only has one connected component. When A1 is a 1, q-invariant strip for T p and the sets Ai := T i?1 (A1 ) (i = 2, . . . , p) are pairwise p disjoint, then A := i=1 Ai is called a p, q-invariant strip. (b) Similarly, an open invariant set U is called a 1, q-invariant open tube, if it consists of q disjoint open and nonempty intervals on every ?bre and only has one connected component. When When U 1 is a 1, q-invariant open tube for T p and the sets U i := p T i?1 (U 1 ) (i = 2, . . . , p) are pairwise disjoint, then U := i=1 U i is called a p, q-invariant strip. Of course a continuous invariant graph is a special case of an invariant strip. By some elementary topological arguments one can also see that the above de?nitions are indeed complementary: Remark 3.8 (a) The complement of a p, q-invariant strip is always a p, q-invariant open tube and vice versa. (b) If U = p U i is a p, q-invariant open tube, then any of the connected components i=1 i U i contains a q-curve γ i such that each of the q intervals of Uθ contains exactly one of i i 1 p the points γ1 (θ), . . . , γq (θ). As the q-curves γ , . . . , γ cannot intersect, they all have the same winding number (c) A winding and a jumping number can be assigned to invariant open tubes in almost exactly the same way as to continuous invariant graphs: The winding number of a q, pinvariant open tube U is that of the q-curves it contains, and the jumping number can be de?ned compleatly analogous to Def. 3.4 by looking at the di?erent lifts of the U i instead of those of the ?i . Lemma 3.9 Let T ∈ Thom and suppose there exists a p, q-invariant open tube with winding number k and jumping number l. Then l k mod 1 . ρT = ω + q pq Conversely, the numbers p, q, k and l are uniquely determined by ω and ρT . Proof: 1 The proof is almost the same as for Prop. 3.6. Instead of the lift ?1 of ?1 we use a lift γ1 of the 1 1 q-curve γ 1 . Then Ui can be lifted around γ1 in a unique way, and although γ is not invariant the properties of Ui still guarantee
npq 1 1 Tθ (γ1 (0)) ? γ1 (npqω) ? nl < 1 .

9

From that the result follows easily by repeating the proof for continuous invariant graphs, 1 replacing ?1 with γ1 . 1 P Invariant strips and tubes are of course bounded by invariant graphs, which are semi-continuous in a sense (although semi-continuity is hard to de?ne for T1 -valued functions). These invariant graphs could be de?ned by just taking the right (or left) endpoints of the intervals on the ?bres as values, but with the q-curves contained in an invariant open tube this can be made a little bit more precise: De?nition 3.10 (Regular invariant graphs) ? : T1 → Tpq is called a regular p, q-invariant graph if it is the boundary line of a p, q-invariant ? open tube, i.e. i i c ?i (θ) = γj (θ) + inf{x ∈ [0, 1) | γj (θ) + x ∈ Uθ } j or
i i c ?i (θ) = γj (θ) ? sup{x ∈ [0, 1) | γj (θ) ? x ∈ Uθ } , j

where U =

pq i=1

U i and the γ i are q-curves contained in U i .

4

The Denjoy argument

The aim of this section is to carry over the distortion argument used in the proof of the Theorem of Denjoy to the quasiperiodically forced case. Denjoy’s Theorem states, that a circle di?eomorphism with irrational rotation number and bounded variation of the derivative is not only semi-conjugated but conjugated to the corresponding rotation. As a simple consequence of this the system will be minimal. However, the crucial step in the proof is to exclude the possibility of wandering open sets, and the rest then follows quite easily from the fact that a semi-conjugacy to the irrational rotation is already established. This is di?erent in the quasiperiodically forced case, where the absence of invariant graphs does not imply semi-conjugacy to an irrational rotation. Thus the Denjoy argument generalized to the forced case will only yield that there are no wandering open sets. A few simple arguments can then be used to establish transitivity (see Thm. 4.4), but whether such systems are minimal has to be left open here. There are two other problems which make the quasiperiodically forced case more complicated: On the one hand, in the one-dimensional case the semi-conjugacy result provides all the combinatorics which are necessary. Without such a result in the forced case this has to be done “bare hands”, and thus a signi?cant part of the proof of Thm. 4.4 will consist of dealing with the possible combinatorics of wandering open sets. Another fact was already mentioned in Section 2: An invariant graph may correspond to two di?erent dynamical situations. Thus the distortion argument will have to be applied twice, ?rst in the proof of Lemma 4.3 and then in the proof of Thm. 4.4, each time dealing with one of the two possibilities. To get a better picture of this, one should recall Denjoy’s examples of di?eomorphisms which are not conjugate to the corresponding irrational rotation. The construction of these starts with an orbit of the irrational rotation, which is then “blown up” to yield wandering intervals. Now, in the quasiperiodically forced case there are two objects which might be used for such a construction. One is just the orbit of a constant line, similar to the one-dimensional case, but instead one could also use an in?nite invariant line for a ?ow on the torus. If the ?bre maps Tθ of T ∈ Thom are di?erentiable for m-a.e. θ, let V (T ) :=
T1

Vθ dθ ,
n

where Vθ denotes the variation of log DTθ , i.e. Vθ = sup{ i=1 | log DTθ (xi ) ? log DTθ (xi?1 )| | 0 ≤ x1 ≤ . . . ≤ xn ≤ 0}. By TBV we will denote the set of all such T with V (T ) < ∞. 10

Bounded variation usually implies bounded distortion of the derivative (Compare Lemma 12.1.3 in [KH97] or Corally 2 to Lemma 2.1 in [dMvS93]). In the forced case, however, any such statement must always be an integrated version of the one-dimensional. To obtain ?brewise bounds will hardly be possible, as an equal amount of distortion might be picked up from a di?erent ?bre in each iteration step. For the formulation of the lemma below we will also consider graphs ?, ψ : I → T1 which are de?ned only on a subinterval I ? T1 and use the notation [?, ψ] := {(θ, x) ∈ T2 | θ ∈ I, x ∈ [?(θ), ψ(θ)]} . Then we have Lemma 4.1 Let T ∈ TBV , I ? T1 an interval and ?, ψ : I → T1 be such that [?, ψ], T ([?, ψ]), . . . , T n?1 ([?, ψ]) are pairwise disjoint. Then ?s > 0
n DTθ (ψ(θ)) n DTθ (?(θ)) s

dθ ≥ |I|e?

sV (T ) |I|

.

I

Proof:
n DTθ (ψ(θ)) n DTθ (?(θ)) s

log
I

·

1 dθ |I|

Jensen



log
I

n DTθ (ψ(θ)) n DTθ (?(θ))

s

·

1 dθ = |I|

= =

s |I| s |I|

n?1 i i log DTθ+iω (Tθ (ψ(θ)) ? log DTθ+iω (Tθ (?(θ)) dθ = I i=0 i i log DTθ (Tθ?iω (ψ(θ ? iω)) ? log DTθ (Tθ?iω (?(θ ? iω)) dθ ≥ ? 0≤i<n θ?iω∈I ≥?Vθ

T1

s V (T ) |I|

P In order to apply this lemma, we will now look at the combinatorics of wandering boxes. To that end, some notation must be introduced (see also Fig. 2): ? In all of the following we will assume that W = I ×K is a wandering set, i.e. T n (W )∩W = ? ?n ∈ N, where I, K ? T1 are open intervals and |I| < 1 . For the sake of simplicity we 2 will assume that I is centered around 0. Iα will denote the symmetric middle part of I with length α|I|, i.e. Iα := (? α|I| , α|I| ). 2 2 ? n1 , . . . , nk ∈ N are called comparable over an interval J, if J ? ? If k ≥ 3, then n1 ? . . . ? nk ? n1 over J means that n1 , . . . , nk are comparable over J and ?θ ∈ J ?xi ∈ (T ni W )θ : x1 < . . . < xk < x1 (if this last statement is satis?ed for some θ ∈ J and xi ∈ (T n W )θ , then it is true for all). We will usually omit the last ?n1 and only write n1 ? . . . ? nk . ? Let n1 and n2 be comparable over J. Then (n1 , n2 )J := {(θ, x) ∈ T2 | θ ∈ J and ?x1 ∈ (T n1 W )θ , x2 ∈ (T n2 W )θ : x1 ≤ x ≤ x2 } .
k i=1

I + ni ω .

11

? The set of return times (with respect to Iα ) is de?ned as N (α) := {n ∈ Z | n comparable over Iα } = {n ∈ Z | |nω mod 1| ≤ ? n ∈ N (α) is called a closest return time (with respect to Iα ), if ?n ? 0 ? n over Iθ and ?k ∈ N (α) \ {0} : |k| < |n|, 0 ? k ? n over Iα or n ? 0 ? ?n over Iθ and ?k ∈ N (α) \ {0} : |k| < |n|, n ? k ? 0 over Iα . Note that it is usually not possible to use the “ordering” ? just in a formal way, as the interval with respect to which it is used always has to be taken into account. Therefore, a little bit of care has to be taken and in the following remark a few simple facts are collected. They might all seem trivial, but as they are used frequently in di?erent combinations later on this should help to avoid confusion. 1?α |I|} . 2

Tn(W) (0,n) I’ W

T?n(W)

I

I’

Figure 2: Here n is comparable over I ′ = I 1 and ?n ? 0 ? n. 3 Remark 4.2 (i) If J is symmetric in I (i.e. J = Iα for some α) and n, k are comparable over J, then either k is comparable over J + nω (this is the case when nω mod 1 and kω mod 1 are in the same half of I), or k is comparable over J ? nω and n + k is comparable over J (when nω mod 1 and kω mod 1 are in opposite halfs of I they “cancel each other out”). Thus 0, n, k and n + k will always be comparable over one of the two intervals J or J + nω. (ii) n1 ? . . . ? nk over J ? n1 + i ? . . . ? nk + i over J + iw ?i ∈ Z. (iii) n1 ? n2 ? n3 over J and n1 ? n3 ? n4 over J ? n1 ? n2 ? n3 ? n4 over J. (iv) Let n1 , . . . , nk be comparable both over J1 and J2 . Then n1 ? . . . ? nk over J1 ? n1 ? . . . ? nk over J2 . (v) ?n ? 0 ? k ? n over Iα ? ?n ? ?k ? 0 ? n over Iα . In particular, if n is a closest return time then so is ?n. Apart from the last all these statements should be obvious, and (v) can be seen as follows: Due 12

to the symmetry of Iα , k is either comparable over Iα + nω or over Iα ? nω (compare (i)). Suppose k is comparable over Iα + nω. Then by (ii) ?n ? k ? n ? 0 over Iα and this can be extended to ?n ? k ? n ? 0 ? k over Iα by (iii). Thus k ? n ? 0 ? k over Iα + kω by (iv) and applying (ii) and (iii) for a second time yields ?n ? ?k ? 0 ? n over Iα . The case where k is comparable over Iα ? nω is treated similarly: Then ?n ? k ? n ? 0 ? k over Iα ? nω by (ii) and (iii) and therefore k ? n ? 0 ? k over Iα + kω by (iv) as before. With these notions it is now possible to give a lower bound for the size of the images of W at closest return times: Lemma 4.3 Let T ∈ TBV and suppose n ∈ N is a closest return time with respect to Iα , w.l.o.g. ?n ? 0 ? n over Iα , β := min{α, 1?α }. Then 2 (i) T k (0, n)Iβ ∩ (0, n)Iβ = ? ?k : |k| < |n| (ii) λ(T n W ∪ T ?n W ) ≥ |K| · |Iβ | · e
? V (T ) 2|I |
β

.

In particular, there can only be ?nitely many closest returns. Proof: (i) Suppose for a contradiction that T k (0, n)Iβ ∩ (0, n)Iβ = ?. Of course, this can only be if Iβ ∩ (Iβ + kω) = ?, i.e. |kω mod 1| < β, and this implies k ∈ N (α). Further, there are in principle three possibilities: 0 ? k ? n or 0 ? n + k ? n or k ? 0 ? n ? n + k. Of course we have to specify with respect to which intervals these numbers can be compared, in particular in the last case which has to be split up once more (according to Rem. 4.2(i)). Case 1: 0 ? k ? n over Iα . Case 2: 0 ? n + k ? n over Iα + nω. But then by Rem. 4.2(ii) ?n ? k ? 0 over Iα . Case 3(a): k ? 0 ? n ? n + k over Iα . Then 0 ? ?k ? n ? k ? n over Iα ? kω and thus also 0 ? ?k ? n over Iα (Rem. 4.2(ii) and (iv) respectively). Case 3(b): k ? 0 ? n ? n + k over Iα + kω. Then 0 ? ?k ? n ? k ? n over Iα (Rem. 4.2(ii)). In all cases this contradicts the fact that n (and thus ?n by Rem. 4.2(v)) is a closest return time. (ii) Note that Iβ + nω ? I. Thus λ(T n W ∪ T ?n W ) =
K I ?n n DTθ (x) + DTθ (x) dθdx ≥ ?n n DTθ (x) · DTθ+nω (x)
1 2


K Iβ

?n n DTθ (x) + DTθ+nω (x) dθdx ≥ n DTθ (x) ?n n DTθ (Tθ+nω (x))
1 2

dθdx =
? V (T ) 2|I |
β

K



=
K Iβ

dθdx

Lem. 4.1



|Iβ |e
K

? V (T ) 2|I |
β

dx = |K||Iβ |e

.

As W is wandering we have that must be ?nite.

n∈Z

λ(T n W ) ≤ 1, and therefore the number of closest returns P

This lemma will turn out to be crucial in the proof of the next theorem, which is the main result of this section. Theorem 4.4 If T ∈ TBV is not topologically transitive, then there exists a regular p, q-invariant graph for T . In particular, ρT depends rationally on ω.

13

Proof: The proof consists of two steps. The ?rst is to show that the existence of a wandering open set implies the existence of a regular invariant graph. The second will then be to show, that the non-existence of wandering sets already implies the transitivity of the system. For this, we will use the fact that no iterate of T can have wandering open sets in the absence of regular invariant graphs either, as all these iterates are also in TBV and therefore Step 1 applies to them as well. Step 1: By Lemma 4.3 there can only be ?nitely many closest return times with respect to I 1 . 2 Let p be the maximum of these, w.l.o.g. ?p ? 0 ? p over I 1 . As a consequence of Rem. 4.2(v) 2 1 we have ?n ∈ Z \ {0} : ?p ? n ? p over I 2 , otherwise the minimum of such n > 0 would be a closest return time greater than p. Of course T p (0, p)I 1 ∩ (0, p)I 1 = ?, but as in the proof of Lemma 4.3(i) we can conclude that afterwards
2 2 2 2

T np (0, p)I 1 ∩ (0, p)I 1 = ?

?n ≥ 2 .

(4.1)

It follows that n∈Z T np (0, p)I 1 is an T p -invariant open set and looks like a “tube” which winds 2 around the circle in?nitely many times in the θ-direction. To make this more precise, we will construct a in?nite invariant line γ inside of this set which never intersects itself. Let ω := pω mod 1. For the sake of simplicity we assume that ω is close to 0 from the right, ? ? 1 e.g. ω ∈ [0, 4 ). The open set (0, p)I 1 ∪ T p (0, p)I 1 is connected, thus there exists a continuous ?
p ω function γ0 : [0, ω] → T1 , such that (0, γ0 (0)) ∈ (0, p)I 1 , γ0 (? ) = T0 (γ0 (0)) and (θ, γ0 (θ)) ∈ ?
2 2 2

(0, p)I 1 ∪ T p (0, p)I 1 ?θ ∈ [0, ω]. Let γ0 : [0, ω] → R be a lift of γ0 , T : R2 → R2 a lift of T p . ? ? 2 2 Then by n γ(θ) := Tθ?n? (γ0 (θ ? n? )) if θ ∈ [n? , (n + 1)? ) ω ω ω ω a continuous T -invariant curve γ : R → R can be de?ned. γ projects down to an in?nite T p invariant curve γ : R → T1 , and the point set Γ := {π(θ, γ(θ)) | θ ∈ R} ? T2 lies inside of ∪n∈Z T np (0, p)I 1 . Further, (4.1) implies that γ does never intersect itself. γ can be strobed at θ = 0 by setting xn := γ(n) ?n ∈ Z. Let Λ := {xn | n ∈ Z}. Then the map Λ → Λ, xn → xn+1 is order-preserving and bijective, but this means that the sequence (xn )n∈N is combinatorically equivalent to the orbit of a circle homeomorphism. In addition, to each n ∈ Z there exists a unique number a(n) ∈ Z which satis?es (0, xn ) ∈ T a(n)p (0, p)J and a(n) ∈ A := {a ∈ Z | |a? ω
2 2

mod 1| ≤

ω ? }. 2

(0, xn ) may be contained in several successive iterates of (0, p)I 1 , but the second requirement
1 ? ? makes the choice of a(n) unique (note that |a? mod 1| = 2 ω is not possible since ω is irrational). ω If we keep in mind the way in which the iterates of (0, p)I 1 are attached to and moving along γ, 2 it is obvious that the map a : Z → A de?ned in this way is bijective, monotone and symmetric (i.e. a(?n) = ?a(n)).

Similar as before we will call n ∈ Z a closest intersection time (in order to distinguish it from the closest return times), whenever x?n < x0 < xn and ?k ∈ Z \ {0} : |k| < |n|, x?n < xk < xn or xn < x0 < x?n and ?k ∈ Z \ {0} : |k| < |n|, xn < xk < x?n . Again we want to apply Lemma 4.3 to conclude that there can only be ?nitely many such n. To that end, consider the rectangle W ′ = I ′ × K, where I ′ is the symmetric middle part of 1 3 ? ω I with lenght exactly 2 ω. Note that I ′ ? I 1 , because |? | = |pω mod 1| ≤ 4 |I| as p ∈ N ( 1 ). W ′ 2 2 is a wandering rectangle for T p , and we now want to look at the closest return times of W ′ with respect to I ′1 . However, we have already described them quite neatly: First of all, the set of return times N ′ ( 1 ) is exactly the set A de?ned above. As T a(n) W ′ is contained in T a(n) (0, p)I 1 , 3
2 3

14

the ordering of these sets over I ′ ( 1 ) coincide and are determined by the ordering of the points 3 xn ∈ Λ. Thus k ∈ Z is a closest return time (with respect to W ′ and I ′1 ) if and only if k = a(n) 3 for some closest intersection time n. As there cannot be in?nitely many closest return times by Lemma 4.3, the same is true for the closest intersection times. Let q be the maximum of the closest intersection times. First assume q = 1 and (w.l.o.g.) x?1 < x0 < x1 . Then (xn )n∈N is strictly monotonically increasing in the sense that x0 < x1 < . . . < xn < x?n < . . . x?1 < x0 ?n ∈ N . A sequence of graphs which increases monotonically in the same sense can be obtained by ?n (θ) := γ(θ) with π(θ) = θ and θ ∈ [n, n + 1) (such that ?n can be identi?ed with γ|[n,n+1) in a very natural way). We have
p Tθ (?n (θ)) =

?n (θ + ω ) ? if θ ∈ [0, 1 ? ω ) ? ?n+1 (θ + ω ) if θ ∈ [1 ? ω , 0) ? ?

Further the graphs ?n are continuous, except at θ = 0 where they are continuous to the right and limθ?0 ?n (θ) = ?n+1 (0). From all this it follows easily that U := n∈Z (??n , ?n ) is an 1, 1-invariant open tube for T p (and thus some p′ , 1-invariant tube for T with p′ ≤ p must exist as well). Now if q > 1 we can lift both γ and T p to the “blown-up” torus R/qZ × T1 . If we then ? strobe the lift γ at θ = 0, the set Λ we obtain contains exactly the points xnq (n ∈ Z) and ? ? the construction works as before. The p′ , 1-invariant open tube U on R/qZ will then project ′ down to a p , q-invariant open tube for the original system. That this projected tube will not “intersect itself” (i.e. the projection restricted to the tube is injective) follows from the fact ? that for no k, n ∈ Z xk can be contained in (xnq , x(n+1)q ). Thus Uθ=0 = n∈Z (x?nq , xnq ) ?θ=j = and U n∈Z (x?(nq+j) , xnq+j ) are disjoint ?j = 1, . . . , q ? 1. As strobing γ at θ = 0 was ? arbitrary (U could have been obtained in the same way by strobing at any other θ), the same ? ? is true for any θ ∈ R/qZ, i.e. Uθ and Uθ+j are disjoint ?j = 1, . . . , q ? 1. Step 2: Absence of regular invariant graphs implies transitivity. We have to show that for all open sets U, V ? T2 ?n ∈ N : T ?n U ∩ V = ?. Of course it su?ces to restrict to the case where U and V are rectangles (in particular connected). First suppose T n U ∩ V = ? ?n ∈ Z . (4.2)

? As U is non-wandering we have T m U ∩ U = ? for some m ∈ N. Thus U := n∈N T nm U will be a kind of “in?nite open tube”, similar to the situation in Step 1. But in this case, as U is non-wandering for any iterate of T , the tube has to intersect itself after winding around the circle a certain number, say q, of times. By passing to the q-fold torus again if necessary, we ? can assume q = 1. This means that for any point (θ, x) ∈ U there exists a simple closed curve ? γ(θ,x) passing through (θ, x), which is completely contained in U . ? is a T m -invariant set, and by using some multiple p of m we can repeat this conNow U ? struction to obtain a T p -invariant set V := n∈N T np V with exactly the same properties. (4.2) ? ∩ V = ?. ? implies U ? ? U and V are already very close to p, q-invariant open tubes, but they may still have some ? ? “holes”. However, points in U and V cannot alternate on the ?bres: Suppose u1 , u2 ∈ Uθ and v1 , v2 ∈ Vθ . The curves γ(θ,u1 ) and γ(θ,u2 ) divide the torus into at least two connected ? components. As V is connected (which is obvious from the construction) it must be completely contained in one of them, i.e. v1 and v2 must be contained in the same interval (u1 , u2 ) or (u2 , u1 ). Therefore u1 < v1 < u2 < v2 < u1 is not possible. With this it is easy to see, that ? ? U := {(θ, x) | ?u1 , u2 ∈ Uθ : x ∈ [u1 , u2 ] and [u1 , u2 ] ∩ Vθ = ?}

15

is a 1, 1-invariant open tube with respect to T p , contradicting the assumption that there are no regular invariant graphs. Thus we have ?n ∈ Z : T n U ∩ V = ?. If n < 0 we are done. Otherwise U ∩ T ?n V is nonwandering with respect to T n , i.e. ? = (U ∩ T ?n V ) ∩ T kn (U ∩ T ?n V ) ? U ∩ T (k?1)n V for some k ≥ 1. Therefore T ?(k?1)n U ∩ V = ?. P Recall that for a single circle di?eomorphism transitivity implies minimality. For quasiperiodically driven circle di?eomorphisms we do not expect this to be true. But the preceding proof can be modi?ed a bit to give information about the structure of all minimal sets of the system. Indeed, what we proved is that n∈N T ?n U is dense in T2 for each open set U ? T2 . Equivalently, the set of points (θ, x) whose orbit never enters U is closed and nowhere dense so that the set N of points (θ, x) whose orbit is not dense is meager (i.e. of ?rst Baire category). If T is not minimal, then obviously N contains every minimal subset of T2 . The structure of such minimal sets is described in the following theorem. Theorem 4.5 Suppose that T ∈ TBV has no regular invariant graph (so that it is transitive) but that it is not minimal, and let M be a minimal subset of T2 . Then each connected component of M is ?1 contained in a single ?ber π1 (θ). Proof: Fix some rectangle U in the complement of a minimal set M . As in Step 2 of the preceding ? proof the “in?nite open tube” U = n∈N T nm U intersects itself after winding around the torus q times where we can assume as above that q = 1, and there exists a simple closed curve γ ? winding around the torus which is completely contained in U so that it has distance δM > 0 to the minimal set M . If we construct this curve as in Step 1 of the preceding proof, except that in the last step of winding around once we close it, then Γ ∩ T m Γ = ?. Denote by Q the complement of Γ ∪ T m Γ in T2 and observe that M ? Q. Now suppose for a contradiction that C is a connected component of the minimal set M such that π1 (C) is a nontrivial subinterval of T1 . Denote its length by δC . Let Q′ be any connected component of Q and assume that Q′ ∩ M = ?. Then Q′ ∩ M is closed and so is π1 (Q′ ∩ M ). We claim that π1 (Q′ ∩ M ) is also open: Let (θ, x) ∈ Q′ ∩ M and ?x some point (η, y) ∈ C such that η is in the middle part of length δC /2 of the interval π1 (C). As the orbit of (η, y) is dense in M there is some iterate k such that d(T k (η, y), (θ, x)) < min{δM , δC /4} and this implies at once that T k C ? Q′ ∩ M and that π1 (Q′ ∩ M ) ? T1 contains an interval neighbourhood of θ. Hence the nonempty set π1 (Q′ ∩ M ) is closed and open so that indeed π1 (Q′ ∩ M ) = T1 . But since Γ ∩ T m Γ = ?, the complement Q of Γ ∪ T m Γ can have at most one such connected component. Equipped with this information we will now construct an invariant open tube which contra? ? dicts the assumption that T has no regular invariant graph: De?ne φ, ψ : T1 → R by ? φ(θ) := inf{t > 0 : γ(θ) ? t ∈ Mθ } ? ψ(θ) := inf{t > 0 : γ(θ) + t ∈ Mθ } ? ? As γ is a continuous curve and as M is closed, the function φ is u.s.c. and ψ is l.s.c. and γ ? φ 1 1 and γ + ψ are u.s.c. respectively l.s.c. continuous functions from T to T . (γ ? φ, γ + ψ) is an ? open tube disjoint from M , and we will show that it is T m -invariant: By de?nition of φ,
m m m Tθ (γ(θ) ? φ(θ)) = Tθ (γ(θ)) ? inf{t > 0 : Tθ (γ(θ)) ? t ∈ Mθ+mω } whereas

γ(θ + mω) ? φ(θ + mω) = γ(θ + mω) ? inf{t > 0 : γ(θ + mω) ∈ Mθ+mω } . But the two right hand sides have the same value, because M is contained in a single connected component of the complement of Γ ∪ T m Γ so that for any two points u, v ∈ Mθ+mω it is m impossible that γ(θ + mω) < u < Tθ (γ(θ)) < v < γ(θ + mω). 16

P The kind of total disconnectedness of minimal sets in θ-direction that is expressed in Theorem 4.5 is in some contrast to the following simple observation. Lemma 4.6 Let T ∈ TBV and let ? = M be a minimal strict subset subset of T2 . Then every point (θ, x) ∈ M has the following property: If U is any neighbourhood of (θ, x), then π1 (U ) contains a nontrivial interval which contains θ. (It may happen that θ is an endpoint of this interval.) Proof: Denote by G the set of theose (θ, x) ∈ T2 which have a neighbourhood U such that π1 (U ∩ M ) is nowhere dense in T1 . This set is obviously open and invariant under T . Hence M \ G it is either empty or all of M . Suppose ?rst that M ? G. Then the compact set M can be covered by ?nitely many sets U for which π1 (U ∩ M ) is nowhere dense, and it follows that π1 (M ) is a nowhere dense closed subset of T1 invariant under the irrational rotation by ω. Hence π1 (M ) = ? in contradiction to the assumption M = ?. Therefore M ∩ G = ? so that each point (θ, x) ∈ M has the property claimed in the lemma. P We close this section with an example which seems a good candidate for transitive but nonminimal behaviour. It is the critical Harper map T (θ, x) = θ + ω, ?1 x ? E + λ cos(2πθ) with E = 0 and λ = 2

with ?ber maps Tθ acting on P 1 R. After a change of co-ordinate x′ = arctan(x) the map T belongs to the class TBV . The map T has Lyapunov exponent zero [BJ02] and rotation number 1 2 Numerical evidence suggests that T 2 has no invariant tube, and invariant tubes for higher 2. powers T q which are not invariant for any smaller power of T are uncompatible with rotation 1 number 2 . Therefore T should be transitive. Numerical evidence again, but also the general classi?cation of ergodically driven M¨bius maps [Thi97, ACO99] together with the close relation o of the dynamics of this family of maps with spectral properties of the almost Mathieu operator 3 suggests that T has a unique invariant probability measure ?(A) = T1 1A (θ, φ(θ)) dθ where φ : T1 → T1 is a measurable invariant graph. (This is called the parabolic case in [Thi97].) Then the topological support of ? would be the only minimal set. Figure 3 shows the plot of a trajectory of length 105 , and Figure 4 displays the result of a numerical reconstruction of the graph φ based on the assumption that the map is indeed parabolic. More details can be found in the forthcoming note [DJKR].

2 This

? follows easily from the symmetry Tθ+ 1 (x) = ?Tθ (?x) of the map on T1 × R: This implies that
2 2

whenever T : T1 × R → T1 × R is a lift of T , then so is the map T de?ned by T θ (x) := ?Tθ+ 1 (?x). Thus the rotation number must be 0 or 1 , and this is true for any λ. Now ρ(T ) = 1 for λ = 0 and the rotation number 2 2 depends continuously on λ. 3 See [KS97, PRSS99] for this relation and [Jit99] for the relevant spectral properties.

17

References
[ACO99] L. Arnold, N.D. Cong, and V.I. Oseledets. Jordan normal form for linear cocycles. Random Operators and Stochastic Equations, 7(4):303–358, 1999. [Arn98] [BJ02] L. Arnold. Random Dynamical Systems. Springer, 1998. J. Bourgain and S. Jitomimirskaya. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. Journal of Statistical Physics, 108(5– 6):1203–1218, 2002. S. Datta, T. J¨ger, G. Keller, and R. Ramaswamy. Preprint. a

[DJKR]

[dMvS93] W. de Melo and S. van Strien. One-dimensional dynamics. Springer, 1993. [Fur61] [Her83] H. Furstenberg. Strict ergodicity and transformation of the torus. American Journal of Mathematics, 83:573–601, 1961. Michael R. Herman. Une m?thode pour minorer les exposants de Lyapunov et e quelques exemples montrant le caract`re local d’un th?or`me d’Arnold et de Moser e e e sur le tore de dimension 2. Commentarii Mathematici Helvetici, 58:453–502, 1983. S. Y. Jitomirskaya. Metal-insulator transition for the almost Mathieu operator. Annals of Mathematics (2), 150(3):1159–1175, 1999. Gerhard Keller. A note on strange nonchaotic attractors. Fundamenta Mathematicae, 151(2):139–148, 1996. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1997. Jukka A. Ketoja and Indubala I. Satija. Harper equation, the dissipative standard map and strange nonchaotic attractors: Relationship between an eigenvalue problem and iterated maps. Physica D, 109:70–80, 1997.

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arctan(x)

θ Figure 3: A trajectory of T

arctan(x)

θ Figure 4: The invariant graph

19


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