Measures of Intermediate Entropies for Skew Product Diffeomorphisms
In this paper we study a skew product map with a measure of positive entropy. We show that if on the fibers the map are diffeomorphisms with nonzero Lyapunov exponents, then has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximum entropy of these measures can be arbitrarily close to .
Key words and phrases:entropy, ergodic measure, Lyapunov exponents, skew product, horseshoe, return map, Pesin theory
1991 Mathematics Subject Classification:Primary: 37D25, 37C40; Secondary: 37A05.
Department of mathematics
The Pennsylvania State University
University Park, PA 16802, USA
Entropy has been one of the centerpieces in dynamics. It reflects the complexity of the system. For smooth systems, positive topological entropy comes from some (partially) hyperbolic structure, and is conjectured to be accompanied with plenty of invariant measures. The work presented here explores answers to such conjectures in a case of skew product maps.
For a compact Riemannian manifold and a (, this means the derivatives are -Hölder continuous) diffeomorphism on , if preserves an ergodic measure , then there are real numbers , , such that for -almost every point , there are subspaces of the tangent space such that for any vector ,
Here is some integer number no more than the dimension of and these subspaces are invariant of , the derivative of . These numbers are called Lyapunov exponents. We say is a hyperbolic measure if all Lyapunov exponents are nonzero. Three decades ago, A. Katok established a well-known result as following:
This theorem has an interesting corollary: Under the conditions of the theorem, there are ergodic measures such that for any real number , i.e. all possible entropies of ergodic measures form an interval. Since the horseshoe map is a full shift, these measures can be constructed by taking sub-shifts or properly assigning weights to different symbols. Existence of these measures of intermediate entropies exhibits the complicated structure of the system.
Till now it is not known whether every smooth system ( diffeomorphism) on a compact manifold with positive (topological) entropy has this property. In general, such a system may not have a hyperbolic measure. Herman constructed a well-known example, which is a minimal diffeomorphism with positive topological entropy . So in the general case even a closed invariant subset should not be expected. Fortunately, minimality may not prevent the system from having measures of intermediate entropies, which is the case in Herman’s example. Herman suggested the question whether for every smooth system positive topological entropy violates unique ergodicity. Katok conjectured more ambitiously: these systems must have measures of arbitrary intermediate entropies. We are working towards this goal.
In this paper we deal with skew product maps with nonzero Lyapunov exponents along fibers. Let be a skew product map on the space preserving an ergodic measure . We assume that is an invertible (mod 0) measure preserving transformation on the probability space . Then is ergodic. For every , is a diffeomorphism on the compact Riemannian manifold , sending to .
Assume that and . If for almost every and every , the Lyapunov exponent
then has ergodic invariant measures of arbitrary intermediate entropies.
Our proof also works when . In this case it concludes that has ergodic measures of arbitrary entropies between and .
In addition, we assume has no periodic point. Otherwise the problem is reduced to Theorem 1.1. We have shown in  that under the conditions there are measures of zero entropy. Some lemmas are generalized and adapted in this paper.
The author would like to thank Anatole Katok for numerous discussions and encouragement.
2. Entropy and Separated Sets on Fibers
In this section we discuss the entropy of the skew product and obtain an estimate on the cardinality of the -separated set on each fiber, which is analogous to the definition of metric entropy by Katok .
Let be a measurable partition of the fiber with finite entropy for almost every , where . Let us put
(Abramov and Rohlin, ) For every , put
The limit exists and it is finite. Let
is called the fiber entropy. We have
For the skew product map we consider, is ergodic and, for almost every , . we have for almost every ,
The following is a version of Shannon-McMillan-Breiman Theorem for skew product maps.
(Belinskaja, ) Let be the element of containing . For almost every ,
Fix a Riemannian metric on . Let be an increasing system of metrics defined for on the same fiber by:
For and , on the fiber let be the minimal number of -balls in the -metric needed to cover a set of -measure at least , and let be the maximal number of -separated points we can find inside every set of -measure at least .
We can follow exactly Katok’s argument for [5, Theorem 1.1] and obtain the analogous result:
If is ergodic, then for almost every and every ,
In this section we discuss some properties related to nontrivial recurrence of the map.
Theorem 3.1 is a generalization of [8, Proposition 3.2]. It says that, for a subset of positive measure, if the conditional measures are uniformly bounded from below, then on each fiber we can find points that return relatively faster, such that the return time is integrable. In this paper we still use the special version for the first return.
Theorem 3.2 is crucial in the proof of the main theorem. For a complicated or even randomly selected return map on some set of positive measure, if the return time is integrable, then this return map is one-to-one and measure preserving on a smaller subset of positive measure. We call this subset the kernel of the return map. Moreover, we have an estimate of the integral of the return time, which may be used to estimate the size (measure) of the kernel.
(Integrability of Return Time) Let be a measurable subset. is the projection of on the base. For , denote by . Assume that and there is such that for (almost) every , . Hence . For (almost) every , denote by the -th return time of . Let
is the longest return time for the -th returns of the points in a subset of conditional measure no less than in . Or equivalently, is the smallest number such that the set of points in with -th return times greater than has conditional measure at most .
Since is -invariant and , we have:
Let be the first return map on , which preserves , then for each ,
where consists of points with -th return time no less than . Note .
For every , let . By definition, for every , . So iff . We have
Let be a measure preserving transformation on a probability space . is invertible and has no periodic point. is a subset of with . is a measurable function such that for almost every . Assume
Then there is a subset of such that the following holds:
, and is invertible and -preserving.
We call the kernel of .
This theorem is nontrivial because the map is not necessarily the first return, but just some return. is just a measurable transformation on which may be neither injective nor surjective. However, we are able to find a subset of on which it is invertible, provided integrability of return time.
The assumption that has no periodic point is not necessary in this theorem. Periodic orbits may be removed as a null set or we can easily find a subset consisting of periodic orbits on which is invertible.
In particular, may coincide with in the theorem. This is an interesting corollary.
With possible loss of a null set we may assume that the first return map on is defined everywhere and invertible. has no periodic point since has not. As is some return map and is measurable, there is a measurable function such that .
Define a partial order on : iff there is such that , i.e. is an image of under iterates of (and ). Since is invertible and has no periodic point, this partial order is well defined.
Let be the forward -orbit of . We define an equivalence relation on : iff , i.e. there are such that . Note or implies , but the converse is not true. Also note within an equivalence class the partial order is a total order, since is invertible and implies for some and . We write if and .
Here we use some facts we showed in :
(, Proposition 4.1) For almost every , there is such that for every , , i.e.
Moreover, if then .
(, Proposition 4.2) For almost every , there is a point such that and . Hence for almost every , there are infinitely many such that .
Excluding a null set and its (full) -orbit (the union is still a null set), we can assume the results in the last two lemmas hold for every .
Let . Then is a measurable subset of , consisting of elements that lie in the forward -orbits of infinitely many elements of . For every , let
Then for every , there is such that and . By Lemma 3.4, the intersection is of infinitely many forward -orbits, which implies . So for every .
For , there are infinitely many elements in such that their forward -orbits pass through . They also pass through one of the elements in the pre-image . But by integrability of return time (1), the pre-image consists of finite number of elements. So there must be some element in which lies in infinitely many forward -orbits. Such an element belongs to , hence is nonempty. The function
is a well-defined measurable function on .
Define . For note and hence . So is a measurable transformation on . We also note that for every .
Let . Then is measurable. On one hand, by definition we have . For every , and . This implies . On the other hand,
For every and every , . We claim
If , then .
Assume . Since , there is such that lies on the forward -orbit of every such that , while lies on infinitely many forward -orbits of the elements in the same equivalence class with . As is invertible, there are only finitely many elements between (in the sense of the partial order) and . So there must be some (in fact, infinitely many) such that and both and lie on the forward -orbit of . Assume and . Then . As has no periodic orbit and as well, we must have and , which is a contradiction. ∎
From the lemma we know for every , , hence for every positive integer . This yields
for every .
In particular, is nonempty and has positive measure. We shall show is invertible on .
is surjective since we have showed .
is injective. If then and there is such that . But . This implies that for every . So if and , then .
preserves . Let for . Then and . for since is invertible. preserves . For any measurable subset , , where . We have
This completes the proof of the first part.
For the second part, consider
is -invariant, and
First note for each , . Then
From (2) and , we have
Lemma 3.7 yields
Let be a subset of of positive measure such that . If is ergodic, then
Let for . Consider
Similar argument shows . If is ergodic, then . Hence
4. Proof of the Main Theorem
4.1. Regular Tube
If on the fiber direction there is no zero Lyapunov exponents, then from Pesin theory  we know for almost every point there is a regular neighborhood around on the fiber. Inside each regular neighborhood we can introduce a local chart and identify a ”rectangle” with the square ( with ) in Euclidean space (in higher dimension this should be recognized as the product of unit balls in dimensions corresponding to contracting and expanding directions).
Fix some small number , we can define admissible -curves as the graphs and admissible -curves as , if is a map with . There is some such that, if in addition then admissible -curves are mapped by , while -curves by , to the admissible curves of the same types, respectively.
Consider admissible -rectangles defined as the sets of points
if and are admissible -curves. Admissible -rectangles are defined analogously. Like admissible curves, these admissible rectangles are also mapped to admissible rectangles of the same types by and , respectively.
Let us fix small numbers and .
There is a ”Regular Tube” , which is a measurable subset of satisfying the following properties:
Let be the projection to the base and let . Then .
Let . There is some number such that, for every , .
For every , there is a rectangle on the fiber whose diameter is less than . where is the Lyapunov regular neighborhood of some point on the fiber .
For every and , if for some , returns to , i.e. and , then the connected component of the intersection containing , denoted by
is an admissible -rectangle in and
is an admissible -rectangle in . Moreover, for , on the fiber we have
Applying Theorem 2.3, we may assume that there is some such that for every and , inside any set of -measure at least on the fiber , we can find a -separated set with cardinality at least .
This regular tube can be obtained with the following steps.
On almost every fiber, find a regular point and its regular neighborhood . Take with diameter less than .
Find satisfying property (5). There is some such that . For , shrink the size of properly such that .
Find and such that also satisfies property (6) and . is as required.
4.2. Control of Return Time
We start with a regular tube . Applying Theorem 3.1, we can find a measurable section , such that
where is the first return time of .
Denote by the characteristic function of the measurable set . Consider the sets
(throughout this paper, numbers like are rounded to the nearest integer, if needed). Since is ergodic, Birkhoff Theorem tells us for almost every ,
Let . Then as ,
There is and a measurable subset with the following properties
. Let .
For , let . For , .
Let . .
Now let us fix . For convenience, denote by the integer part of . For large ,
For every , , by property (6) of the regular tube, there is a -separated set with cardinality
For , the -th return time of to is an integer number between and . So there is with cardinality
and the -th return times for points in are the same, denoted by . The set can be chosen to be the union of measurable sections over .
is a measurable function on . We extend to a measurable function on : for , let . Consider the map . is well defined on and . Moreover,
Applying Theorem 3.2 we can find the kernel of positive measure such that the restricted on is invertible and preserves . We can assume that is ergodic (with respect to the measure induced by ) by taking an ergodic component of positive -measure.
Let and . Let be the first return map on with respect to . Then is invertible and preserves . Define measurable functions and on such that