Zero entropy invariant measures for some skew product diffeomorphisms
In this paper we study some skew product diffeomorphisms with nonuniformly hyperbolic structure along fibers. We show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. This gives affirmative answer for these diffeomorphisms to the question suggested by Herman that a smooth diffeomorphism of positive topological entropy fails to be uniquely ergodic. The proof is based on some techniques analogous to those developed by Pesin () and Katok (, ) with investigation on some combinatorial properties of the projected return map on the base.
Let be a () diffeomorphism of a compact -dimensional smooth manifold and the derivative of . preserves a Borel probability measure . For every in a set of full measure, the Lyapunov exponent
exists for every nonzero vector . This functional takes on at most values on and is independent of if is ergodic. If all Lyapunov exponents are nonzero, then is called a hyperbolic measure. Smooth systems with hyperbolic measures are called nonuniformly hyperbolic. The theory for studying such systems was developed by Pesin and then combined with some powerful techniques by A. Katok to look for invariant orbits and produced a number of profound results. These techniques serve as cornerstones for our discussion. For all necessary definitions, theorems and background facts relevant to this paper, one may see  for quick reference or  for detailed proofs.
In  Katok showed:
Let be a diffeomorphism of a compact manifold M, and a Borel probability -invariant hyperbolic measure. Then
Where is the set of all periodic points of and the number of periodic points of with period . is the metric entropy with respect to .
In particular, if the manifold is 2-dimensional, then by Ruelle inequality , every ergodic invariant measure with positive metric entropy must be hyperbolic. Taking also the variational principle into account, we have
For any diffeomorphism of a 2-dimensional compact manifold with positive topological entropy,
Hence is not minimal or uniquely ergodic.
In general, Equation (1) is not true for high dimensional cases. There can be no periodic orbit for a diffeomorphism with positive topological entropy. Herman  constructed a remarkable example as following:
Consider the map defined by
where is a fixed number. Let be the rotation by .
(Herman, ) There is a dense subset W of , such that for every , the smooth diffeomorphism on , given by , is minimal and has positive topological entropy.
Herman’s example prompted a fruitful research, for example, on generic linear cocycles over compact systems. The phenomenon he discovered turned out to be common for extension over rotations .
However, the diffeomorphisms in Herman’s example fail to be uniquely ergodic. We can find a measurable transformation such that for almost every , is diagonal. Then for every measure preserved by the geodesic flow which corresponds to the left action by , is -invariant, where is the Lebesgue measure on . In particular, if is supported on a periodic orbit of the geodesic flow, then .
Whether a smooth diffeomorphism of positive topological entropy can be uniquely ergodic is still in question (For homeomorphisms, the answer is yes. See for examples, ). We studied some skew product diffeomorphisms and found some invariant measures similar to those in Herman’s example.
Let be a probability measure space. is an invertible transformation (mod 0) preserving . is an -dimensional compact Riemannian manifold. For every , is a diffeomorphism. Assume on preserves a measure . Let for . In this paper We prove:
(Main Theorem) If for almost every , The Lyapunov exponent for all , then f has an invariant measure whose conditional measure on each fiber is atomic.
Now suppose that we have a diffeomorphism on . Assume that and is 2-dimensional. If , we must have for some ergodic invariant measure . Then by Ledrappier-Young’s formula , the Lyapunov exponents along fiber direction must be nonzero almost everywhere. Hence by Theorem 1.4 has an invariant measure with atomic conditional measures along fibers. The following statement avoids any mention of exponents.
If has positive topological entropy, has zero topological entropy, and is 2-dimensional, then has a measure of zero entropy and is not uniquely ergodic.
2. Shadowing lemma
Now that we have a diffeomorphism on that has nonzero exponents along fiber direction. We may assume that is ergodic by considering an ergodic component. Almost all results in [6, 8, 10] can be adapted in this setting with careful modification. By considering the derivative for as a linear cocycle over , we have:
Assume . Denote by the standard Euclidean r-ball in centered at the origin. There exists a set of full measure such that for every sufficiently small and some :
There exists a tempered function and a collection of embeddings for each such that and .
There exist a constant and a measurable function such that for ,
The map has the form
where , and
For , :
The points are called regular points. For each regular point , the set is called a regular neighborhood of . Let be the radius of the maximal ball contained in the regular neighbor hood . We say is the size of .
For each and each sufficiently small , there is a set which has compact intersection (may be empty) with each fiber , such that and the following conditions hold:
The functions , and as in Theorem 2.1 for , and are all continuous on for each .
The decomposition depends continuously on in .
On , there are bounds: , , .
With similar definitions and properties for admissible manifolds, we are able to derive the following version of Shadowing Lemma:
Given , for , set . Given and , a sequence is called an -pseudo orbit for if there are and such that for every , and . Then there exists such that for every -pseudo orbit, there is a unique point such that for all .
3. Integrability of return time
Now we would like to take a proper Pesin set on which the shadowing techniques can be carried out.
Let be the projection to the base. A measurable subset is called a ”Regular Tube”, if for some , , and as in Theorem 2.4, there exists for every , a point such that , and .
The existence of such a ”Regular Tube” is guaranteed in Section 2. In this ”Regular Tube”, we can take a measurable section , . Let . We are then going to consider the first return map on .
can be chosen in such a way that the first return time from to is integrable with respect to . In particular, we may assume every point in returns to in finite times.
For every , denote by the return time of . Since is -invariant and , we have:
We may choose such that for every , only if . Let . By the way is chosen and the assumption , we have , hence
The return time is integrable. g
4. Projected return map on the base
Now let us try to find the invariant measure described in Theorem 1.4. We may assume that has no periodic point, or else the problem can be reduced to the case considered by Katok . Moreover, is invertible as we assumed earlier.
We are looking for an invariant set which has finite intersection with almost every fiber . The measure supported on is the delta counting measure on . Then an invariant measure for can be given by .
To find the invariant set , we start with a fixed ”Regular Tube” and a measurable section as specified in section 3. Let be the first return map on and the first return map for on . Define by . is the projection of the return map on the base. is invertible but may not. For every , let be such that .
We can define a partial order on : iff there is such that , i.e. is an image of under iterates of . Since is invertible and has no periodic point, this partial order is well defined. Moreover, if there is such that then we write , which implies . This is also a partial order.
We can define an equivalence relation on : iff , i.e. there are such that . If then for all . If , we must have or , denoted by or . If , define as the minimal (with respect to , always in this paper) element in . In particular, if then and .
The equivalence relation defined here is crucial in this paper. If , then and return to the same fiber after iteration of . However, does not necessarily return to , i.e. two points in may return to on the same fiber and this may happen all the time. We had trouble dealing with this situation while looking for pseudo orbits. Introduction of this equivalence relation solved this problem. We can then, in each equivalence class, find a unique orbit of (a sequence of returns, lifted to a pseudo orbit) to construct the invariant set.
For almost every , is finite. Denote by the maximal element of . Then for all . Let , then . Moreover, if , then .
For every , define the set of ”jumps” . By integrability of return times (Proposition 3.2), must be finite for almost every . To see this, we can consider the set
Let . We can count the return times and get
and for almost every .
For every but , there must be such that and . So for different elements , there must be different elements such that , . Hence .
By definition, for every , and if since they are both images of under iteration of . So for every , we have .
Since , there is some such that . Then for every , . But . .
If , then , because the second set is contained in the first one. But and , from previous discussion we must have . g
Let . Then . Hence by replacing by and accordingly, we may assume that for every , there is at least one element such that but .
If , then there must be an element such that has infinitely many elements by Poincaré Recurrence Theorem because is invertible and -preserving. From the proof of Proposition 4.1, has finitely many elements. But for every , there must be such that . Hence there is an element such that has infinitely many elements. But for all because , which is a contradiction. g
For every , let and . If , then we must have and .
Pick in the following way:
If is not properly defined: Let . (only for in a set of measure zero)
If has a minimal element : Let .
If has no minimal element: By Proposition 4.1, can be completed to a full orbit of , . In each equivalence class , is a sequence of returns and is uniquely defined in the sense . ordered by ”” can be viewed as a sequence and for all . Then the sequence is in fact a -pseudo orbit. Let us call it the pseudo orbit associated to . Note the pseudo orbits associated to equivalent elements coincide. By the way the ”Regular Tube” was chosen, we can find as specified in Theorem 2.4. Let .
Let . By definition, is invariant for all . Hence is -invariant.
For almost every , is nonempty and contains finitely many elements.
For almost every , by definition. Note that if . For different elements , there are such that , . We must have and . But , . So we have .
Recall that , so by ergodicity, must be nonempty and contain finitely many elements for almost every . g
Since is invariant and is finite, is an invariant measure as requested, where is the delta counting measure on supported on , for almost every . The entropy of this measure is the same as the entropy of the transformation on the base.
5. Measures of intermediate entropies
In , Katok showed a stronger result:
If is a diffeomorphism of a compact smooth manifold and an ergodic hyperbolic measure for with , then for any there exists a hyperbolic horseshoe such that . Hence for any number between zero and , there is an ergodic invariant measure such that .
We are looking for analogous result to the theorem for our skew product diffeomorphisms. As we know, for skew product diffeomorphisms there may not be any proper closed invariant sets. However, we may expect to have an invariant set that has closed intersection with almost every fiber, on which acts like a horseshoe map. Moreover, this horseshoe should carry an entropy arbitrarily close to in order to produce invariant measures with arbitrary intermediate entropies. This work is in progress.
We may also ask the question if theorem 5.1 holds for any diffeomorphism without the assumption that is hyperbolic. We have not yet found even a zero entropy measure in this general case.
-  A. Avila, R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schr¡§odinger cocycles, Ann. of Math. (2), 164 (2006), 911–940.
-  L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, Handbook of dynamical systems, Vol. 1B, 57–263, Elsevier B. V., Amsterdam, 2006.
-  L. Barreira and Y. Pesin, Nonuniform hyperbolicity: Dynamics of systems with nonzero Lyapunov exponents, Cambridge University Press, 2007.
-  F. Béguin, S. Crovisier, F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. Ann. Sci. École Norm. Sup. (4) 40 (2007), 251–308.
-  M. Herman, Construction d’un difféomorphisme minimal d’entropie topologique non nulle, Ergodic Theory Dyn. Syst. 1 (1981), 65–76.
-  A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S. 51 (1980), 137–173.
-  A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 1245–1253, PWN, Warsaw, 1984.
-  A. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior, supplement to Introduction to the Modern Theory of Dynamical Systems, by A. Katok and B. Hasselblatt, Cambridge University Press, 1995.
-  F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Ann. Math. (2) 122 (1985) , 509–574.
-  Y. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv, 40 (1976), 1261–1305.
-  D. Ruelle, An inequality for the entropy of differentiable map, Bol. Soc. Brasil. Mat. 9 (1978), 83–87.