# zbMATH — the first resource for mathematics

Topological entropy and partially hyperbolic diffeomorphisms. (English) Zbl 1143.37023
The authors consider continuity properties for the topological entropy of partially hyperbolic diffeomorphisms on compact manifolds. Fur this purpose, they define the notion of unstable and stable foliations stably carrying some nontrivial homologies. Under this topological assumption, the following two results are shown:
(1)
If the center foliation is one-dimensional, then the topological entropy is locally a constant.
(2)
If the center foliation is two-dimensional, then the topological entropy is continuous on the set of $$C^\infty$$-diffeomorphisms.
The proof uses a topological invariant, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, as well as a refined Pesin-Ruelle inequality, which is shown for partially hyperbolic diffeomorphisms. In the final section, several examples illustrate that the above results do not hold without an appropriate assumption on the homology.

##### MSC:
 37D30 Partially hyperbolic systems and dominated splittings 37B40 Topological entropy
Full Text:
##### References:
 [1] Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds (1993) · Zbl 0772.58001 · doi:10.1017/CBO9780511752537 [2] Katok, Introduction to the Modern Theory Dynamical Systems (1955) [3] Barreria, Smooth Ergodic Theory and Non-Uniformly Hyperbolic Dynamics (2005) [4] DOI: 10.1007/BF02766215 · Zbl 0641.54036 · doi:10.1007/BF02766215 [5] Walters, An Introduction to Ergodic theory (1982) · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2 [6] Walters, Ergodic Theory (1975) [7] Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (1999) · Zbl 0914.58021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.