Entropy of Physical Measures for $$C^\infty $$ Dynamical Systems

Abstract

For a C ∞ map on a compact manifold we prove that for a Lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the sum of positive Lyapunov exponents at x. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii's work [22] relating physical and Sinai-Ruelle-Bowen measures.

SRB measures for C $\infty$ surface diffeomorphisms

“…In this section we use tools introduced firstly in [14] to bound the distorsion of unstable manifolds. As these manifolds are one-dimensional, the reparametrizations in Yomdin's theory are just affine.…”

“…As these manifolds are one-dimensional, the reparametrizations in Yomdin's theory are just affine. In fact we can in this case completely avoid the use of the classical Yomdin's Reparametrization Lemma by applying the Landau-Kolmogorov inequality as in [14]. 5.1.…”

“…Finally we emphasize the necessity of C ∞ smoothness. We prove in a forthcoming paper [12] that the Main Theorem is false in finite smoothness by building for any finite r > 1 a C r surface diffeomorphism (f, M ) with a periodic saddle hyperbolic point p such that χ(x) > 0 for all x ∈ U for some set U ⊂ B(µ p ) with Leb(U ) > 0, where µ p denotes the periodic measure associated to p (see [14] for such an example of interval maps). However, as in the examples of C r surface diffeomorphisms without measure of maximal entropy [15], the construction is local.…”

“…Assume lim sup n 1 n log d x f n > 0 on a set of positive Lebesgue measure as in the Main Theorem or Corollary 2. Then by using Fubini's theorem as in [14] there is a C ∞ smooth embedded curve σ : I → M and a subset A of σ * with Leb σ * (A) > 0, such that we have lim sup n…”

Volume growth and topological entropy of certain partially hyperbolic systems

“…• Also in C ∞ setting, Burguet [4] has shown that for Lebesgue almost every point x ∈ M , there is some invariant probability measure in the limit set of { 1 n n−1 j=0 δ f j (x) } n∈N with entropy larger than (or equal to) the volume growth rate on the direction with positive upper Lyapunov exponents. He also gave some counterexample to show that it is not true if the system only has finite regularity.…”