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For a $C^\\infty$ map $f$ on a compact manifold $M$ 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 top Lyapunov exponent of $\\Lambda\\, df:\\Lambda \\,TM\\circlearrowleft$ at $x$. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii's work relating physical and Sinai-Ruelle-Bowen measures.
Building on the theory of symbolic extensions and uniform generators for discrete transformations we develop a similar theory for topological regular flows. In this context a symbolic extension is given by a suspension flow over a subshift.
We prove that periodic asymptotic expansiveness implies the equidistribution of periodic points to measures of maximal entropy. Then following Yomdin's approach we show by using semi-algebraic tools that $C^1$ interval maps and $C^1$ surface diffeomorphisms satisfy this expansiveness property respectively for repelling and saddle hyperbolic points with Lyapunov exponents uniformly away from zero.
We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded. This improves, with elementary tools, …
For a topological dynamical system $(X,T)$ we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal …
For a given metrizable space X, we study continuity properties of the entropy as function not only of the measure but also of the dynamical system on X. We introduce the notion of robust tail entropy, which implies upper semicontinuity of the …
We study the jumps of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f\in C^r([0,1])$ with $$h\_{top}(f)\\frac{\log^+\|f'\|_{\infty}}{r}.$$ To this end we study the …
A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if \( h_{top}(T)
We study the rate of convergence to zero of the tail entropy of $C^{\\infty}$ maps. We give an upper bound of this rate in terms of the growth in $k$ of the derivative of order $k$and give examples showing the optimality of the established rate …