Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time
Jérôme Buzzi Véronique Maume-Deschamps
Discrete & Continuous Dynamical Systems - A 2005, 12(4): 639-656 doi: 10.3934/dcds.2005.12.639
We establish upper bounds on the rate of decay of correlations of tower systems with summable variation of the Jacobian and integrable return time. That is, we consider situations in which the Jacobian is not Hölder and the return time is only subexponentially decaying. We obtain a subexponential bound on the correlations, which is essentially the slowest of the decays of the variation of the Jacobian and of the return time.
keywords: tower extension. equilibrium states transfer operator Absolutely continuous invariant measures decay of correlations
Entropic stability beyond partial hyperbolicity
Jérôme Buzzi Todd Fisher
Journal of Modern Dynamics 2013, 7(4): 527-552 doi: 10.3934/jmd.2013.7.527
We analyze a class of $C^0$-small but $C^1$-large deformations of Anosov diffeomorphisms that break the topological conjugacy and structural stability, but unexpectedly retain the following stability property. The usual semiconjugacy mapping the deformation to the Anosov diffeomorphism is in fact an isomorphism with respect to all ergodic, invariant probability measures with entropy close to the maximum. In particular, the value of the topological entropy and the existence of a unique measure of maximal entropy are preserved. We also establish expansiveness around those measures. However, this expansivity is too weak to ensure the existence of symbolic extensions.
    Many constructions of robustly transitive diffeomorphisms can be done within this class. In particular, we show that it includes a class described by Bonatti and Viana of robustly transitive diffeomorphisms that are not partially hyperbolic.
keywords: dominated splitting. ergodic theory robust ergodicity Measures of maximal entropy topological entropy
Large entropy implies existence of a maximal entropy measure for interval maps
Jérôme Buzzi Sylvie Ruette
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 673-688 doi: 10.3934/dcds.2006.14.673
We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that htop(f)  >  2log || f'||∞ / r possesses measures with maximal entropy.
keywords: Maximal entropy measure Markov shift. interval map

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