Entropy-expansiveness for partially hyperbolic diffeomorphisms
Lorenzo J. Díaz Todd Fisher M. J. Pacifico José L. Vieitez
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4195-4207 doi: 10.3934/dcds.2012.32.4195
We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
keywords: symbolic extension. Entropy-expansive dominated splitting equilibrium state partially hyperbolic
Symbolic extensions and partially hyperbolic diffeomorphisms
Lorenzo J. Díaz Todd Fisher
Discrete & Continuous Dynamical Systems - A 2011, 29(4): 1419-1441 doi: 10.3934/dcds.2011.29.1419
We show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing nonhyperbolic robustly transitive sets with a center indecomposable bundle of dimension at least 2. Similarly, $C^1$-generically homoclinic classes with a center indecomposable bundle of dimension at least 2 that satisfy a technical assumption called index adaptation have no symbolic extensions.
keywords: partially hyperbolic homoclinic tangency. dominated splitting Symbolic extensions robust transitivity
Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle
David Burguet Todd Fisher
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2253-2270 doi: 10.3934/dcds.2013.33.2253
We relate the symbolic extension entropy of a partially hyperbolic dynamical system to the entropy appearing at small scales in local center manifolds. In particular, we prove the existence of symbolic extensions for $\mathcal{C}^2$ partially hyperbolic diffeomorphisms with a $2$-dimensional center bundle. 200 words.
keywords: Symbolic extensions partial hyperbolicity.
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
Hyperbolic sets with nonempty interior
Todd Fisher
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 433-446 doi: 10.3934/dcds.2006.15.433
In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.
keywords: Anosov. Dynamical systems interior hyperbolic set
Super-exponential growth of the number of periodic orbits inside homoclinic classes
Christian Bonatti Lorenzo J. Díaz Todd Fisher
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 589-604 doi: 10.3934/dcds.2008.20.589
We show that there is a residual subset $\S (M)$ of Diff$^1$ (M) such that, for every $f\in \S(M)$, any homoclinic class of $f$ containing periodic saddles $p$ and $q$ of indices $\alpha$ and $\beta$ respectively, where $\alpha< \beta$, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown that the super-exponential growth occurs for hyperbolic periodic points of index $\gamma$ inside the homoclinic class for every $\gamma\in[\alpha,\beta]$.
keywords: index of a saddle Artin-Mazur diffeomorphism heterodimensional cycle chain recurrence class homoclinic class symbolic extensions.

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