Entropy-expansiveness for partially hyperbolic diffeomorphisms
Lorenzo J. Díaz Todd Fisher M. J. Pacifico José L. Vieitez
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
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
How do hyperbolic homoclinic classes collide at heterodimensional cycles?
Lorenzo J. Díaz Jorge Rocha
We present a model illustrating heterodimensional cycles (i.e., cycles associated to saddles having different indices) as a mechanism leading to the collision of hyperbolic homoclinic classes (of points of different indices) and thereafter to the persistence of (heterodimensional) cycles. The collisions are associated to secondary (saddle-node) bifurcations appearing in the unfolding of the initial cycle.
keywords: thickness of a Cantor set. homoclinic class saddle-node bifurcation heterodimensional cycle partial hyperbolicity
Pseudo-orbit shadowing in the $C^1$ topology
Flavio Abdenur Lorenzo J. Díaz
We prove that the shadowing property does not hold for diffeomorphisms in an open and dense subset of the set of $C^1$-robustly non-hyperbolic transitive diffeomorphisms (i.e., diffeomorphisms with a $C^1$-neighborhood consisting of non-hyperbolic transitive diffeomorphisms).
keywords: Heterodimensional cycle Periodic point Transitivity. Index Pseudo-orbit Shadowing Homoclinic class
Super-exponential growth of the number of periodic orbits inside homoclinic classes
Christian Bonatti Lorenzo J. Díaz Todd Fisher
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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