## Journals

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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS

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.

JMD

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.

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.

DCDS

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.

DCDS

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 $\gamma\in[\alpha,\beta]$.

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