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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 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.
DCDS

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

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

## Year of publication

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