## Journals

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### Open Access Journals

JMD

Let $M$ be a closed $3$-manifold, and let $X_t$ be a transitive Anosov flow.
We construct a diffeomorphism of the form $f(p)=Y_{t(p)}(p)$, where $Y$ is an
Anosov flow equivalent to $X$. The diffeomorphism $f$ is structurally
stable (satisfies Axiom A and the strong transversality
condition); the non-wandering set of $f$ is the union of a transitive
attractor and a transitive repeller; and $f$ is also partially
hyperbolic (the direction $\RR.Y$ is the central bundle).

keywords:
partial hyperbolicity
,
AxiomA diffeomorphism
,
Birkhoff
sections
,
Anosov flows
,
perturbations.

JMD

We prove that the spaces of $C^1$ symplectomorphisms and of $C^1$ volume-preserving diffeomorphisms of connected manifolds contain residual subsets of diffeomorphisms whose centralizers are trivial.

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

JMD

We show that for every compact $3$-manifold $M$ there exists an open
subset of $Diff^1(M)$ in which every generic diffeomorphism admits
uncountably many ergodic probability measures that are hyperbolic
while their supports are disjoint and admit a basis of attracting
neighborhoods and a basis of repelling neighborhoods. As a
consequence, the points in the support of these measures have no
stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and
unstable manifolds with complementary dimensions at almost every
point. We also give such an example in dimension two, without local
genericity.

DCDS

We give a sufficient criterion for the hyperbolicity of a homoclinic
class. More precisely, if the homoclinic class $H(p)$ admits a
partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where
$E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all
periodic points homoclinically related with $p$ are

*uniformly $E^u$-expanding at the period*, then $H(p)$ is hyperbolic. We also give some consequences of this result.## Year of publication

## Related Authors

## Related Keywords

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