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DCDS

We establish upper bounds on the rate of decay of correlations of
tower systems with summable variation of the Jacobian and integrable
return time. That is, we consider situations in which the
Jacobian is not Hölder and the return time is only subexponentially
decaying.
We obtain a subexponential bound on the correlations, which is essentially the
slowest of the decays of the variation of the Jacobian and of the return time.

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

We give a new type of sufficient condition for the existence of
measures with maximal entropy for an interval map $f$, using some non-uniform
hyperbolicity to compensate for a lack of smoothness of $f$.
More precisely, if the topological entropy of a $C^1$ interval map is
greater than the sum of the local entropy and the entropy of the critical
points, then there exists at least one measure with maximal entropy. As a
corollary, we obtain that any $C^r$ interval map $f$ such that
h

_{top}(f) > 2log || f'||∞ / r possesses measures with maximal entropy.## Year of publication

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