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### Open Access Journals

DCDS

In this note we derive an upper bound for the Hausdorff and box
dimension of the stable and local stable set of a hyperbolic set
$\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold.
As a consequence we obtain that dim$_H W^s(\Lambda)=n$ is equivalent
to the existence of a SRB-measure. We also discuss related
results for expanding maps.

DCDS

For conformal hyperbolic flows, we establish explicit formulas for
the Hausdorff dimension and for the pointwise dimension of an
arbitrary invariant measure. We emphasize that these measures are
not necessarily ergodic. The formula for the pointwise dimension is
expressed in terms of the local entropy and of the Lyapunov
exponents. We note that this formula was obtained before only in the
special case of (ergodic) equilibrium measures, and these always
possess a local product structure (which is not the case for
arbitrary invariant measures). The formula for the pointwise
dimension allows us to show that the Hausdorff dimension of a
(nonergodic) invariant measure is equal to the essential supremum of
the Hausdorff dimension of the measures in an ergodic decomposition.

DCDS

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the
periodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle,
and others in the case of hyperbolic diffeomorphisms.
We show that the topological pressure P

_{top}$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.## Year of publication

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