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DCDS

We establish lower bounds for the
topological entropy expressed in terms of the exponential growth
rate of $k$-volumes. This approach provides the sharpest possible
bounds when no further geometric information is available.
In particular, our methods apply to (partially) volume-expanding
dynamics with not necessarily compact phase space, including
a large class of geodesic flows. As an
application, we conclude that the topological entropy of these
systems is positive.

DCDS

We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface.
We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.

keywords:
Lyapunov exponents
,
geodesic flow
,
rank 1 surfaces
,
Hausdorff dimension
,
shadowing.
,
entropy
,
pressure
,
multifractal formalism

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

## Related Authors

## Related Keywords

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