Lower bounds for the topological entropy
Katrin Gelfert
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 555-565 doi: 10.3934/dcds.2005.12.555
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.
keywords: Hausdorff dimension geodesic flow. Topological entropy
Lyapunov spectrum for geodesic flows of rank 1 surfaces
Keith Burns Katrin Gelfert
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1841-1872 doi: 10.3934/dcds.2014.34.1841
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
On the distribution of periodic orbits
Katrin Gelfert Christian Wolf
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 949-966 doi: 10.3934/dcds.2010.26.949
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 Ptop$(\varphi)$ 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.
keywords: equilibrium states large deviation. topological pressure non-uniformly hyperbolic dynamics Pesin theory

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