Stable sets, hyperbolicity and dimension
Rasul Shafikov Christian Wolf
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 403-412 doi: 10.3934/dcds.2005.12.403
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.
keywords: box dimension stable sets. SRB measure Hausdorff dimension Hyperbolic sets
Dimension and ergodic decompositions for hyperbolic flows
Luis Barreira Christian Wolf
Discrete & Continuous Dynamical Systems - A 2007, 17(1): 201-212 doi: 10.3934/dcds.2007.17.201
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.
keywords: hyperbolic flows pointwise dimension. Ergodic decompositions
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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