2007, 17(1): 201-212. doi: 10.3934/dcds.2007.17.201

Dimension and ergodic decompositions for hyperbolic flows

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

2. 

Department of Mathematics, Wichita State University, Wichita, Kansas, 67260

Received  February 2006 Revised  October 2006 Published  October 2006

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.
Citation: Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201
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