Pointwise Dimension for Hyperbolic Dynamics
Sometimes a given global invariant can be built with the help of a local quantity. For example, the Kolmogorov-Sinai entropy and the Hausdorff dimension, which are quantities of global nature, can be built (in a rigorous mathematical sense) respectively with the help of the local entropy and the pointwise dimension. In the case of the entropy this is due to Shannon-McMillan-Breiman’s theorem: the Kolmogorov-Sinai entropy is obtained integrating the local entropy. In this chapter we are mostly interested in the Hausdorff dimension of an invariant measure. In particular, for repellers and hyperbolic sets of conformal maps we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and the Lyapunov exponents. This allows us to whow that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition.
KeywordsInvariant Measure Hausdorff Dimension Invariant Probability Measure Local Entropy Ergodic Component
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