Pointwise Dimension and Applications
Part of the
Springer Monographs in Mathematics
book series (SMM)
In this chapter, again for conformal hyperbolic flows, we establish an explicit formula for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and the Lyapunov exponents. In particular, this formula allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in each ergodic decomposition. We also discuss the problem of the existence of invariant measures of maximal dimension. These are measures at which the supremum of the Hausdorff dimensions over all invariant measures is attained.
KeywordsLyapunov Exponent Invariant Measure Explicit Formula Hausdorff Dimension Smooth Manifold
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L. Barreira and C. Wolf, Measures of maximal dimension for hyperbolic diffeomorphisms
, Commun. Math. Phys. 239
L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows
, Discrete Contin. Dyn. Syst. 17
H. Federer, Geometric Measure Theory
, Grundlehren der mathematischen Wissenschaften 153, Springer, Berlin, 1969.
Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications
, Chicago Lectures in Mathematics, Chicago University Press, Chicago, 1997.
Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows
, Commun. Math. Phys. 216
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