Pointwise Dimension and Applications

  • Luís Barreira
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.


Lyapunov Exponent Invariant Measure Explicit Formula Hausdorff Dimension Smooth Manifold 
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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Luís Barreira
    • 1
  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal

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