Dimension Theory of Hyperbolic Dynamics

  • Luis Barreira
Part of the Universitext book series (UTX)


We study in this chapter the dimension of hyperbolic invariant sets of conformal transformations, both invertible and noninvertible. This means that the derivative of the map along the stable and unstable directions is a multiple of an isometry at every point. More precisely, we compute the Hausdorff dimension and the lower and upper box dimensions of repellers and hyperbolic sets for a conformal dynamics. The dimension of the invariant sets is expressed, as explicitly as possible, in terms of the topological pressure. It turns out that Markov partitions are a principal element of the proofs. In particular, they allow us to reduce effectively some of the arguments and computations to the special case of symbolic dynamics.


Invariant Measure Unstable Manifold Hausdorff Dimension Dimension Theory Symbolic Dynamic 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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