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Invariant Manifolds and Markov Partitions

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Abstract

We continue in this chapter our study of hyperbolic dynamics, starting with the construction of stable and unstable manifolds for any point of a hyperbolic set. The optimal regularity of the invariant manifolds is obtained using invariant cone families. In particular, we use the stable and unstable manifolds to define a product structure on any locally maximal hyperbolic set. In addition, we construct Markov partitions with a substantial elaboration of the corresponding construction for repellers. The shadowing property is also presented and is used as a tool in the construction of the Markov partitions. Moreover, we describe Hopf’s argument to establish the ergodicity of Lebesgue measure for a hyperbolic toral automorphism.

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  1. Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal

    Luis Barreira

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Barreira, L. (2012). Invariant Manifolds and Markov Partitions. In: Ergodic Theory, Hyperbolic Dynamics and Dimension Theory. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28090-0_7

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