Expanding Maps

  • Min QUIAN
  • Jian-Sheng XIE
  • Shu ZHU
Part of the Lecture Notes in Mathematics book series (LNM, volume 1978)


In the previous chapter, we showed the Margulis-Ruelle inequality, which says that the measure-theoretic entropy is bounded above by the sum of positive Lyapunov exponents.

In 1977, Pesin [63] showed that for a C 2 diffeomorphism f if its invariant Borel probability measure µ is absolutely continuous with respect to the Lebesgue measure on the manifold, then the equality in (II.II.1) holds, i.e.

$$h_\mu(f) = \int_M {\sum\limits_i {\lambda _i (x)^ +m_i (x)d\mu(x).} }$$

Now identity (PEF) is known as Pesin’s entropy formula.

In this chapter, we consider a simple case that this identity holds, namely expanding maps. It is simpler than the case of diffeomorphisms as the local unstable manifold of an expanding map at a point is a neighborhood of the point. The results of this chapter are from [27].


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Min QUIAN
    • 1
  • Jian-Sheng XIE
    • 2
  • Shu ZHU
    • 3
  1. 1.School of Mathematical SciencesPeking UniversityBeijingP. R. China
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiP. R. China
  3. 3.MississaugaCanada

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