Inventiones mathematicae

, Volume 205, Issue 2, pp 363–382 | Cite as

The Lyapunov exponent of holomorphic maps

Article

Abstract

We prove that for any polynomial map with a single critical point its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.

Mathematics Subject Classification

37F10 37F15 37F50 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Institute of MathematicsPolish Academy of ScienceWarsawPoland
  3. 3.Department of MathematicsNational University of SinaporeSingaporeSingapore
  4. 4.Shanghai Center for Mathematical SciencesFudan UniversityShanghaiChina

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