Communications in Mathematical Physics

, Volume 149, Issue 1, pp 31–69 | Cite as

Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps

  • Gerhard Keller
  • Tomasz Nowicki
Article

Abstract

We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DT n (Tc)|≧ c n for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form\(S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)\), and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.

This is achieved by proving quasicompactness of the Perron Frobenius operator and of similar transfer operators for the Markov extension ofT and relating the isolated eigenvalues of these operators to the poles of the corresponding Ruelle zeta functions.

Keywords

Periodic Orbit Central Limit Central Limit Theorem Zeta Function Periodic Point 

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Gerhard Keller
    • 1
  • Tomasz Nowicki
    • 2
  1. 1.Department of MathematicsUniversity of ErlangenErlangenFRG
  2. 2.University of WarsawPoland

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