Statistical properties of topological Collet–Eckmann maps

By F. Przytycki, Juan Rivera-Letelier. 2006

67 citations recorded by the source corpus.

Abstract We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet–Eckmann”. This condition is weaker than the “Collet–Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem. We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet–Eckmann condition: a rational map satisfies the Topological Collet–Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/576bf4b008625e62ff17680073d708158d257a21

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Citations in the corpus

Exponential estimates for plurisubharmonic functions and stochastic dynamics (2008, unknown): “C V ] Classification AMS 2000: 32U, 37F. Keywords: Monge-Ampère measure, Green measure, exponential mixing, central limit theorem, large deviations theorem.”
Nice Inducing Schemes and the Thermodynamics of Rational Maps (2008, methodology): “map. These results are mainly taken or deduced from results in [Prz99, PRLS03, PRLS04]. To prove Theorem A and its stronger version (Theorem A’ in §7) we make use of the inducing scheme introduced in [PRL07], which is developed in the more general setting considered here in §§3, 4. In §3.1 we recall”