The Lyapunov exponent of holomorphic maps

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 uni...

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Bibliographic Details
Published in:Inventiones mathematicae 2016-08, Vol.205 (2), p.363-382
Main Authors: Levin, Genadi, Przytycki, Feliks, Shen, Weixiao
Format: Article
Language:English
Subjects:
Critical point
Density
Dynamical systems
Lyapunov exponents
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
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Summary: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.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-015-0637-1