Nice Inducing Schemes and the Thermodynamics of Rational Maps

We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential , and the analytic dependence on t of the cor...

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Bibliographic Details
Published in:Communications in mathematical physics 2011-02, Vol.301 (3), p.661-707
Main Authors: Przytycki, Feliks, Rivera-Letelier, Juan
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Operator theory
Ordinary differential equations
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Theoretical
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in , with at most two exceptions.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-010-1158-9