Communications in Mathematical Physics

, Volume 301, Issue 3, pp 661–707 | Cite as

Nice Inducing Schemes and the Thermodynamics of Rational Maps

  • Feliks PrzytyckiEmail author
  • Juan Rivera-Letelier
Open Access


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 \({-t {\rm ln} \left|f^{\prime} \right|}\) , 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 \({\mathbb{R}}\) , with at most two exceptions.


Lyapunov Exponent Periodic Point Pressure Function Conformal Measure Pointwise Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are grateful to Weixiao Shen and Daniel Smania for their help with references, Weixiao Shen again and Genadi Levin for their help with the non-renormalizable case and Henri Comman for his help with the large deviations results. We also thank Neil Dobbs, Godofredo Iommi, Jan Kiwi and Mariusz Urbanski for useful conversations and comments. Finally, we are grateful to Krzysztof Baranski for making Fig. 1 and the referee for his suggestions and comments that helped to clarify some of the concepts introduced in the paper.

Open Access

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Facultad de Matemáticas, Campus San JoaquínPontificia Universidad Católica de ChileSantiagoChile

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