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Nice inducing schemes and the thermodynamics of rational maps
"... Abstract. We consider 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 (non-)existence of equilibrium states of f for the potential −tln |f ′ |, and the analy ..."
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Abstract. We consider 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 (non-)existence of equilibrium states of f for the potential −tln |f ′ |, and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism of a rational map that is “expanding away from critical points ” and that has arbitrarily small “nice sets ” with some additional properties. Our results apply in particular to non-renormalizable polynomials without indifferent periodic points, infinitely renormalizable quadratic polynomials with a priori bounds, real quadratic polynomials, topological Collet-Eckmann rational maps, and to backward contracting rational maps. As an application, for these maps we describe the dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the measure of
EXTREME VALUE LAWS IN DYNAMICAL SYSTEMS FOR NON-SMOOTH OBSERVATIONS
, 2010
"... We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely conti ..."
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Cited by 18 (5 self)
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We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.
A characterization of hyperbolic potentials of rational maps
- Bull. Braz. Math. Soc. (N.S
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SYMBOLIC DYNAMICS FOR SURFACE DIFFEOMORPHISMS WITH POSITIVE ENTROPY
"... 1.2. Symbolic dynamics 343 1.3. Markov partitions 344 ..."
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Cited by 6 (1 self)
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1.2. Symbolic dynamics 343 1.3. Markov partitions 344
TRANSIENCE IN DYNAMICAL SYSTEMS
"... Abstract. We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide examples of new kinds of transient behaviour. 1. ..."
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Cited by 5 (1 self)
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Abstract. We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide examples of new kinds of transient behaviour. 1.
4 EQUILIBRIUM STATES, PRESSURE AND ESCAPE FOR MULTIMODAL MAPS WITH HOLES
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"... Abstract. We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also ..."
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Abstract. We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also
