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13
Equilibrium states for interval maps: the potential −tlog Df
"... Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use ..."
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Cited by 36 (9 self)
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Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of PerronFrobenius operators. We demonstrate that this ‘bounded range ’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context. 1.
Invariant manifolds and equilibrium states for nonuniformly hyperbolic horseshoes
, 2008
"... In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder cont ..."
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Cited by 8 (1 self)
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In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials. 1 Introduction and statement of results The goal of this paper is to study some dynamical and ergodic properties of a special class of nonuniformly hyperbolic horseshoes. The nonuniform hyperbolicity, for the systems studied here, comes as a consequence of the presence of a single orbit of homoclinic tangency inside the horseshoe, that is, accumulated by periodic orbits of it.
EXPANDING MEASURES
, 2008
"... We prove that any C 1+α transformation, possibly with a (nonflat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a ..."
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Cited by 8 (1 self)
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We prove that any C 1+α transformation, possibly with a (nonflat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller [37] for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding
Semicontinuity of entropy, existence of equilibrium states and continuity of . . .
, 2006
"... We obtain results on existence and continuity of physical measures through equilibrium states and apply these to nonuniformly expanding transformations on compact manifolds with nonflat critical sets, deducing sufficient conditions for continuity of physical measures and, for local diffeomorphis ..."
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Cited by 6 (1 self)
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We obtain results on existence and continuity of physical measures through equilibrium states and apply these to nonuniformly expanding transformations on compact manifolds with nonflat critical sets, deducing sufficient conditions for continuity of physical measures and, for local diffeomorphisms, necessary and sufficient conditions for stochastic stability. In particular we show that, under certain conditions, stochastically robust nonuniform expansion implies existence and continuous variation of physical measures.
ON THE DISTRIBUTION OF PERIODIC ORBITS
, 2009
"... Let f: M → M be a C 1+εmap on a smooth Riemannian manifold M and let Λ ⊂ M be a compact finvariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of fΛ. These results are noninvertible and, in particular, nonuniformly hyperbolic ..."
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Cited by 3 (0 self)
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Let f: M → M be a C 1+εmap on a smooth Riemannian manifold M and let Λ ⊂ M be a compact finvariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of fΛ. These results are noninvertible and, in particular, nonuniformly hyperbolic versions of wellknown results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop(ϕ) can be computed by the values of the potential ϕ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is wellapproximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.