Equilibrium states for interval maps: the potential $-t\log |Df|$

Bruin, Henk; Todd, Michael J.

doi:10.24033/asens.2103

Cited by 51 publications

(111 citation statements)

References 46 publications

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“…The liftability problem for general piecewise invertible maps is addressed in detail in [PSZ08], see also [BT07a] where the problem of comparing equilibrium measures obtained by different inducing schemes is addresses for certain multimodal maps and for the potentials −t log |df (x)| with t close to 1, see also [BTar,BT07b].…”

Section: Ergodic Propertiesmentioning

confidence: 99%

“…Recently, Bruin and Todd [BT07a] applied the results presented here (see also [PS05]) to certain multimodal maps and prove the existence and uniqueness of equilibrium measure with respect to all invariant measures. They can deal with the liftability problem by building various inducing schemes and comparing the equilibrium measures associated to these schemes.…”

Section: Introductionmentioning

confidence: 99%

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Equilibrium measures for maps with inducing schemes

Pesin¹,

Senti²

2008

Journal of Modern Dynamics

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Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of real-valued potential functions ϕ on I, which admit a unique equilibrium measure µ ϕ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the central limit theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions ϕ t = −t log |df | with t ∈ (t 0 , t 1 ) for some t 0 < 1 < t 1 . In the particular case of Sunimodal maps we show that one can choose t 0 < 0 and that the class of measures under consideration consists of all invariant Borel probability measures.

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Section: Ergodic Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Equilibrium measures for maps with inducing schemes

Pesin¹,

Senti²

2008

Journal of Modern Dynamics

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“…There have been several recent results on the thermodynamic formalism of multimodal interval maps with non-flat critical points, by Bruin and Todd [BT08,BT09] and Pesin and Senti [PS08]. Besides [BT08, Theorem 6], that gives a complete description of the thermodynamic formalism for t close to 0 and for a general topologically transitive multimodal interval map with non-flat critical points, all the results that we are aware of are restricted to non-uniformly hyperbolic maps.…”

Section: Introductionmentioning

confidence: 99%

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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Abstract: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 ln f , 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 R, with at most two exceptions.

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“…We will not supply a proof of the above theorem, since it follows rather easily from [BrT,Theorem 5]. We will focus our attention on the following related theorem dealing with the potential −t log |Df |.…”

Section: Potentials With Summable Variationsmentioning

confidence: 99%

“…In Section 2.2, we explain the procedure of lifting measures µ to Hofbauer tower (Î,f ), which is behind the construction in this proposition. The full proof of Proposition 1 is given in [BrT,Theorem 3 and Lemma 2]. Proposition 1.…”

Section: Potentials With Summable Variationsmentioning

confidence: 99%

Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < h top (f)

Bruin

Todd

2008

Commun. Math. Phys.

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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 sup ϕ − 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.

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Equilibrium states for interval maps: the potential $-t\log |Df|$

Cited by 51 publications

References 46 publications

Equilibrium measures for maps with inducing schemes

Equilibrium measures for maps with inducing schemes

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < h top (f)

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