On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

Hofbauer, Franz

doi:10.1007/bf02760884

Cited by 155 publications

(173 citation statements)

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“…This extends the results of Bruin and Keller [BK98] for the parameters under consideration. In particular, this also establishes the existence and uniqueness of the measure of maximal entropy by a different method than Hofbauer [Hof79,Hof81].…”

Section: Introductionmentioning

confidence: 94%

“…To prove that the equilibrium measure is unique with respect to all invariant measures in M(f, X i ), we remark that Theorem 7.4 holds for any piecewise continuous piecewise monotone interval map provided the basic elements of the inducing scheme accumulate to the critical point (see [PSZ08, Section 7] for details and more general results). This implies that the class M L (f, X i ) includes all f -invariant measures on X i of positive entropy ( [Hof79]). By [BLVS03, Theorem 1.2], every invariant measure has Lyapunov exponent bounded away from 0 and hence no invariant measure of zero entropy can be an equilibrium measure for the function ϕ t .…”

Section: Remark 71 the Negative Schwarzian Derivative Assumption Ismentioning

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: Introductionmentioning

confidence: 94%

Section: Remark 71 the Negative Schwarzian Derivative Assumption Ismentioning

confidence: 99%

Equilibrium measures for maps with inducing schemes

Pesin¹,

Senti²

2008

Journal of Modern Dynamics

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“…The classical papers are [15], [13] and [10]. We consider the piecewise monotone continuous maps of the following type.…”

Section: Preliminariesmentioning

confidence: 99%

“…If the coding map is injective, one can show that the map ϕ is continuous (see The strings u j and v j are called critical orbits and (see for instance [10])…”

Section: B Faller and C-e Pfistermentioning

confidence: 99%

A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

Faller¹,

Pfister²

2009

Ergod. Th. Dynam. Sys.

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Abstract. We consider the map T α,β (x) := βx + α mod 1, which admits a unique probability measure of maximal entropy µ α,β . For x ∈ [0, 1], we show that the orbit of x is µ α,β -normal for almost all (α, β) ∈ [0, 1) × (1, ∞) (Lebesgue measure). Nevertheless we construct analytic curves in [0, 1) × (1, ∞) along them the orbit of x = 0 is at most at one point µ α,β -normal. These curves are disjoint and they fill the set [0, 1) × (1, ∞). We also study the generalized β-transformations (in particular the tent map). We show that the critical orbit x = 1 is normal with respect to the measure of maximal entropy for almost all β. IntroductionIn this paper, we consider a dynamical system (X, d, T ) where (X, d) is a compact metric space endowed with its Borel σ-algebra B and T : X → X is a measurable map. Let C(X) denote the set of all continuous functions from X into R. The set M (X) of all Borel probability measures is equipped with the weak * -topology.is the subset of all T -invariant probability measures. For µ ∈ M (X, T ), let h(µ) denote the measure-theoretic entropy of µ. For all x ∈ X and n ≥ 1, the empirical measure of order n at x iswhere δ x is the Dirac mass at x. Let V T (x) ⊂ M (X, T ) denote the set of all cluster points of {E n (x)} n≥1 in the weak * -topology.Prepared using etds.cls [Version: 1999/07/21 v1.0]

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“…Hofbauer [10], using his Markov diagram, introduced previously in [7], has shown that all continuous piecewise monotonic maps have a weak specification property. This weak property implies the generic properties of invariant measures which we alluded to, but not the result about equilibrium states.…”

Section: Introductionmentioning

confidence: 99%

Specification on the interval

Buzzi¹

1997

Trans. Amer. Math. Soc.

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Abstract. We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the β-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure.Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133-134) (for which we give a proof).

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On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

Cited by 155 publications

References 1 publication

Equilibrium measures for maps with inducing schemes

Equilibrium measures for maps with inducing schemes

A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

Specification on the interval

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