Decay of correlations in one-dimensional dynamics

Bruin, Henk; Luzzatto, Stefano; Strien, Sebastian van

doi:10.1016/s0012-9593(03)00025-9

Cited by 83 publications

(140 citation statements)

References 32 publications

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“…It would be interesting to see if our approach can be extended to unimodal maps with slower rates of mixing. See the recent study by Bruin, Luzzatto, and van Strien [12], based on Young's coupling argument [26].…”

Section: Setting and Statement Of Resultsmentioning

confidence: 99%

Almost sure rates of mixing for i.i.d. unimodal maps

Baladi¹,

Benedicks

Maume-Deschamps³

2002

Annales Scientifiques de l’École Normale Supérieure

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Section: Setting and Statement Of Resultsmentioning

confidence: 99%

Almost sure rates of mixing for i.i.d. unimodal maps

Baladi¹,

Benedicks

Maume-Deschamps³

2002

Annales Scientifiques de l’École Normale Supérieure

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“…By [AM03, Theorem A], [AM05], almost every non-regular parameter of a family of unimodal maps satisfying our hypothesis also satisfies the Collet-Eckmann condition. By [BLVS03], any unimodal map, satisfying the Collet-Eckmann condition, admits an inducing scheme satisfying Conditions (H1)-(H5). The result now follows from Theorem 6.3.…”

Section: Remark 71 the Negative Schwarzian Derivative Assumption Ismentioning

confidence: 99%

“…Multimodal maps. We follow [BLVS03]. Consider a C 3 interval or circle map f with a finite set C of critical points and no stable or neutral periodic point.…”

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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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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“…Thus, it seems that the method converges for a wider class of maps than those specified in Theorem 5.5 and Remark 5.7. In fact, recently much progress has been made in establishing the existence of invariant densities for general interval maps which are not expanding, see for example [1,5,6]. However, it remains to be explored whether the theory developed in these papers yields the tools in order to prove Ulam's conjecture or the convergence of the scheme developed in this paper.…”

Section: ±2mentioning

confidence: 98%

Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis

Aston

Junge²

2013

Math. Comp.

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We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree n which are defined by the requirement that they preserve the measure on n + 1 neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.

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Decay of correlations in one-dimensional dynamics

Abstract: 08/10/12 meb. Witing for arXiv response and then will ask ascademic if its been peer-reviewed. 10/10/12 meb. All OK to pub and peer reviewe

Cited by 83 publications

References 32 publications

Almost sure rates of mixing for i.i.d. unimodal maps

Almost sure rates of mixing for i.i.d. unimodal maps

Equilibrium measures for maps with inducing schemes

Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis

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