Alsedà–Misiurewicz systems with place-dependent probabilities*

Czudek, Klaudiusz

doi:10.1088/1361-6544/aba094

Cited by 5 publications

(3 citation statements)

References 35 publications

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“…The dynamics of AM systems and related ones has already gained some interest in recent years, being studied in e.g. [1][2][3][4][5][6]25]. Within the class of uniformly contracting iterated function systems, piecewise linear maps and the dimension of their attractors were recently studied in [22].…”

Section: Introductionmentioning

confidence: 99%

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

Barański

Śpiewak

2023

Ergod. Th. Dynam. Sys.

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We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà–Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than $1$ . In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat.58(suppl.) (2014), 15–36].

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Section: Introductionmentioning

confidence: 99%

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

Barański

Śpiewak

2023

Ergod. Th. Dynam. Sys.

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“…One is the so-called place-dependent measures, which were studied by Fan and Lau [12], Hu, Lau and Wang [16], Jaroszewska [18], Jaroszewska and Rams [19], Kwiecińska and W. Słomczyński [22], Czudek [8] and others. Let {p i } i∈A be a family of Hölder continuous maps p i : I → [0, 1] such that i∈A p i ≡ 1.…”

Section: Introductionmentioning

confidence: 99%

Typical absolute continuity for classes of dynamically defined measures

Bárány¹,

Simon²,

Solomyak³

et al. 2021

Preprint

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We consider one-parameter families of smooth uniformly hyperbolic and contractive iterated function systems {f λ j } λ∈U on the real line. Given a family of parameter dependent measures {µ λ } λ∈U on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps Π λ corresponding to the IFS. The main novelty of our work is that the measures µ λ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure (Π λ ) * µ λ , where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if µ λ are Gibbs measures for a family of Hölder continuous potentials φ λ , with a Hölder continuous dependence on the parameter and {Π λ } satisfy the transversality condition, then the projected measure (Π λ ) * µ λ is absolutely continuous for Lebesgue a.e. parameter λ, such that the ratio of entropy over the Lyapunov exponent is strictly greater than 1. We deduce it from a more general almost sure lower bound on the Sobolev dimension of (Π λ ) * µ λ for families of measures with regular enough dependence on the parameter. Under less restrictive regularity assumptions, we also obtain an almost sure formula for the Hausdorff dimension of projected measures. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).

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“…The dynamics of AM-systems and related ones has already gained some interest in recent years, being studied in e.g. [1,2,3,4,5,6,25]. In the class of uniformly contracting iterated function systems, the piecewise linear maps and the dimension of their attractors were studied recently in [22].…”

Section: Introductionmentioning

confidence: 99%

Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

Barański

Śpiewak

2019

J Dyn Diff Equat

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We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.

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Alsedà–Misiurewicz systems with place-dependent probabilities*

Cited by 5 publications

References 35 publications

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

Typical absolute continuity for classes of dynamically defined measures

Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

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