Dimension of ergodic measures projected onto self‐similar sets with overlaps

Jordan, Thomas; Rapaport, Ariel

doi:10.1112/plms.12337

Cited by 11 publications

(9 citation statements)

References 11 publications

(45 reference statements)

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“…Shmerkin [14] proved an analogous result for the L q dimension of self-similar measures in 2019. Using similar ideas, Jordan and Rapaport [8] extended Hochman's result mentioned above from self-similar measures to the projections of ergodic measures. Using this result, Prokaj and Simon [9,Corollary 7.2] proved that ESC also implies that formula (1.11) holds for graph-directed self-similar attractors on the line.…”

Section: Terminologymentioning

confidence: 87%

“…We point out that we can select a suitable subsystem S F ⊂ S N F and form a graph-directed self-similar IFS from the functions of S F such that the attractor of this graph-directed system and Λ F coincide. Then we use the Jordan Rapaport theorem [12] to compute the dimension of this graph-directed attractor which, as we just mentioned, is the same as Λ F . In this way we obtain that dim H Λ F = s F .…”

Section: Introducing Continuous Piecewise Linear Ifssmentioning

confidence: 99%

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Piecewise linear iterated function systems on the line of overlapping construction

Prokaj

Simon

2021

Nonlinearity

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In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

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Section: Terminologymentioning

confidence: 87%

Section: Introducing Continuous Piecewise Linear Ifssmentioning

confidence: 99%

Piecewise linear iterated function systems on the line of overlapping construction

Prokaj

Simon

2021

Nonlinearity

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“…To understand the general structure of the measure or the self-similar set K , one often considers basic dimensional quantities such as the Hausdorff dimension and analogous statements for measures, or other notions of dimension. Computing these values can be highly non-trivial for general iterated function systems of similarities and there is significant literature on this matter (see, for example, [2, 12, 16, 23, 26, 29, 32, 36]). In this paper, we focus on a more fine-grained notion of dimension known as the local dimension.…”

Section: Introductionmentioning

confidence: 99%

Geometric and combinatorial properties of self-similar multifractal measures

Rutar

2022

Ergod. Th. Dynam. Sys.

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For any self-similar measure $\mu $ in $\mathbb {R}$ , we show that the distribution of $\mu $ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of $\mu $ to certain compact subsets of $\mathbb {R}$ , determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some $q\in \mathbb {R}$ , there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

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“…Further progress on absolute continuity of Bernoulli convolutions was obtained by Varjú [48]. Jordan and Rapaport [19] showed that the dimension of the push-forward measure of any ergodic shift-invariant measure equals to the entropy over Lyapunov exponent ratio under the exponential separation condition. However, such strong results are unknown in the case when the IFS consists of general conformal maps.…”

Section: Introductionmentioning

confidence: 99%

“…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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Dimension of ergodic measures projected onto self‐similar sets with overlaps

Cited by 11 publications

References 11 publications

Piecewise linear iterated function systems on the line of overlapping construction

Piecewise linear iterated function systems on the line of overlapping construction

Geometric and combinatorial properties of self-similar multifractal measures

Typical absolute continuity for classes of dynamically defined measures

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