On the dimension spectrum of infinite subsystems of continued fractions

Chousionis, Vasileios; Leykekhman, Dmitriy; Urbański, Mariusz

doi:10.1090/tran/7984

Cited by 16 publications

(25 citation statements)

References 29 publications

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“…It follows from [30,Theorem 3.15] that dim H F I = h; the Hausdorff dimension of such limit sets has been studied in [5,21,23,26,29] Case 2: Assume 1 ∈ A. Since the map x → 1/(1 + x) is bi-Lipschitz on [0, 1] and dim θ is stable under bi-Lipschitz maps,…”

Section: Continued Fraction Setsmentioning

confidence: 99%

Intermediate dimensions of infinitely generated attractors

Amlan¹,

Fraser²

2021

Preprint

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We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter θ ∈ [0, 1] which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbański concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries.We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a 'generic' infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where 'generic' can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.

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Section: Continued Fraction Setsmentioning

confidence: 99%

Intermediate dimensions of infinitely generated attractors

Amlan¹,

Fraser²

2021

Preprint

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“…We will return to this in §4.9, where we verify and improve estimates of Hensley [17] and Jenkinson [21] for some iterated function schemes and estimates of Chousionis at al. [6] for some countable iterated function systems.…”

Section: Application Iii: Schottky Group Limit Setsmentioning

confidence: 99%

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Pollicott¹,

Vytnova²

2020

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In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov 1 spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [31]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain-Kontorovich [2], Huang [18] and Kan [27]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [36].In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement. * The first author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1 the second author is partly supported by EPSRC grant EP/T001674/1.1 Markov's name will appear in this article in two contexts, namely those of Markov spectra in number theory and the Markov condition from probability theory. Since the both notions are associated with the same person (A.A. Markov, 1856(A.A. Markov, -1922, we have chosen to use the same spelling, despite the conventions often used in these different areas.

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“…In particular we will get an almost cost free, effective, lower estimate for the Hausdorff dimension of such limit sets. More about conformal graph directed Markov systems can be found in many papers and books; we bring up here some of them: [MU3]- [MU6], [MPU], [MSzU], [U4], [U5], [CTU], [CLU1], [CLU2], and [CU].…”

Section: Graph Directed Markov Systemsmentioning

confidence: 99%

“…In particular we get an almost cost free, effective, lower estimate for the Hausdorff dimension of such limit sets. More about conformal graph directed Markov systems can be found in many papers and books; we bring up here some of them: [MU3]- [MU6], [MPU], [MSzU], [U4], [U5], [CTU], [CLU1], [CLU2], and [CU]. Afterwards, in Part 3 and especially in Part 4, we apply the techniques developed here to get a quite good, explicit estimate from below of Hausdorff dimensions of Julia sets of all elliptic functions and to explore stochastic properties of invariant versions of conformal measures for parabolic and subexpanding elliptic functions.…”

mentioning

confidence: 99%

Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

Kotus,

Urbanski

2020

Preprint

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On the dimension spectrum of infinite subsystems of continued fractions

Cited by 16 publications

References 29 publications

Intermediate dimensions of infinitely generated attractors

Intermediate dimensions of infinitely generated attractors

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

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