Inhomogeneous random coverings of topological Markov shifts

Seuret, Stéphane

doi:10.1017/s0305004117000512

Cited by 11 publications

(11 citation statements)

References 21 publications

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“…Both of these conditions are often satisfied if

μ

is a Gibbs measure, for instance in the setting of Seuret's paper. Moreover, in that case it is well known that

G_{μ} false(s false) = F_{μ} false(s false) = {normalΨ}_{μ} false(s false) : = error false{x; {\underset{̲}{d}}_{μ} (x) = s false}

for

s ⩽ s_{*}

, where

s_{*}

is the value of

s

that maximises

{normalΨ}_{μ} false(s false)

, and

F_{μ} false(s false) ⩾ G_{μ} false(s false)

for

s ⩾ s_{*}

(see [, Propositions 1 and 2]). Thus in this case, the upper bound in Proposition is equal to

{\bar{F}}_{μ}

, and

f_{μ} false(α false) = {\overset{F}{true}}_{μ} false(1 / α false)

for all

α > 0

, that is, our bounds specialise to Seuret's result (see Figure ).…”

Section: Resultsmentioning

confidence: 99%

“…Proof We will use the same method to prove the statement as was used by Seuret [, Proposition 5]. Let

{scriptD}_{n} = false{[k_{1} 2^{- n}, false(k_{1} + 1 false) 2^{- n}) \times \dots \times [k_{d} 2^{- n}, false(k_{d} + 1 false) 2^{- n}); k_{j} \in boldZ false},

and given a point

x

and an integer

n

, let

D_{n} false(x false)

be the unique

D \in {scriptD}_{n}

such that

x \in D

.…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…We are interested in the almost sure value of the Hausdorff dimension of the set of points covered infinitely often by balls of radius

n^{- α}

, when the centres are chosen independently and distributed according to the measure

μ

. This problem was also considered by Seuret , but he restricted to the case when

μ

is a Gibbs measure invariant under an expanding Markov map. Because of this assumption, it is possible to use techniques from thermodynamic formalism, and with such tools Seuret gave the almost sure value of the Hausdorff dimension in terms of multifractal spectra of the measure

μ

.…”

Section: Introductionmentioning

confidence: 99%

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Hausdorff dimension of random limsup sets

Ekström

Persson

2018

J. London Math. Soc.

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We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in R d whose centres are independent, identically distributed random variables. The formulas obtained involve the rate of decrease of the radii of the balls and multifractal properties of the measure according to which the balls are distributed, and generalise formulas that are known to hold for particular classes of measures.

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“…Both of these conditions are often satisfied if

μ

is a Gibbs measure, for instance in the setting of Seuret's paper. Moreover, in that case it is well known that

G_{μ} false(s false) = F_{μ} false(s false) = {normalΨ}_{μ} false(s false) : = error false{x; {\underset{̲}{d}}_{μ} (x) = s false}

for

s ⩽ s_{*}

, where

s_{*}

is the value of

s

that maximises

{normalΨ}_{μ} false(s false)

, and

F_{μ} false(s false) ⩾ G_{μ} false(s false)

for

s ⩾ s_{*}

(see [, Propositions 1 and 2]). Thus in this case, the upper bound in Proposition is equal to

{\bar{F}}_{μ}

, and

f_{μ} false(α false) = {\overset{F}{true}}_{μ} false(1 / α false)

for all

α > 0

, that is, our bounds specialise to Seuret's result (see Figure ).…”

Section: Resultsmentioning

confidence: 99%

“…Proof We will use the same method to prove the statement as was used by Seuret [, Proposition 5]. Let

{scriptD}_{n} = false{[k_{1} 2^{- n}, false(k_{1} + 1 false) 2^{- n}) \times \dots \times [k_{d} 2^{- n}, false(k_{d} + 1 false) 2^{- n}); k_{j} \in boldZ false},

and given a point

x

and an integer

n

, let

D_{n} false(x false)

be the unique

D \in {scriptD}_{n}

such that

x \in D

.…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…We are interested in the almost sure value of the Hausdorff dimension of the set of points covered infinitely often by balls of radius

n^{- α}

, when the centres are chosen independently and distributed according to the measure

μ

. This problem was also considered by Seuret , but he restricted to the case when

μ

μ

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hausdorff dimension of random limsup sets

Ekström

Persson

2018

J. London Math. Soc.

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show abstract

“…This brings us to the randomly generated limsup-sets or random covers, studied in [FW04, Dur10, JJK + 14, Per15,FJJS18,Seu18,EP18]. In this paper we will build on ideas from [Per15] to develop a new method for analysing the Hausdorff dimension of limsup-sets.…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Persson¹

2019

Annales Henri Lebesgue

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Generalising a construction of Falconer, we consider classes of G -subsets of R d with the property that sets belonging to the class have large Hausdor dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.As applications of this theory, we calculate, or at least estimate, the Hausdor dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for ! 1, dimHf y : jT n a (x) yj < n innitely often g = 1 ;for almost every x P [1 a; 1], where Ta is a quadratic map with a in a set of parameters described by Benedicks and Carleson.

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“…Dimensional properties of random limsup sets have been actively studied, see for example [12,15,17,18,20,23,24,26,27,29]. Combining the results of these papers, the almost sure value of dimension of random limsup sets is known in the following cases:…”

Section: Introductionmentioning

confidence: 99%