Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

Sun, Yu; Cao, Chun-Yun

doi:10.1090/proc/13420

Cited by 6 publications

(9 citation statements)

References 24 publications

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“…• for any U w * v ω ∈ ∆ w,r , using (24), (25) and (26) and the definition of ζ ω,w and ζ ω,w , we can get…”

Section: Now We Can Get (See Lemma 52 In General Situation)mentioning

confidence: 99%

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Shrinking Target Problem for Random Ifs

Yuan

2018

Fractals

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We describe the shrinking target problem for random iterated function systems which semi-conjugate to a random subshifts of finite type. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to an more general targets and the sets based on the quantitative Poincaré recurrence properties.

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“…• for any U w * v ω ∈ ∆ w,r , using (24), (25) and (26) and the definition of ζ ω,w and ζ ω,w , we can get…”

Section: Now We Can Get (See Lemma 52 In General Situation)mentioning

confidence: 99%

“…First, since we have choose the smallest p, and θ(i+2, ω, p)−θ(i+2, ω, p−1) is smaller than θ(i + 2, ω, p)ε 3 i+1 . Second, use (24), (25) and (26). We can skip the details of the proof at first sight.…”

Section: Now We Can Get (See Lemma 52 In General Situation)mentioning

confidence: 99%

Shrinking Target Problem for Random Ifs

Yuan

2018

Fractals

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“…We now define our measure on K η which we will eventually see satisfies (23). For each level we distribute mass according to the following rules.…”

Section: Existence Of K ηmentioning

confidence: 99%

“…In particular, the fields of Number Theory, Dynamical Systems, and Fractal Geometry have all benefited significantly from these results. For further applications of the Mass Transference Principle and Theorem MTP* see [2,4,5,15,23].…”

Section: Introductionmentioning

confidence: 99%

“…Within Diophantine Approximation, it is reasonable to expect that Theorem 1 will enable the establishment of further Hausdorff measure statements relating to approximation on manifolds (see [3,4] and the references therein for more on this problem). Within Dynamical Systems, it is also reasonable to expect that Theorem 1 will allow one to study a wider class of shrinking target problems, in particular when our target is allowed to have a more exotic structure (see [20,22,23] and the references therein for more on this problem). We hope to return to these topics in a later work.…”

Section: Introductionmentioning

confidence: 99%

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A general mass transference principle

Allen

Baker

2019

Sel. Math. New Ser.

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The Mass Transference Principle proved by Beresnevich and Velani in 2006 is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for lim sup sets of balls in R n from a priori weaker Lebesgue measure statements. The Mass Transference Principle and subsequent generalisations have had a profound impact on several areas of mathematics, especially Diophantine Approximation. In the present paper, we prove a considerably more general form of the Mass Transference Principle which extends known results of this type in several distinct directions. In particular, we establish a mass transference principle for lim sup sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in R n . Furthermore, our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff g-measure statements to Hausdorff f -measure statements. We conclude the paper with an application of our mass transference principle to a general class of random lim sup sets.2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11K60.

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The entropy formula of shrinking target problem in nonautonomous dynamical systems

Lin,

Ma,

et al. 2024

Dynamical Systems

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Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

Cited by 6 publications

References 24 publications

Shrinking Target Problem for Random Ifs

Shrinking Target Problem for Random Ifs

A general mass transference principle

The entropy formula of shrinking target problem in nonautonomous dynamical systems

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