On eventually always hitting points

Ganotaki, Charis; Persson, Tomas

doi:10.1007/s00605-021-01616-7

Monatsh Math

2021

DOI: 10.1007/s00605-021-01616-7

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On eventually always hitting points

Charis Ganotaki

Tomas Persson

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2024

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Cited by 3 publications

References 16 publications

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Uniform Diophantine approximation and run-length function in continued fractions

TAN,

ZHOU

2024

Ergod. Th. Dynam. Sys.

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We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set $$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$ for a class of quadratic irrational numbers $y\in [0,1)$ . These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

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Uniform Diophantine approximation and run-length function in continued fractions

TAN,

ZHOU

2024

Ergod. Th. Dynam. Sys.

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Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Holland,

Kirsebom,

Kunde

et al. 2024

Trans. Amer. Math. Soc.

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Suppose ( f , X , μ ) (f,\mathcal {X},\mu ) is a measure preserving dynamical system and ϕ : X → R \phi \colon \mathcal {X}\to \mathbb {R} a measurable function. Consider the maximum process M n ≔ max { X 1 , … , X n } M_n≔\max \{X_1,\ldots ,X_n\} , where X i = ϕ ∘ f i − 1 X_i=\phi \circ f^{i-1} is a time series of observations on the system. Suppose that ( u n ) (u_n) is a non-decreasing sequence of real numbers, such that μ ( X 1 > u n ) → 0 \mu (X_1>u_n)\to 0 . For certain dynamical systems, we obtain a zero–one measure dichotomy for μ ( M n ≤ u n i.o. ) \mu (M_n\leq u_n\,\text {i.o.}) depending on the sequence u n u_n . Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences u n u_n . Our results on the permitted sequences u n u_n are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

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Metric Results for the Eventually Always Hitting Points and Level Sets in Subshift With Specification

WANG,

2024

Fractals

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We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity [Formula: see text] which describes the maximal length of string of zeros of the prefix among the first [Formula: see text] iterations of [Formula: see text] in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined.

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On eventually always hitting points

Cited by 3 publications

References 16 publications

Uniform Diophantine approximation and run-length function in continued fractions

Uniform Diophantine approximation and run-length function in continued fractions

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Metric Results for the Eventually Always Hitting Points and Level Sets in Subshift With Specification

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