Metric and Dimensional Properties of the Badly Approximable Set for Beta-Transformations

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

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Badly Approximable and Nonrecurrent Sets for Expanding Markov Maps

Yuan

2021

Fractals

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We consider the asymptotic behaviors of the orbits of an expanding Markov system [Formula: see text], and prove that the badly approximable set [Formula: see text] is of full Hausdorff dimension for any given sequence [Formula: see text]. Consequently, the Hansdorff dimension of the set of nonrecurrent points in the sense that [Formula: see text] is also full. The results can be applied to [Formula: see text]-transformations, Gauss maps and Lüroth maps, etc.

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Metric and Dimensional Properties of the Badly Approximable Set for Beta-Transformations

Cited by 3 publications

References 14 publications

The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

The Distal and Asymptotic Sets for Continued Fractions

Badly Approximable and Nonrecurrent Sets for Expanding Markov Maps

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