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A Survey on the Dimension Theory in Dynamical Diophantine Approximation

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Recent Developments in Fractals and Related Fields (FARF3 2015)

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Abstract

Dynamical Diophantine approximation studies the quantitative properties of the distribution of the orbits in a dynamical system. More precisely, it focuses on the size of dynamically defined limsup sets in the sense of measure and dimension. This quantitative study is motivated by the qualitative nature of the density of the orbits and the connections with the classic Diophantine approximation. In this survey, we collect some recent progress on the dimension theory in dynamical Diophantine approximation. This includes the systems of rational maps on its Julia set, linear map on the torus, beta dynamical system, continued fractions as well as conformal iterated function systems.

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Acknowledgements

This work is supported by NSFC (grant no. 11225101, 11471130) and NCET-13-0236.

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  1. Huazhong University of Science and Technology, Wuhan, 430074, China

    Baowei Wang & Jun Wu

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  1. Baowei Wang
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  2. Jun Wu
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Correspondence to Jun Wu .

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  1. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Julien Barral

  2. Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est, UPEMLV, UPEC, CNRS, Créteil, France

    Stéphane Seuret

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Wang, B., Wu, J. (2017). A Survey on the Dimension Theory in Dynamical Diophantine Approximation. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. FARF3 2015. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-57805-7_12

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