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A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN $\unicode[STIX]{x1D6FD}$ -DYNAMICAL SYSTEMS

Published online by Cambridge University Press:  16 May 2016

Michael Coons
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email michael.coons@newcastle.edu.au
Mumtaz Hussain
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email mumtaz.hussain@newcastle.edu.au, drhussainmumtaz@gmail.com
Bao-Wei Wang
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China email bwei_wang@hust.edu.cn

Abstract

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$
and
$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$
where $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$ is a positive function such that $\unicode[STIX]{x1D6F9}(n)\rightarrow 0$ as $n\rightarrow \infty$ . In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

Type
Research Article
Copyright
Copyright © University College London 2016 

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A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN $\unicode[STIX]{x1D6FD}$ -DYNAMICAL SYSTEMS
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