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The $\unicode[STIX]{x1D6FD}$-transformation with a hole at 0

Part of: Elementary number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Low-dimensional dynamical systems Topological dynamics Real functions Discrete mathematics in relation to computer science Measure-theoretic ergodic theory

Published online by Cambridge University Press:  25 March 2019

CHARLENE KALLE ,
DERONG KONG ,
NIELS LANGEVELD  and
WENXIA LI
CHARLENE KALLE
Affiliation:
Mathematical Institute, University of Leiden, PO Box 9512, 2300RA Leiden, The Netherlands email [email protected], [email protected]
DERONG KONG
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing401331, PR China email [email protected]
NIELS LANGEVELD
Affiliation:
Mathematical Institute, University of Leiden, PO Box 9512, 2300RA Leiden, The Netherlands email [email protected], [email protected]
WENXIA LI
Affiliation:
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai200062, PR China email [email protected]
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Abstract

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For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by

$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$
In this paper we characterize the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ of all parameters $t\in [0,1)$ for which the set-valued function $t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that $E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all $\unicode[STIX]{x1D6FD}\in (1,2)$. We prove that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$ the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\unicode[STIX]{x1D6FD}\in (1,2)$ for which $E_{\unicode[STIX]{x1D6FD}}$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_{2}$, the bifurcation set of the doubling map. Finally, we give for each $\unicode[STIX]{x1D6FD}\in (1,2)$ a lower and an upper bound for the value $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of $K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if $t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$. We show that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}\in (1,2)$.

Keywords

dimension theorylow-dimensional dynamicssymbolic dynamics

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 26A30: Singular functions, Cantor functions, functions with other special properties 37B10: Symbolic dynamics 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37E15: Combinatorial dynamics (types of periodic orbits)
Secondary: 11A63: Radix representation; digital problems 68R15: Combinatorics on words 28D05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2482 - 2514
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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