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The doubling map with asymmetrical holes

Published online by Cambridge University Press:  14 November 2013

PAUL GLENDINNING  and
NIKITA SIDOROV
PAUL GLENDINNING
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK email p.a.glendinning@manchester.ac.uksidorov@manchester.ac.uk
NIKITA SIDOROV
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK email p.a.glendinning@manchester.ac.uksidorov@manchester.ac.uk
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Abstract

Let $0\lt a\lt b\lt 1$ and let $T$ be the doubling map. Set $ \mathcal{J} (a, b): = \{ x\in [0, 1] : {T}^{n} x\not\in (a, b), n\geq 0\} $. In this paper we completely characterize the holes $(a, b)$ for which any of the following scenarios hold: (i) $ \mathcal{J} (a, b)$ contains a point $x\in (0, 1)$; (ii) $ \mathcal{J} (a, b)\cap [\delta , 1- \delta ] $ is infinite for any fixed $\delta \gt 0$; (iii) $ \mathcal{J} (a, b)$ is uncountable of zero Hausdorff dimension; (iv) $ \mathcal{J} (a, b)$ is of positive Hausdorff dimension. In particular, we show that (iv) is always the case if

$$\begin{eqnarray*}b- a\lt \frac{1}{4} { \mathop{\prod }\nolimits}_{n= 1}^{\infty } (1- {2}^{- {2}^{n} } )\approx 0. 175\hspace{0.167em} 092\end{eqnarray*}$$
and that this bound is sharp. As a corollary, we give a full description of first- and second-order critical holes introduced by N. Sidorov [Supercritical holes for the doubling map. Preprint, see http://arxiv.org/abs/1204.1920] for the doubling map. Furthermore, we show that our model yields a continuum of ‘routes to chaos’ via arbitrary sequences of products of natural numbers, thus generalizing the standard route to chaos via period doubling.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1208 - 1228
Copyright
© Cambridge University Press, 2013 

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