The doubling map with asymmetrical holes

Abstract

Let $0<a<b<1$ and let $T$ be the doubling map. Set $\mathcal{J}(a,b):=\{x\in [0,1]:T^{n}x\notin (a,b),n\ge 0\}$. In this paper we completely characterize the holes $(a,b)$ for which any of the following scenarios holds: {enumerate} $\mathcal{J}(a,b)$ contains a point $x\in (0,1)$; $\mathcal J(a,b)\cap [\de,1-\de]$ is infinite for any fixed $\de>0$; $\mathcal{J}(a,b)$ is uncountable of zero Hausdorff dimension; $\mathcal{J}(a,b)$ is of positive Hausdorff dimension. {enumerate} In particular, we show that (iv) is always the case if \[ b-a<\frac14\prod_{n=1}^\infty \bigl(1-2^{-2^n}\bigr)\approx 0.175092 \] and that this bound is sharp. As a corollary, we give a full description of first and second order critical holes introduced in \cite{SSC} 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.

26 pages, 3 figures

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