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arXiv · 1302.2486

The doubling map with asymmetrical holes

Paul Glendinning·Nikita Sidorov
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Abstract

Let $0 0$; $\mathcal J(a,b)$ 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.

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Paul Glendinning, Nikita Sidorov. 2013-02-11. The doubling map with asymmetrical holes. https://doi.org/10.1017/etds.2013.98

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