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arXiv · 1204.1920

Supercritical holes for the doubling map

Nikita Sidorov
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Abstract

For a map $S:X\to X$ and an open connected set ($=$ a hole) $H\subset X$ we define $\mathcal J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids $H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such that $\bar{H_0}\subset H$ the set $\mathcal J_H(S)$ is either empty or contains only fixed points of $S$; (ii) for any hole $H$ such that $\barH\subset H_0$ the Hausdorff dimension of $\mathcal J_H(S)$ is positive. The purpose of this note to completely characterize all supercritical holes for the doubling map $Tx=2x\bmod1$.

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Nikita Sidorov. 2013-10-17. Supercritical holes for the doubling map. https://doi.org/10.1007/s10474-014-0403-7

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