Supercritical holes for the doubling map

TL;DR

This paper characterizes all supercritical holes for the doubling map, which are special open sets where the set of points avoiding the hole exhibits specific fixed point and positive dimension properties.

Contribution

It provides a complete characterization of supercritical holes for the doubling map, clarifying their structure and properties.

Findings

Identifies conditions for supercritical holes in the doubling map.

Shows the relationship between hole size and the orbit avoidance set.

Provides a classification of all such holes for the doubling map.

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 $Tx=2x\bmod1$.