Supercritical 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 $\overline{H_0}\subset H$ 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 $Tx=2x\mathrm{mod}1$.

This is a new version, where a full characterization of supercritical holes for the doubling map is obtained

Cited by in corpus (5)

• The $\beta$-transformation with a hole
• On Transfer Operators and Maps with Random Holes
• Subshifts of finite type and matching for intermediate $\beta$-transformations
• Topological expansive Lorenz maps with a hole at critical point
• The baker's map with a convex hole