The doubling map with asymmetrical holes

TL;DR

This paper characterizes the properties of holes in the doubling map where orbits avoid the hole, revealing sharp bounds for Hausdorff dimension and connecting to chaos routes.

Contribution

It provides a complete characterization of holes in the doubling map affecting orbit structure and Hausdorff dimension, including sharp bounds and generalizations of chaos routes.

Findings

Hausdorff dimension positive if hole size exceeds approximately 0.175

Sharp bounds for when the set contains points or is uncountable

Generalization of chaos routes via product sequences

Abstract

Let $0<a<b<1$ and let $T$ be the doubling map. Set $\mathcal J(a,b):=\{x\in[0,1] : T^nx\notin (a,b), n\ge0\}$. 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.