Open maps: small and large holes with unusual properties

TL;DR

This paper demonstrates the existence of small, overlapping unions of cylinders with unusual properties in subshifts and applies these findings to hyperbolic algebraic automorphisms of tori.

Contribution

It introduces novel constructions of overlapping unions of cylinders with specific orbit and entropy properties in subshifts.

Findings

Existence of arbitrarily small unions intersecting all orbits.

Construction of unions whose survivor sets contain any given subshift.

Application to hyperbolic algebraic automorphisms of tori.

Abstract

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.