Simultaneously Non-dense Orbits Under Different Expanding Maps

TL;DR

This paper investigates the size and intersection properties of sets of points with orbits avoiding a given point under various expanding maps, including nonlinear and non-integer multiplication maps, revealing their large intersection characteristics.

Contribution

It establishes that such sets have full Hausdorff dimension and large intersection properties under a broad class of expanding maps, extending previous results to nonlinear and non-integer cases.

Findings

Sets have full Hausdorff dimension.

Countable intersections of these sets also have full Hausdorff dimension.

Results apply to nonlinear maps and multiplication by dense non-integer sets.

Abstract

Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. For maps like multiplication by an integer modulo 1, such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. that countable intersections of such sets also have full Hausdorff dimension. This result applies to maps like multiplication by integers modulo 1, but also to nonlinear maps like $x \mapsto 1/x$ modulo 1. We prove that the same thing holds for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.