Non-typical points for $\beta$-shifts

TL;DR

This paper investigates the properties of non-typical points in $eta$-shifts, showing that certain sets of points with divergent ergodic averages or orbit avoidance have large intersection properties, removing previous technical restrictions.

Contribution

It extends prior results on $eta$-shifts by removing the $eta>1.541$ condition and analyzing large intersection properties of non-typical point sets.

Findings

Sets of points avoiding certain Cantor sets have large intersection properties.

Sets of points with divergent ergodic averages also have large intersection properties.

The technical condition on $eta$ was removed, broadening applicability.

Abstract

We study sets of nontypical points under the map $f_\beta \mapsto \beta x $ mod 1, for non-integer $\beta$ and extend our results from [F\"arm, Persson, Schmeling, 2010] in several directions. In particular we prove that sets of points whose forward orbit avoid certain Cantor sets, and set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition $\beta>1.541$ found in [F\"arm, Persson, Schmeling, 2010] can be removed.