Non-Typical Points for $\beta $-Shifts
Volume 61 / 2013
Bulletin Polish Acad. Sci. Math. 61 (2013), 123-132
MSC: 37E05, 37C45, 11J83.
DOI: 10.4064/ba61-2-5
Abstract
We study sets of non-typical points under the map $f_\beta \mapsto \beta x $ mod 1 for non-integer $\beta$ and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition $\beta>1.541$ found in the above paper can be removed.
