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Non-Typical Points for β-Shifts

David Färm;  Tomas Persson — 2013

Bulletin of the Polish Academy of Sciences. Mathematics

We study sets of non-typical points under the map f β ↦ β x mod 1 for non-integer β 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 β > 1.541 found in the above paper can be removed.

Dimension of countable intersections of some sets arising in expansions in non-integer bases

David Färm;  Tomas Persson;  Jörg Schmeling — 2010

Fundamenta Mathematicae

We consider expansions of real numbers in non-integer bases. These expansions are generated by β-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.

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