Data: 29/08/2018 - Quarta-feira
Local: IM-UFRJ, CT, sala C119
Palestrante: Dominik Kwietniak (Jagiellonian University in Krakow)
Resumo: We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base b-expansions, and their various generalisations: generalised Luroth series expansions and beta-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in [0,1). Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a so called pi^0_3-complete set, meaning that it is a countable intersection of F_sigma-sets, but it is not possible to write it as a countable union of G_delta-sets). We also solve Sharkovsky--Sivak problem on Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence sets of normal numbers under consideration are pi^0_3-complete. The talk is based on a joint work with: Dylan Airey, Steve Jackson, and Bill Mance.