On the $r$-continued fraction expansions of reals
speaker: Erblin Mehmetaj (Georgetown University)abstract: In this talk, we will consider $r$-continued fraction expansions, which are a type of generalized continued fraction expansions derived via the iterations of the map of the interval $Tr(x) = rx\ (\textrm{mod}\ 1),$ where $r$ is a non-integer real number greater than $1.$ In 1960, Parry proved that a sequence is admissible as the $\beta$-expansion of a real number if and only if all of the shifts of the sequence are dominated lexicographically by the sequence obtained from the $\beta$-expansion of $1.$ We prove a similar result for the $r$-continued fraction expansions. We prove that a sequence is admissible, that is, it comes from the $r$-continued fraction map, if and only if all of its shifts are alternating-lexicographically less than the sequence obtained from the $r$-continued fraction expansion of $1.$ Finally, we show that the $r$-continued fraction map admits an ergodic absolutely continuous invariant measure that is equivalent to Lebesgue measure.
timetable:
Tue 11 Dec, 15:00 - 15:45, Aula Dini
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