On the lack of equidistribution on fat Cantor sets

Dia 2023-12-15 14:30:00-03:00
Hora 2023-12-15 14:30:00-03:00
LugarSalón de seminarios del IMERL

On the lack of equidistribution on fat Cantor sets

Gabriel Fuhrmann (Durham University)

Given an irrational rotation, it is straightforward to see that for every Cantor set C, there is a dense set of points whose orbit doesn't intersect C. On the other hand, if C is a fat Cantor set (that is, of positive Lebesgue measure), almost every point visits C with a frequency equal to the measure of C. But what other frequencies of visits to C may occur? In the words of a recent MathOverflow post [1], what is the Birkhoff spectrum of fat Cantor sets?

We give a first answer to this question by showing that every irrational rotation allows for certain fat Cantor sets C whose Birkhoff spectrum is maximal, that is, equal to the interval [0,Leb(C)]. In this talk, I will focus on discussing some of the basic tools behind this result and extensions of it.

[1] D. Kwietniak, Possible Birkhoff spectra for irrational rotations, MathOverflow (2020), https://mathoverflow.net/q/355860 (version: 2020-03-27).