221124 Karma Dajani

November 24, 2022

Title: beta-expansions in one and higher dimensions: an ergodic view (applied MI-talk)

Speaker: Karma Dajani

In this talk we give an exposition on one of the interactions between ergodic theory and number theory. We will concentrate on the concept of $beta$-expansions, which are representations of numbers of the form $x=displaystylesum_{i=1}^{infty}displaystylefrac{a_i}{beta^i}$ with $beta>1$ a real number, and $a_iin{0,1,cdots, lceil beta rceil -1}$.

We explain first simple concepts in ergodic theory that can help us understand the asymptotic behaviour of a typical expansion. What typical is depends on the stationary measure under consideration, and each such measure highlights a particular property of points in its support, i.e. the world that the measure sees. We extend the one-dimensional ideas to higher dimensions and show how they can be used to study multiple codings of points in an overlapping Sierpinski gasket.

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