Interactions between ergodic theory and number theory: from beta-expansions to the Sierpinski gasket

07.03.2023, 16:00  –  Online Seminar
Arbeitsgruppenseminar Analysis

Karma Dajani (Utrecht University)

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 β-expansions, which are representations of numbers of the form x = ∞∑ i=1 ai βi with β > 1 a real number, and ai ∈ {0, 1, · · · , dβe − 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.

This talk is part of the colloquium "Mathematics in Lebanon and beyond," of the partnership between the University of Potsdam and two universities in Beirut, the Lebanese University (UL) and the American University of Beirut (AUB). The colloquium will serve as a meeting place for scientific exchange, bringing together mathematicians and physicists from the Lebanese diaspora, friends of Lebanon from around the world, and researchers based in Lebanon.

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