Interactions between ergodic theory and number theory: from β--expansions to the Sierpinski gasket dajani
Mathematical Institute
Utrecht University
The Netherlands
Abstract
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
\[x=\sum_{i=1}^{\infty\frac{a_i}{\beta^i}\] with \[\beta>1\] a real number, and \[a_i\in\{0,1,\cdots, \lceil \beta \rceil -1\}\]. 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.
