Random maps and random β-expansions in two dimensions

Abstract

In this thesis, we show a method to dynamically generate expansions of numbers in an arbitrary base β > 1 using maps called the lazy and greedy maps. We introduce concepts such as the Frobenius-Perron operator, which we then use to find the unique absolutely continuous invariant measure for the greedy map in the case where the base is equal to the golden mean. We provide some intuition about most concepts and results as they are introduced. We introduce a two-dimensional random map K which simultaneously generates two random β-expansions and show that it can be essentially identified with the left shift. We then find an invariant measure of maximal entropy for K. We introduce a skew product transformation based on K and prove that there exists an absolutely continuous invariant measure. We prove some properties of digit sequences that give a simultaneous expansion of two numbers x and y in bases β_1 and β_2 . Finally, we introduce a random map G that generates these sequences, after which we show that it can be essentially identified with the left shift.