## Random maps and random β-expansions in two dimensions

##### Summary

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.