Expansions and Extensions : Ergodic, combinatorial and geometric properties of β-expansions with arbitrary digits

Abstract

Let beta be a real number bigger than 1 and A a finite set of arbitrary real numbers. A beta-expansion with digits in A of a real number x is an expression for x by an infinite sum of fractions with powers of beta in the denominators and elements from A in the numerators. Such expansions can be obtained by iterating certain dynamical systems. In this way we can obtain properties of the expansions by studying the dynamical systems and use ergodic theory as a tool. This is well-known in case the digit set contains the integers from 0 to the largest integer not exceeding beta. In this thesis we first define a dynamical system that generates beta-expansions with arbitrary digits for digit sets A of which the digits are not too far apart. The transformation from this system is called the greedy beta-transformation, since the expansions it produces have the largest digit possible in each step. We study various properties of this system. We establish the existence of a unique invariant measure, absolutely continuous wrt Lebesgue. We show that this measure is ergodic and give its support. For digit sets with three digits we construct a measure theoretical natural extension of the greedy dynamical system. Through this natural extension, we obtain an expression for the density of the invariant measure of the greedy transformation for all digit sets with three elements. From this construction we can deduce that the greedy transformation is exact and weakly Bernoulli. In general almost all numbers have infinitely many different beta-expansions with arbitrary digits. There is also not one, but a whole family of transformations that generate these expansions by iteration. For each of these transformations we have a characterization of the expansions it generates. The opposite of the greedy transformation is the lazy transformation, that produces expansions which have the smallest possible digit in each step. We show that each lazy beta-transformation with digits in a certain set is isomorphic to a greedy beta-transformation with digits in a different set. Therefore, also the lazy transformation has a unique, ergodic invariant measure, absolutely continuous wrt Lebesgue. Using the greedy and lazy transformations we can give a random transformation that for a given base beta and a given digit set A produces for each x all possible beta-expansions with digits in A. This random transformation is isomorphic to a uniform Bernoulli shift. Through this isomorphism we obtain an invariant measure for the random transformation. If the base beta is a Pisot unit and the digits are of a certain type, then we can define a natural extension of which the transformation is easy to define and the underlying space is a subset of a certain Euclidean space. Related to this natural extension is a multiple tiling of a Euclidean space of lower dimension. We show that this multiple tiling is a tiling if and only if the natural extension domain gives a tiling of the torus. We also give ways to find the covering degree of the multiple tiling.