Greedy expansions and sets with deleted digits

Random Β-expansions with Deleted Digits

In this paper we define random β-expansions with digits taken from a given set of real numbers A = {a1, . . . , am}. We study a generalization of the greedy and lazy expansion and define a function K, that generates essentially all β-expansions with digits belonging to the set A. We show that K admits an invariant measure ν under which K is isomorphic to the uniform Bernoulli shift on A.

A Note on the Greedy Β-transformation with Deleted Digits

In this article we give the attractor and the absolutely continuous, invariant measure of the greedy and lazy β-transformation with deleted digits and show their ergodicity. We will consider two specific examples of greedy β-transformations of which the invariant measure can be explicitly calculated.

Β-expansions with Deleted Digits for Pisot Numbers Β

An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta

A Natural Extension for the Greedy Β-transformation with Three Deleted Digits Karma Dajani and Charlene Kalle

We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy βtransformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli.