Invariant measures, matching and the frequency of 0 for signed binary expansions

Abstract

We introduce a family of maps {Sη}η∈[1,2] defined on [−1,1] by Sη(x)=2x−dη, where d∈{−1,0,1}. Each map Sη generates signed binary expansions, i.e., binary expansions with digits −1, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter η. The transformations Sη have an ergodic invariant measure μη that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure μη([−12,12]) by the Ergodic Theorem. We show that the density of μη is a step function except for a set of parameters of zero Lebesgue measure and full Hausdorff dimension and we give a full description of the maximal parameter intervals on which the density has the same number of steps. We give an explicit formula for the frequency of the digit 0 in typical signed binary expansions on each of these parameter intervals and show that this frequency depends continuously on the parameter η. Moreover, it takes the value 23 only on the interval [65,32] and it is strictly less than 23 on the remainder of the parameter space.