Invariant Measures, Matching and the Frequency of 0 for Signed Binary Expansions

Abstract

We introduce a family of maps $\{S_{\eta }{\}}_{\eta \in [1,2]}$ defined on $[-1,1]$ by $S_{\eta }(x)=2x-d\eta$, where $d\in \{-1,0,1\}$. Each map $S_{\eta }$ 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 $\eta$. The transformations $S_{\eta }$ have an ergodic invariant measure ${\mu }_{\eta }$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure ${\mu }_{\eta }([-\frac{1}{2},\frac{1}{2}])$ by the Ergodic Theorem. We show that the density of ${\mu }_{\eta }$ 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 $\eta$. Moreover, it takes the value $\frac{2}{3}$ only on the interval $[\frac{6}{5},\frac{3}{2}]$ and it is strictly less than $\frac{2}{3}$ on the remainder of the parameter space.