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}(\[-\frac12,\frac12])$ 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 $\frac23$ only on the interval $\big\[ \frac65, \frac32\big]$ and it is strictly less than $\frac23$ on the remainder of the parameter space.