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

We introduce a family of maps {Sη}η∈[1,2] defined on [-1, 1] by S_η (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 S_η have an ergodic invariant measure µ_η that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure µη (Formula Presented) by the ergodic theorem. We show that the density of µ_η 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 depe[ds c]ntinuously on the parameter η. Moreover, it takes the value (Formula Presented) only on the interval (Formula Presented) and it is strictly less than (Formula Presented) on the remainder of the parameter space.