Dynamical Systems Seminar: Invariant measures, matching and the frequency of 0 for sighted binary expansions

Abstract: Abstract: We introduce a parametrized family of maps S_\alpha, the so called symmetric doubling maps, defined on [-1,1] by S(x) = 2x-d\alpha, where d\in{-1,0,1} and \alpha\in [1,2]. Each map S_\alpha generates binary expansions with digits -1,0 and 1. The transformations S_\alpha have a natural invariant measure that is absolutely continuous with respect to Lebesgue measure. We show that for a set of parameters of full measure, the invariant measure of the symmetric doubling map is piecewise smooth. We also study the frequency of the digit 0 in typical expansions, as a function of the parameter. In particular, we investigate the self similarity displayed by the function \alpha\to\mu_\alpha([-1/2,1/2]) where \mu_\alpha([-1/2,1/2]) denotes the measure of the cylinder where digit zero occurs. This is joint work with Charlene Kalle.