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

  • Karma Dajani

    Utrecht University, The Netherlands
  • Charlene Kalle

    Leiden University, The Netherlands
Invariant Measures, Matching and the Frequency of 0 for Signed Binary Expansions cover

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Abstract

We introduce a family of maps defined on by , where . Each map generates signed binary expansions, i.e., binary expansions with digits , 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter . The transformations have an ergodic invariant measure that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure by the Ergodic Theorem. We show that the density of 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 . Moreover, it takes the value only on the interval and it is strictly less than on the remainder of the parameter space.

Cite this article

Karma Dajani, Charlene Kalle, Invariant Measures, Matching and the Frequency of 0 for Signed Binary Expansions. Publ. Res. Inst. Math. Sci. 56 (2020), no. 4, pp. 701–742

DOI 10.4171/PRIMS/56-4-2