2017/03/18 by Karma Dajani, Dajani, Karma, Charlene Kalle +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #28D05 #37A05 #37A45 #37E05 #37E15 #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1703.06335
openalex publication_date 2017/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a parametrised family of maps S\η \η \∈ [1,2],\ncalled symmetric doubling maps, defined on [-1,1] by S_\η (x)=2x-d\η,\nwhere d\∈ -1,0,1 . Each map S_\η generates binary expansions with\ndigits -1, 0 and 1. We study the frequency of the digit 0 in typical\nexpansions as a function of the parameter \η. The transformations S_\η\nhave a natural ergodic invariant measure \μ_\η that is absolutely\ncontinuous with respect to Lebesgue measure. The frequency of the digit 0 is\nrelated to the measure \μ\η([- frac12, frac12]) by the Ergodic Theorem.\nWe show that the density of \μ_\η is piecewise smooth except for a set of\nparameters of zero Lebesgue measure and full Hausdorff dimension and give a\nfull description of the structure of the maximal parameter intervals on which\nthe density is piecewise smooth. We give an explicit formula for the frequency\nof the digit 0 in typical signed binary expansions on each of these parameter\nintervals and show that this frequency depends continuously on the parameter\n\η. Moreover, it takes the value frac23 only on the interval \[\n frac65, frac32\] and it is strictly less than frac23 on the remainder\nof the parameter space.\n