2020/08/10 by Karma Dajani, Charlene Kalle, Marta Maggioni
Computer Science · Mathematics · #Algorithms and Data Compression #Binary number #Combinatorics #Discrete mathematics #Dynamical systems theory #Interval (graph theory) #Invariant (physics) #Lebesgue integration #Matching (statistics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Piecewise #Pure mathematics #Random compact set #Random element #Random variable #Statistics #math.DS #msc:11K55 #msc:37A05 #msc:37A45 #msc:37E05 #msc:60G10 #semigroups and automata theory
paper · pdf · doi:10.1088/1361-6544/abebc6
arxiv created 2020/08/10 · openalex publication_date 2021/06/01 · arxiv updated 2021/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [−1, 1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated signed binary expansions never exceeds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> .