2025/05/14 by Stanisław Kasjan, Gerhard Keller, Kasjan, Stanisław +1
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Cellular Automata and Applications
paper · pdf · doi:10.48550/arxiv.2505.09253
For a finite alphabet A define by d1(x,y):=\limsupn→∞(1)/(2n+1)#\|i|≤ n: xi≠ yi\ the Besicovitch pseudo-metric on A\mathbb Z. It is well known that a closed subshift of A\mathbb Z has finite covering numbers w.r.t. d1 if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure μ on A\mathbb Z with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing tools to determine these covering numbers for various classes of \mathcal B-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures μs, all with the same discrete spectrum generated by a single number, but such that μs- and μs'-typical x resp. x'∈ A\mathbb Z have sufficiently different growth of covering numbers such that there are no finite block codes mapping x→ x' and x'→ x. (Indeed, both orbits have different amorphic complexities.)