2022/03/29 by Ronnie Pavlov, Pavlov, Ronnie, Scott Schmieding +1 · 2 citations
Computer Science · Materials Science · Mathematics · #37B05 #37B10 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2203.15159
openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate generic properties (i.e. properties corresponding to residual sets) in the space of subshifts with the Hausdorff metric. Our results deal with four spaces: the space S of all subshifts, the space S′ of non-isolated subshifts, the closure T′ of the infinite transitive subshifts, and the closure TT′ of the infinite totally transitive subshifts. In the first two settings, we prove that generic subshifts are fairly degenerate; for instance, all points in a generic subshift are biasymptotic to periodic orbits. In contrast, generic subshifts in the latter two spaces possess more interesting dynamical behavior. Notably, generic subshifts in both T′ and TT′ are zero entropy, minimal, uniquely ergodic, and have word complexity which realizes any possible subexponential growth rate along a subsequence. In addition, a generic subshift in T′ is a regular Toeplitz subshift which is strongly orbit equivalent to the universal odometer.