On the structure of generic subshifts

TL;DR

This paper studies the typical properties of subshifts under the Hausdorff metric, revealing that generic subshifts are often degenerate but can also exhibit complex behaviors like zero entropy and minimality in certain spaces.

Contribution

It characterizes the generic properties of subshifts in various spaces, showing degeneracy in some and rich dynamical features in others, including connections to Toeplitz subshifts and odometers.

Findings

Generic subshifts in certain spaces are degenerate, with points biasymptotic to periodic orbits.

In other spaces, generic subshifts are zero entropy, minimal, and uniquely ergodic.

Generic subshifts can have subexponential word complexity growth along subsequences.

Abstract

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 $\mathbf{S}$ of all subshifts, the space $\mathbf{S}^{\prime}$ of non-isolated subshifts, the closure $\overline{\mathbf{T}^{\prime}}$ of the infinite transitive subshifts, and the closure $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ 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 $\overline{\mathbf{T}^{\prime}}$ and $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ 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 $\overline{\mathbf{T}^{\prime}}$ is a regular Toeplitz subshift which is strongly orbit equivalent to the universal odometer.