Paper detail

On the structure of generic subshifts

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.