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.

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