TY - JOUR

T1 - On the structure of generic subshifts

AU - Pavlov, Ronnie

AU - Schmieding, Scott

N1 - Funding Information:
Ronnie Pavlov gratefully acknowledges the support of a Simons Foundation Collaboration Grant.
Publisher Copyright:
© 2023 IOP Publishing Ltd & London Mathematical Society

PY - 2023/9/1

Y1 - 2023/9/1

N2 - 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 T T ′ ‾ 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 T T ′ ‾ 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.

AB - 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 T T ′ ‾ 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 T T ′ ‾ 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.

UR - http://www.scopus.com/inward/record.url?scp=85167920989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85167920989&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/acea25

DO - 10.1088/1361-6544/acea25

M3 - Article

AN - SCOPUS:85167920989

SN - 0951-7715

VL - 36

SP - 4904

EP - 4953

JO - Nonlinearity

JF - Nonlinearity

IS - 9

ER -