TY - JOUR

T1 - The mapping class group of a minimal subshift

AU - Schmieding, Scott

AU - Yang, Kitty

N1 - Funding Information:
This work was supported in part by the National Science Foundation grant ‘RTG: Analysis on Manifolds’ at Northwestern University.
Publisher Copyright:
© Instytut Matematyczny PAN, 2021.

PY - 2021

Y1 - 2021

N2 - For a homeomorphism T: X → X of a Cantor set X, the mapping class group M(T ) is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension ΣT X. The group M(T ) can be interpreted as the symmetry group of the system (X, T ) with respect to the flow equivalence relation. We study M(T ), focusing on the case when (X, T ) is a minimal subshift. We show that when (X, T ) is a subshift associated to a substitution, the group M(T ) is an extension of Z by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of (X, T ). When (X, T ) is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that M(T ) is virtually abelian. We also show that when (X, T ) is minimal, M(T ) embeds into the Picard group of the crossed product algebra C(X) ⋊T Z.

AB - For a homeomorphism T: X → X of a Cantor set X, the mapping class group M(T ) is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension ΣT X. The group M(T ) can be interpreted as the symmetry group of the system (X, T ) with respect to the flow equivalence relation. We study M(T ), focusing on the case when (X, T ) is a minimal subshift. We show that when (X, T ) is a subshift associated to a substitution, the group M(T ) is an extension of Z by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of (X, T ). When (X, T ) is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that M(T ) is virtually abelian. We also show that when (X, T ) is minimal, M(T ) embeds into the Picard group of the crossed product algebra C(X) ⋊T Z.

UR - http://www.scopus.com/inward/record.url?scp=85104185544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85104185544&partnerID=8YFLogxK

U2 - 10.4064/CM7933-2-2020

DO - 10.4064/CM7933-2-2020

M3 - Article

AN - SCOPUS:85104185544

SN - 0010-1354

VL - 163

SP - 233

EP - 265

JO - Colloquium Mathematicum

JF - Colloquium Mathematicum

IS - 2

ER -