Algebraic Invariants of Symbolic Systems

Yang, Kitty

doi:https://doi.org/10.21985/n2-02a7-2q80

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Algebraic Invariants of Symbolic Systems

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We study the symmetry groups with respect to various equivalence relations defined on subshifts, and more generally, on Cantor systems. Two basic notions of equivalence for dynamical systems are conjugacy and flow equivalence. In this dissertation, we focus on the well-studied automorphism group, which is the group of self-conjugacies, and the more recently defined mapping class group, the group of self-flow equivalences up to isotopy, of different classes of subshifts. In Chapter 1, the introduction, we go into more detail about the historical context and motivation for the results that appear in this thesis. In Chapter 2, we give precise definitions and the background necessary for the rest of the dissertation. We also include some illuminating examples of subshifts, automorphism groups, and mapping class groups. Chapter 3 extends Ryan's Theorem, which computes the center of the automorphism group for an important class of subshifts of subshifts, shifts of finite type. We strengthen the result to show that any normal amenable subgroup of the automorphism group, which includes the center, must be contained in the subgroup generated by the shift map. We also generalize to the class of sofic shifts, which include shifts of finite type. In Chapter 4, we work in the setting of minimal subshifts of linear complexity. The condition of linear complexity leads to restrictions on dynamical properties of the subshift, and this corresponds to constraints on the automorphism group and mapping class group. We show that for the special class of substitution subshifts, the mapping class group is a finite extension of the integers. More generally, if a minimal subshift of linear complexity satisfies a technical condition of trivial infinitesimals, then the mapping class group is the finite extension of an abelian group. Chapter 5 builds on the work in Chapter 4 in the more general context of Cantor systems. We can associate a C*-algebra to any Cantor system, and we study Morita equivalences of these C*-algebras. The group of self-Morita equivalences is called the Picard group. We show that there is a well-defined map from the mapping class group to the Picard group of a Cantor system. Furthermore, if the system is minimal, the mapping class group embeds in the Picard group.

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