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Eigenvalues and the stabilized automorphism group

Bastián Espinoza, Jennifer N. Jones-Baro

Abstract

We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves developing new algebraic tools to extract information about the rational eigenvalues of these systems from their stabilized automorphism groups. In particular, we prove that if two minimal system have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues. Using these tools, we also extend Schmieding's result on the recovery of entropy from the stabilized automorphism group to include irreducible shifts of finite type.

Eigenvalues and the stabilized automorphism group

Abstract

We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves developing new algebraic tools to extract information about the rational eigenvalues of these systems from their stabilized automorphism groups. In particular, we prove that if two minimal system have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues. Using these tools, we also extend Schmieding's result on the recovery of entropy from the stabilized automorphism group to include irreducible shifts of finite type.
Paper Structure (15 sections, 35 theorems, 124 equations)

This paper contains 15 sections, 35 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. Symbolic dynamics
  5. Algebraic background
  6. Rational eigenvalues and the stabilized automorphism group
  7. Properties of cyclic almost-partitions
  8. Description of the stabilized automorphism group
  9. Key algebraic properties of wreath products
  10. Recovering rational eigenvalues
  11. The transitive case
  12. Example 1
  13. Example 2:
  14. Systems with a finite number of asymptotic classes
  15. Irreducible subshifts of finite type

Key Result

theorem 1.1

Let $(X,T)$ and $(Y,S)$ be minimal systems, each having at least one rational eigenvalue other than $1$. If $\operatorname{Aut}^{(\infty)}(X,T)$ is isomorphic to $\operatorname{Aut}^{(\infty)}(Y,S)$, then $(X,T)$ and $(Y,S)$ have the same rational eigenvalues.

Theorems & Definitions (71)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • theorem 1.5
  • proposition 3.1
  • proof
  • remark 3.2
  • definition 3.3
  • lemma 3.4
  • ...and 61 more