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Verification of mixing properties in two-dimensional shifts of finite type

Jung-Chao Ban, Wen-Guei Hu, Song-Sun Lin, Yin-Heng Lin

Abstract

The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width $n$ are introduced for all $n\geq 2$. To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps.

Verification of mixing properties in two-dimensional shifts of finite type

Abstract

The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width are introduced for all . To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps.

Paper Structure

This paper contains 7 sections, 19 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extendabilities and topological mixing
  4. Invariant diagonal cycles and primitive commutative cycles
  5. Invariant diagonal cycles
  6. Primitive commutative cycles
  7. Strong specification

Key Result

Theorem \oldthetheorem

If then $\mathbb{H}_{n}(\mathcal{B})$ and $\mathbb{V}_{n}(\mathcal{B})$ are weakly primitive for all $n\geq 2$ if and only if $\Sigma(\mathcal{B})$ is topologically mixing.

Theorems & Definitions (35)

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  • ...and 25 more