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 $n$ 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.