Verification of mixing properties in two-dimensional shifts of finite type
- 1 July 2021
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (7), 072703
- https://doi.org/10.1063/5.0007365
Abstract
This work introduces constructive and systematic methods for verifying the topological mixing and strong specification (or strong irreducibility) of two-dimensional shifts of finite type. First, we define transition matrices on infinite strips of width n for all n ≥ 2. To determine the primitivity of the transition matrices, we introduce the connecting operators that reduce the high-order transition matrices to lower-order transition matrices. Then, two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. Then, the primitivity, corner-extendability, and crisscross-extendability are used to demonstrate the topological mixing. Finally, we show that the hole-filling condition yields the strong specification property. The application of all the above-mentioned conditions can be verified in a finite number of steps.Other Versions
Funding Information
- Ministry of Science and Technology, Taiwan (MOST 109-2115-M-004-002-MY2, MOST 107-2115-M-009-006)
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