Topological completely positive entropy is no simpler in $\mathbb Z^2$-SFTs

TL;DR

This paper constructs Z^2 subshifts of finite type with topological completely positive entropy at every computable level, demonstrating the complexity of TCPE in two-dimensional symbolic dynamics.

Contribution

It provides the first examples of Z^2-SFTs at all levels of the TCPE hierarchy and proves TCPE is coanalytic complete in this setting.

Findings

Constructed Z^2-SFTs at all computable TCPE levels

Answered an open question about TCPE level 3 in Z^2-SFTs

Proved TCPE property is coanalytic complete in Z^2-SFTs

Abstract

We construct Z^2-SFTs at every computable level of the hierarchy of topological completely positive entropy (TCPE), answering Barbieri and Garc\'{i}a-Ramos, who asked if there was one at level 3. Furthermore, we show the property of TCPE in Z^2-SFTs is coanalytic complete. Thus there is no simpler description of TCPE in Z^2-SFTs than in the general case.