On extensions of subshifts by finite groups

TL;DR

This paper explores how subshifts represented by $mbda$-graph systems can be extended by finite groups, establishing a classification framework based on $G$-strong shift equivalence and cohomology classes.

Contribution

It introduces the concept of skew products of $mbda$-graph systems and characterizes $G$-conjugacy of subshifts via $G$-strong shift equivalence and cohomology.

Findings

Two symbolic matrix systems are $G$-strong shift equivalent iff their subshifts are $G$-conjugate.

$G$-equivalent classes of subshifts are classified by cohomology classes of skewing functions.

The paper provides a classification framework linking algebraic invariants to dynamical properties.

Abstract

$\lambda$-graph systems are labeled Bratteli diagram with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of $\lambda$-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are $G$-strong shift equivalent if and only if their presented subshifts are $G$-conjugate. $G$-equivalent classes of subshifts are classified by the cohomology classes of their associated skewing functions.