On the number of fixed points of sofic flip systems

TL;DR

This paper provides a formula for counting fixed points in sofic flip systems using finite matrices derived from Krieger's joint state chain, linking topological dynamics with algebraic structures.

Contribution

It introduces a method to compute fixed points in sofic flip systems via matrices from Krieger's joint state chain, extending understanding of their structure.

Findings

Fixed points are expressed using finite matrices.

Matrices are derived from Krieger's joint state chain.

The approach applies to conjugate sofic shifts.

Abstract

In the case when $X$ is a sofic shift and $\phi : X \to X$ is a homeomorphism such that $\phi^2 = \text{id}_X$ and $\phi \sigma_X = \sigma_X^{-1} \phi$, the number of points in $X$ that are fixed by $\sigma_X^m$ and $\sigma_X^n \phi$, $m=1,2,...$, $n\in\Bbb Z$, is expressed in terms of a finite number of square matrices: The matrices are obtained from Krieger's joint state chain of a sofic shift which is conjugate to $X$.