Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

TL;DR

This paper characterizes when the shift map on countable Markov shifts can be extended continuously to their compactifications and provides explicit formulas for the spectral radius of associated weighted endomorphisms.

Contribution

It offers an operator algebraic criterion for the continuous extension of shift maps and generalizes spectral radius formulas to countable alphabets.

Findings

Explicit formulas for spectral radius of weighted endomorphisms.

Operator algebraic characterization of shift map extension.

Extension of finite alphabet results to countable alphabets.

Abstract

We obtain an operator algebraic characterization for when we can continuously extend the shift map from a standard countable Markov shift $\Sigma_A$ to its respective generalized countable Markov shift $X_A$ (a compactification of $\Sigma_A$). When the shift map is continuously extendable, we obtain explicit formulas for the spectral radius of weighted endomorphisms $a\alpha$, where $\alpha$ is dual to the shift map and conjugated to $\Theta(f)=f \circ \sigma$ on $C(X_A)$, extending a theorem of Kwa\'sniewski and Lebedev from finite to countable alphabets.