Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

Abstract

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