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 $\hat{I}{\pounds }_A$ to its respective generalized countable Markov shift $X_A$ (a compactification of $\hat{I}{\pounds }_A$). When the shift map is continuously extendable, we obtain explicit formulas for the spectral radius of weighted endomorphisms $a\hat{I}\pm$, where $\hat{I}\pm$ is dual to the shift map and conjugated to $\hat{I}˜(f)=f\circ \ddot{I}ƒ$ on $C(X_A)$, extending a theorem of Kwaśniewski and Lebedev from finite to countable alphabets.

37 pages, 14 figures. Preliminary version, comments are welcome