Ergodic Theory and Dynamical Systems

Research Article

The entropies of topological Markov shifts and a related class of algebraic integers

D. A. Linda1

a1 Department of Mathematics, University of Washington, Seattle, Washington 98195, USA

Abstract

We give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.

(Received October 06 1983)