Entropies of automorphisms of a topological Markov shift

Abstract:

Let $\sigma$ be a mixing topological Markov shift, $\lambda$ a weak Perron number, $q\left ( t \right )$ a polynomial with nonnegative integer coefficients, and $r$ a non-negative rational. We construct a homeomorphism commuting with $\sigma$ whose topological entropy is $\log {\left [ {q\left ( \lambda \right )q\left ( {1/\lambda } \right )} \right ]^r}$. These values are shown to include the logarithms of all weak Perron numbers, and are dense in the nonnegative reals.