Entropies of automorphisms of a topological Markov shift

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.