Entropies of Automorphisms of a Topological Markov Shift

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