Entropies of automorphisms of a topological Markov shift

Source abstract

Let σ \sigma be a mixing topological Markov shift, λ \lambda a weak Perron number, q ( t ) q\left ( t \right ) a polynomial with nonnegative integer coefficients, and r r a non-negative rational. We construct a homeomorphism commuting with σ \sigma whose topological entropy is log ⁡ [ q ( λ ) q ( 1 / λ ) ] r \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.