Computation of Topological Entropy via $\phi$-expansion, an Inverse Problem for the Dynamical Systems $\beta x+\alpha \mod 1$

We give an algorithm, based on the $\phi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $\beta x+\alpha \mod1$.The first part is an exposition of the $\phi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $\phi$-expansion, necessary and sufficient conditions, which are different from those in Parry's paper.