---
abstract: 'We give an algorithm, based on the $\varphi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is to solve an inverse problem for the dynamical systems $\beta x +\alpha\mod 1$. The first part is an exposition of the $\varphi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $\varphi$-expansion, necessary and sufficient conditions, which are different from those in Parry’s paper [@P2].'
author:
- |
    B. Faller[^1] and C.-E. Pfister[^2]\
    EPF-L, Institut d’analyse et calcul scientifique\
    CH-1015 Lausanne, Switzerland
date: '16.05.2008\'
title: |
    Computation of Topological Entropy via $\varphi$-expansion,\
    an Inverse Problem for the Dynamical Systems $\beta
    x+\alpha\mod1$
---

Introduction {#section1}
============

In 1957 Rényi published his paper [@R] about representations for real numbers by $f$-expansions, called hereafter $\varphi$-expansions, which had tremendous impact in Dynamical Systems Theory. The ideas of Rényi were further developed by Parry in [@P1] and [@P2]. See also the book of Schweiger [@Sch]. The first part of the paper, section \[section2\], is an exposition of the theory of $\varphi$-expansions in the setting of piecewise monotone dynamical systems. Although many of the results of section \[section2\] are known, for example see [@Bo] 