# Generalized $β$-transformations and the entropy of unimodal maps

### Daniel J. Thompson

The Ohio State University, Columbus, USA

## Abstract

Generalized $β$-transformations are the class of piecewise continuous interval maps given by taking the $β$-transformation $x↦βx$ (mod 1), where $β>1$, and replacing some of the branches with branches of constant negative slope. If the orbit of 1 is finite, then the map is Markov, and we call $β$ (which must be an algebraic number) a *generalized Parry number*. We show that the Galois conjugates of such $β$ have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. We give a characterization of the closure of all these Galois conjugates, and show that this set is path connected. Our approach is based on an analysis of Solomyak for the case of $β$-transformations. One motivation for this work is that the entropy of a post-critically finite (PCF) unimodal map is the logarithm of a generalized Parry number. Thus, our results give a mild restriction on the set of entropies that can be attained by PCF unimodal maps.

## Cite this article

Daniel J. Thompson, Generalized $β$-transformations and the entropy of unimodal maps. Comment. Math. Helv. 92 (2017), no. 4, pp. 777–800

DOI 10.4171/CMH/424