Generalized $\beta$-transformations and the entropy of unimodal maps

Abstract

Generalized $\beta$-transformations are the class of piecewise continuous interval maps given by taking the $\beta$-transformation $x\mapsto \beta x$ (mod 1), where $\beta >1$, 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 $\beta$ (which must be an algebraic number) a generalized Parry number. We show that the Galois conjugates of such $\beta$ 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 $\beta$-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.