Generalized beta-transformations and the entropy of unimodal maps

TL;DR

This paper studies generalized beta-transformations, revealing properties of their Galois conjugates and linking these to the entropy of post-critically finite unimodal maps, thus constraining possible entropy values.

Contribution

It characterizes the Galois conjugates of generalized Parry numbers and relates their properties to the entropy of PCF unimodal maps, extending understanding of their algebraic and dynamical structure.

Findings

Galois conjugates have modulus less than 2

The set of conjugates is path connected

Entropy of PCF unimodal maps is constrained by these properties

Abstract

Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto \beta x ~\pmod 1$, 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.