  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.
