English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Generalized beta-transformations are the class of piecewise continuousinterval maps given by taking the beta-transformation $x \mapsto \beta x ~\pmod1$, where $\beta>1$, and replacing some of the branches with branches ofconstant 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 Parrynumber. We show that the Galois conjugates of such beta have modulus less than2, and the modulus is bounded away from 2 apart from the exceptional case ofconjugates lying on the real line. We give a characterization of the closure ofall these Galois conjugates, and show that this set is path connected. Ourapproach is based on an analysis of Solomyak for the case ofbeta-transformations. One motivation for this work is that the entropy of apost-critically finite (PCF) unimodal map is the logarithm of a generalizedParry number. Thus, our results give a mild restriction on the set of entropiesthat can be attained by PCF unimodal maps.

Generalized beta-transformations and the entropy of unimodal maps