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On digit frequencies in $ \beta$-expansions


Authors: Philip Boyland, André de Carvalho and Toby Hall
Journal: Trans. Amer. Math. Soc. 368 (2016), 8633-8674
MSC (2010): Primary 11A63; Secondary 37B10, 68R15
DOI: https://doi.org/10.1090/tran/6617
Published electronically: January 27, 2016
MathSciNet review: 3551584
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Abstract: We study the sets $ \operatorname {DF}(\beta )$ of digit frequencies of $ \beta $-expansions of numbers in $ [0,1]$. We show that $ \operatorname {DF}(\beta )$ is a compact convex set with countably many extreme points which varies continuously with $ \beta $; that there is a full measure collection of non-trivial closed intervals on each of which $ \operatorname {DF}(\beta )$ mode locks to a constant polytope with rational vertices; and that the generic digit frequency set has infinitely many extreme points, accumulating on a single non-rational extreme point whose components are rationally independent.


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Additional Information

Philip Boyland
Affiliation: Department of Mathematics, University of Florida, 372 Little Hall, Gainesville, Florida 32611-8105
Email: boyland@ufl.edu

André de Carvalho
Affiliation: Departamento de Matemática Aplicada, IME-USP, Rua Do Matão 1010, Cidade Universitária, 05508-090 São Paulo SP, Brazil
Email: andre@ime.usp.br

Toby Hall
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
Email: tobyhall@liv.ac.uk

DOI: https://doi.org/10.1090/tran/6617
Received by editor(s): August 29, 2013
Received by editor(s) in revised form: September 1, 2014, and October 30, 2014
Published electronically: January 27, 2016
Additional Notes: The authors would like to thank the referee, whose careful reading of the paper led to many significant improvements. We are grateful for the support of FAPESP grants 2010/09667-0 and 2011/17581-0. This research has also been supported in part by EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS)
Article copyright: © Copyright 2016 American Mathematical Society

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