On digit frequencies in $\beta$-expansions
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- by Philip Boyland, André de Carvalho and Toby Hall PDF
- Trans. Amer. Math. Soc. 368 (2016), 8633-8674 Request permission
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.References
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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
- MR Author ID: 652366
- Email: andre@ime.usp.br
- Toby Hall
- Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom
- MR Author ID: 312789
- Email: tobyhall@liv.ac.uk
- 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)
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8633-8674
- MSC (2010): Primary 11A63; Secondary 37B10, 68R15
- DOI: https://doi.org/10.1090/tran/6617
- MathSciNet review: 3551584