Symbolic dynamics for -shifts and self-normalnumbers

JÖRG SCHMELING

doi:10.1017/S0143385797079182

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

More than 30 years ago R\'enyi [1] introduced the representations ofreal numbers with an arbitrary base $\beta > 1$ as a generalization of the$p$-adic representations. One of the most studied problems in this field isthe link between expansions to base $\beta$ and ergodic properties of thecorresponding $\beta$-shift.

In this paper we will follow the bibliography of Blanchard [2] andgivean affirmative answer to a question on the size of the set of real numbers$\beta$ having complicated symbolic dynamics of their $\beta$-shifts.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Takens, Floris and Verbitski, Evgeni 1998. Multifractal Analysis of Local Entropies for Gibbs Measures. Periodica Mathematica Hungarica, Vol. 37, Issue. 1-3, p. 143.

Barreira, Luis Saussol, Benoît and Schmeling, Jörg 2002. Higher-dimensional multifractal analysis. Journal de Mathématiques Pures et Appliquées, Vol. 81, Issue. 1, p. 67.

Petersen, Karl 2002. Dynamics and Randomness. Vol. 7, Issue. , p. 147.

Niederreiter, Harald and Shparlinski, Igor E. 2003. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Vol. 2643, Issue. , p. 6.

NAKADA, HITOSHI and NATSUI, RIE 2003. SOME STRONG MIXING PROPERTIES OF A SEQUENCE OF RANDOM VARIABLES ARISING FROM α-CONTINUED FRACTIONS. Stochastics and Dynamics, Vol. 03, Issue. 04, p. 463.

Chi, Dong Pyo and Kwon, DoYong 2004. Sturmian words, β-shifts, and transcendence. Theoretical Computer Science, Vol. 321, Issue. 2-3, p. 395.

Barreira, Luis and Doutor, Paulo 2004. Birkhoff Averages for Hyperbolic Flows: Variational Principles and Applications. Journal of Statistical Physics, Vol. 115, Issue. 5-6, p. 1567.

Buzzi, Jérôme 2005. Subshifts of quasi-finite type. Inventiones mathematicae, Vol. 159, Issue. 2, p. 369.

Pfister, C-E and Sullivan, W G 2005. Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts. Nonlinearity, Vol. 18, Issue. 1, p. 237.

Aaronson, Jon and Nakada, Hitoshi 2005. On the mixing coefficients of piecewise monotonic maps. Israel Journal of Mathematics, Vol. 148, Issue. 1, p. 1.

Chazottes, J-R Ramirez, L and Ugalde, E 2005. Finite type approximations of Gibbs measures on sofic subshifts. Nonlinearity, Vol. 18, Issue. 1, p. 445.

Verger-Gaugry, Jean-Louis 2006. On gaps in Rényi β-expansions of unity for β>1 an algebraic number. Annales de l'Institut Fourier, Vol. 56, Issue. 7, p. 2565.

ADAMCZEWSKI, BORIS and BUGEAUD, YANN 2007. Dynamics forβ-shifts and Diophantine approximation. Ergodic Theory and Dynamical Systems, Vol. 27, Issue. 6, p. 1695.

LAU, KA-SING and SHU, LIN 2008. The spectrum of Poincaré recurrence. Ergodic Theory and Dynamical Systems, Vol. 28, Issue. 6, p. 1917.

Li, Bing and Wu, Jun 2008. Beta-expansion and continued fraction expansion. Journal of Mathematical Analysis and Applications, Vol. 339, Issue. 2, p. 1322.

SCHNELLMANN, DANIEL 2008. Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents. Ergodic Theory and Dynamical Systems, Vol. 28, Issue. 1, p. 245.

Verger-Gaugry, Jean-Louis 2008. On the dichotomy of Perron numbers and beta-conjugates. Monatshefte für Mathematik, Vol. 155, Issue. 3-4, p. 277.

Gazeau, Jean-Pierre and Verger-Gaugry, Jean-Louis 2008. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. Journal de théorie des nombres de Bordeaux, Vol. 16, Issue. 1, p. 125.

Li, Bing Wu, Jun and Xu, Jian 2008. Metric properties and exceptional sets of β-expansions over formal Laurent series. Monatshefte für Mathematik, Vol. 155, Issue. 2, p. 145.

FALLER, B. and PFISTER, C.-E. 2009. A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations. Ergodic Theory and Dynamical Systems, Vol. 29, Issue. 5, p. 1529.

Download full list

Article contents

Symbolic dynamics for $\beta$-shifts and self-normalnumbers

Abstract

Information

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Symbolic dynamics for $\beta$-shifts and self-normalnumbers

Abstract

Information

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests