Non-normal numbers in dynamical systems fulfilling the specification property

Manfred G. Madritsch; Izabela Petrykiewicz

doi:10.3934/dcds.2014.34.4751

Article Contents

2014, Volume 34, Issue 11: 4751-4764. Doi: 10.3934/dcds.2014.34.4751

This issue Previous Article Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems Next Article The structure of limit sets for $\mathbb{Z}^d$ actions

Non-normal numbers in dynamical systems fulfilling the specification property

Manfred G. Madritsch^1, and
Izabela Petrykiewicz^2,

1.
Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506
2.
Université Joseph Fourier, Institut Fourier, 100 rue des maths, 38402 St Martin d'Hères

Received: October 31, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

In the present paper we want to focus on the dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Cesàro variants diverge.

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Mathematics Subject Classification: Primary: 11K16, 37B10; Secondary: 11A63, 54H20.

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