Non-normal numbers with respect to Markov partitions

Manfred G. Madritsch

doi:10.3934/dcds.2014.34.663

Article Contents

2014, Volume 34, Issue 2: 663-676. Doi: 10.3934/dcds.2014.34.663

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Non-normal numbers with respect to Markov partitions

Manfred G. Madritsch^1,

1.
Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506

Received: November 30, 2012

Revised: November 30, 2012

Published: January 2014

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Abstract

We call a real number normal if for any block of digits the asymptotic frequency of this block in the $N$-adic expansion equals the expected one. In the present paper we consider non-normal numbers and, in particular, essentially and extremely non-normal numbers. We call a real number essentially non-normal if for each single digit there exists no asymptotic frequency of its occurrence. Furthermore we call a real number extremely non-normal if all possible probability vectors are accumulation points of the sequence of frequency vectors. Our aim now is to extend and generalize these results to Markov partitions.

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

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