Non-normal numbers with respect to Markov partitions

Manfred Madritsch (1)

1 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: symbolic dynamical system; non-normal numbers; Markov partitions

Domaines: Théorie des nombres [math.NT]

Manfred Madritsch. Non-normal numbers with respect to Markov partitions. Discrete and Continuous Dynamical Systems - Series A, 2014, 34 (2), pp.663 - 676. ⟨10.3934/dcds.2014.34.663⟩. ⟨hal-00782110⟩