[en] We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional numeration system U with regular language of numeration if and only if its U -kernel is finite.
Disciplines :
Mathematics
Author, co-author :
Massuir, Adeline ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Language :
English
Title :
Automatic sequences based on Parry or Bertrand numeration systems
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