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We study the factor complexity and closure properties of automatic sequencesbased on Parry or Bertrand numeration systems. These automatic sequences can beviewed as generalizations of the more typical $k$-automatic sequences andPisot-automatic sequences. We show that, like $k$-automatic sequences,Parry-automatic sequences have sublinear factor complexity while there existBertrand-automatic sequences with superlinear factor complexity. We prove thatthe set of Parry-automatic sequences with respect to a fixed Parry numerationsystem is not closed under taking images by uniform substitutions or periodicdeletion of letters. These closure properties hold for $k$-automatic sequencesand Pisot-automatic sequences, so our result shows that these properties arelost when generalizing to Parry numeration systems and beyond. Moreover, weshow that a multidimensional sequence is $U$-automatic with respect to apositional numeration system $U$ with regular language of numeration if andonly if its $U$-kernel is finite.

Automatic sequences based on Parry or Bertrand numeration systems