595 research outputs found
Decidability of the HD0L ultimate periodicity problem
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
Multidimensional extension of the Morse--Hedlund theorem
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a
sequence over a finite alphabet is ultimately periodic if and only if, for
some , the number of different factors of length appearing in is
less than . Attempts to extend this fundamental result, for example, to
higher dimensions, have been considered during the last fifteen years. Let
. A legitimate extension to a multidimensional setting of the notion of
periodicity is to consider sets of \ZZ^d definable by a first order formula
in the Presburger arithmetic . With this latter notion and using a
powerful criterion due to Muchnik, we exhibit a complete extension of the
Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of
$\ZZ^d$ definable in in terms of some functions counting recurrent
blocks, that is, blocks occurring infinitely often
Conjugacy of unimodular Pisot substitutions subshifts to domain exchanges
We prove that any unimodular Pisot substitution subshift is measurably
conjugate to a domain exchange in Euclidean spaces which factorizes onto a
minimal rotation on a torus. This generalizes the pioneer works of Rauzy and
Arnoux-Ito providing geometric realizations to any unimodular Pisot
substitution without any additional combinatorial condition.Comment: 29 p. In this new version, a gap in the proof of the main theorem has
been fixe
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