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Do the Properties of an ***S*-adic Representation Determine Factor Complexity?

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Fabien Durand

LAMFA, CNRS UMR 7352

Université de Picardie Jules Verne

UFR des Sciences

33, rue Saint-Leu

80039 Amiens Cedex 1

France

Julien Leroy

Department of Mathematics

University of Liège

Grande Traverse 12 (B37)

B-4000 Liège, Belgium

and

LAMFA, CNRS UMR 7352

Université de Picardie Jules Verne

UFR des Sciences

33, rue Saint-Leu

80039 Amiens Cedex 1

France

Gwenaël Richomme

Université Paul-Valéry Montpellier 3

UFR IV, Dpt MIAp, Case J11

Route de Mende

34199 Montpellier Cedex 5

France

and

LIRMM (CNRS, Univ. Montpellier 2) - UMR 5506 - CC 477

161 rue Ada

34095 Montpellier Cedex 5

France

**Abstract:**

The *S*-adic conjecture postulates the existence of a
condition *C* such that a sequence has linear complexity if and only if it is an
*S*-adic sequence satisfying *C* for some finite set *S* of morphisms.
We present an overview of the factor complexity of *S*-adic sequences and we give some examples that either illustrate some interesting properties,
or that are counterexamples to what might seem to be a "good" condition *C*.

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(Concerned with sequence
A003849
A010060
A049320.)

Received June 23 2012;
revised version received October 9 2012.
Published in *Journal of Integer Sequences*, March 2 2013.

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