Abstract
We study classes of minimal sets defined by restrictions on the possible extensions of the words. These sets generalize the previously studied classes of neutral and tree sets by relaxing the condition imposed on the empty word and measured by an integer called the characteristic of the set. We present several enumeration results holding in these sets of words. These formulae concern return words and bifix codes. They generalize formulae previously known for Sturmian sets or more generally for tree sets. We also give two geometric examples of this class of sets, namely the natural coding of some interval exchange transformations and the natural coding of some linear involutions.
Highlights
Sets of words of linear complexity play an important role in combinatorics on words and symbolic dynamics
A special family of neutral sets is given by tree sets, sets such that E(x) is a tree for every nonempty word and acyclic for every word
We prove that the decoding of any recurrent neutral set S by an S-maximal bifix code is a neutral set
Summary
Sets of words of linear complexity play an important role in combinatorics on words and symbolic dynamics. In several families of sets of linear complexity, the set of return words to x is known to be of fixed cardinality independent of x. This was proved for Sturmian words in [14], for interval exchange sets in [17] (see [11]) and for neutral sets of characteristic zero in [1]. We prove two results which allows one to obtain a large family of neutral sets ( tree sets) of geometric origin, namely using interval exchange transformations or linear involutions.
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