Notes on sets of first return
Val\'erie Berth\'e, Clelia De Felice, Francesco Dolce and, Dominique Perrin, Christophe Reutenauer, Giuseppina Rindone
TL;DR
This paper proves that for a three-letter alphabet, the set of first return words to a given word forms a basis of the free group within a certain set satisfying the tree condition.
Contribution
It establishes a new connection between first return words and free group bases for three-letter alphabets under the tree condition.
Findings
First return set forms a basis of the free group for three-letter alphabets.
The result applies to sets satisfying the tree condition.
Provides a new perspective on the structure of free groups in symbolic dynamics.
Abstract
We prove in this note that, for an alphabet with three letters, the set of first return to a given word in a set satisfying the tree condition is a basis of the free group.
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Taxonomy
Topicssemigroups and automata theory · Computability, Logic, AI Algorithms · Geometric and Algebraic Topology