A geometric point of view on the synchronization of three-Christoffel words
Résumé
We study the synchronization of three Christoffel words (i.e.,
superimposition of words with same length) from a geometric point of
view. We provide a geometrical interpretation of the synchronization in
terms of Reveilles’ discrete lines. We introduce for the synchronization
a technique based on algebra, arithmetic, and Cayley graphs. We define
the orbital matrix, based on the vertices of the Cayley graph, for each
Christoffel word. We provide a characterization of a vector, called the
seed, which is a vector that synchronizes the edges of the three Cayley
graphs. After synchronization, we show that each Christoffel word is
replaced by a 4-connected Reveilles 2D segment whose parameter is given
by the seed.
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