On a generalization of Christoffel words: epichristoffel words

Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the generalization of these sequences are the episturmian sequences, introduced by A. de Luca [dL97a] and studied in particular by Arnoux, Rauzy, Justin, Pirillo, Droubay, Glen [AR91, Rau85, Jus05, JP02, JP04, DJP01, Gle07, GJP08]. There exists a finite version of the Sturmian sequence called the Christoffel words. They are know since the works of Christoffel [Chr75] and have interested many mathematicians: Borel, Laubie, Reutenauer, Berthé, de Luca [BL93, BdLR07, BR06, KR07], etc. In this paper, we introduce a generalization of Christoffel words for an alphabet with 3 letters or more, using the episturmian morphisms. We call them the epichristoffel words. We define this new class of finite words and show how some of the properties of the Christoffel words can be generalized naturally or not for this new class.