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Billiards in regular 2 n ‐gons and the self‐dual induction
Author(s) -
Ferenczi Sébastien
Publication year - 2013
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds075
Subject(s) - interval (graph theory) , mixing (physics) , mathematics , combinatorics , mathematical induction , coding (social sciences) , discrete mathematics , pure mathematics , physics , geometry , statistics , quantum mechanics
We build a coding of the trajectories of billiards in regular 2 n ‐gons, similar but different from the one in J. Smillie and C. Ulcigrai [in ‘Beyond Sturmian sequences: coding linear trajectories in the regular octagon’, Proc. Lond. Math. Soc. (3) 102 (2011) 291–340], by applying the self‐dual induction [S. Ferenczi and L.Q. Zamboni, ‘Structure of K ‐interval exchange transformations: induction, trajectories, and distance theorems’, J. Anal. Math. 112 (2010) 289–328] to the underlying one‐parameter family of n ‐interval exchange transformations. This allows us to show that, in that family, for n =3 non‐periodicity is enough to guarantee weak mixing, and in some cases minimal self‐joinings, and for every n we can build examples of n ‐interval exchange transformations with weak mixing, which are the first known explicitly for n >6.