Interval exchange transformations over algebraic number fields: the cubic Arnoux-Yoccoz model

ISSN： 1468-9375

Source： Dynamical Systems: An International Journal, Vol.22, Iss.1, 2007-03, pp. : 73-106

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Abstract

We apply methods developed for two-dimensional piecewise isometries to the study of renormalizable interval exchange transformations over an algebraic number field [image omitted], which lead to dynamics on lattices. We consider the [image omitted]-module [image omitted] generated by the translations of the map. On it, we define an infinite family of discrete vector fields, representing the action of the map over the cosets [image omitted], which together form an invariant partition of the field [image omitted]. We define a recursive symbolic dynamics, with the property that the eventually periodic sequences coincide with the field elements. We apply this approach to the study of a model introduced by Arnoux and Yoccoz, for which -1 is a cubic Pisot number. We show that all cosets of [image omitted] decompose in a highly non-trivial manner into the union of finitely many orbits.