An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields

LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {T _n } _n=1 ^∞ be any sequence of interval exchange transformations such thatT ₁ =T andT _n is the first return map induced byT _n-1 on one of its exchanged intervals I_n-1. We prove that {T _n } _n=1 ^∞ contains finitely many transformations up to rescaling. If the interval I_n is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n ₀ ∈ ℤ⁺, λ ∈R ⁺ such that for alln ≥n ₀,I _n = λI _n+k andT _n ,T _n+k are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms, applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We also discuss the case of periodic induced sequences. These results have applications to topology. In particular, every projective measured foliation on Thurston’s boundary to Teichmüller space that is minimal and metrically ‘quadratic’ is fixed by a hyperbolic element of the modular group. Moreover, if the foliation is orientable, it covers (via a branched covering) an irrational foliation of the two-torus. We also obtain a new proof, for quadratic irrationals, of Boshernitzan’s result that a minimal rank 2 interval exchange transformation is uniquely ergodic. The first author was supported in part by NSF-DMS-9224667. The second author was supported in part by an NSF-NATO fellowship, held at the Université Paris-Sud, Orsay.