ABSTRACT
____Let T be an i.e.t. (interval exchange transformation) on N intervals whose lengths lie in a quadratic number field. Let T(n) be any sequence of i.e.t.s such that T(1)=T and T(n) is the first return map induced by T(n-1) on one of the intervals exchanged by T(n-1). We prove that the sequence T(n) contains only a finite number of i.e.t.s up to rescaling.
____When N=2 the assertion reduces to Lagrange's classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We discuss some applications. (21 pages.)