An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields. (English) Zbl 0931.28013

Let \(T:[a,b]\to [a,b]\) denote an interval exchange transformation. By children of \(T\) we shall mean the (first return) transformations induced on the exchanged intervals. Children of \(T\) are also interval exchange transformations. We define the family of all descendants of \(T\) as the smallest family of maps containing \(T\) and closed under taking children. We agree to identify transformations conjugate via a linear function (rescaling). The main result in this paper asserts that if the lengths of the exchanged intervals all belong to a quadratic number field \(\{q+ r\sqrt d: q,r\in Q\}\) \((d\in\mathbb{N})\), then the family of all descendants of \(T\) is finite up to rescaling. (In case of two subintervals the above reduces to Lagrange’s theorem: the terms in the continued fraction expansion of a quadratic rational are eventually periodic.)
The proof relies on estimating the maximal and minimal return times to an interval \(J\), namely it is shown that both range between \(c_1|J|^{-1}\) and \(c_2|J|^{-1}\), where the constants \(c_1\), \(c_2\) depend on the transformation only. This is used to estimate certain parameters called complexity and reduced complexity of the descendants, which is a crucial item of the main proof.
By analogy to Galois’ characterization of purely periodic continued fractions, the case of purely periodic interval exchange transformation is also studied, i.e., such that \(T\) is equivalent to one of its descendants. A structure theorem is provided, where purely periodic interval exchange transformations are described using skew product extensions (by finite permutations) over certain two or three interval exchange transformations.
In section 4, we find applications of the main theorem in the theory of measured foliations [see W. P. Thurston, Bull. Am. Math. Soc., New Ser. 19, No. 2, 417-431 (1988; Zbl 0674.57008) for background], namely, it is proved that every quadratic (transitive singular measured) foliation (on a closed orientable surface of genus \(g\geq 2\)) is pseudo-Anosov.


28D05 Measure-preserving transformations
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
11K50 Metric theory of continued fractions


Zbl 0674.57008
Full Text: DOI


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