Interval Identification Systems and Plane Sections of 3-Periodic Surfaces

I. A. Dynnikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 72-84

Résumé

Interval identification systems is a notion that, on the one hand, generalizes interval exchange transformations and, on the other hand, describes special cases of such transformations. In the present paper we overview some elementary facts, address a few questions about interval identification systems, and describe explicitly systems that allow one to construct 3-periodic surfaces in the 3-space whose intersections with planes of a fixed direction have chaotic behavior. The problem of asymptotic behavior of plane sections of 3-periodic surfaces was posed by S. P. Novikov in 1982 and studied then by his students. One of the most interesting remaining open questions about such sections is reduced to the study of interval identification systems.