Very close to the identity, the diffeomorphisms of a Euclidean space with at least 2 dimensions have their periodic orbits on regular polygons

摘要

Let E denote a Euclidean space with dimension p >= 2; O-n = {x, f(x), ..., f(n-1)(x)} a periodic orbit of length n >= 2 for a C-1-diffeomorphism f : E -> E; and rho(f) the number rho(f) = sup{|||D(z)f - Id|||, z is an element of E} is an element of [0, +infinity], defined by means of the classical triple norm associated with a Euclidean norm parallel to.parallel to in the set of linear endomorphisms of E. I have proved in (Dehove 2007 Nonlinearity 20 2191-203) that one has optimally rho(f) >= 2 sin(pi). The object of this paper is to show that if rho(f) = 2 sin(pi), then the orbit O-n is located on the vertices of a regular polygon, on the convex hull of which the diffeomorphism f coincides with a rotation of an angle 2 pi. This involves several new results in geometry, formulated and proved in the paper.