Small denominators. I. Mapping of the circumference onto itself

Abstract

10 the first part of the paper it is shown that analytic mappings of the eire>. cumference t differing little from a rotation, whose rotation number is irrational and satisfies certain arithmetical requirements, may be can'ied into a rotation by an analytic substituti()n of variables •. In the second part we consider the space of mappins,s of the circumference onto itself and the place occupied in this space by .mappings of various types. fie indicate applications to the investigation oftta-jectories on the torus and to the Dirichlet problem for the equation 01 the string. Intt~ucdon Continuous mappings of the circumference onto itself were studied by Paine;> care (see [l]~ Chapter XV, pp. 165-191) in connection with the qualitative ineo vestigation of trajectories on the torus. The problem of Dirichlet for the equation of the string can be reduced to such mappings, but the topological investigation turns out here to be insufficient (see [5]). In the first portion of the present paper we attempt an analytic refinement of the Denjoy theorem completing the theory. of Poincare [2]. Suppose that F(z) is periodic, F(z+ 211) =·F(z), real on the real axis and analytic in its neighborhood, with F' (z) +=.-1 for 1m z =·0. Then to the mapping of a strip of the complex plane defined by z-.. Az := z + F (z) there corresponds an orientation-preserving homeomorphism B of the neighborhood of tbe points w (z) =e iz : w·:a:·W(z)-+ w(Az) == Bw. In this sense we say that A is an analytic mapping of the circumference onto it Q self. * Suppose that the rotation number of A is equal to 21TP.. From Denjoy's theorem it follows that for .irrational JL there exists a continuous inversible real function ifJ(z} of the real variable z, periodic in the sense that ¢(z + 217) = ¢(z) + 217 and such that