Well-approximable Angles and Mixing for Flows on T with Nonsingular Fixed Points

We consider special flows over circle rotations with an asymmetric function having logarithmic singularities. If some expressions containing singularity coefficients are different from any negative integer, then there exists a class of well-approximable angles of rotation such that the special flow over the rotation of this class is mixing. Examples of smooth flows on a two-dimensional torus with a smooth invariant measure and nonsingular hyperbolic fixed points appear naturally in Arnold’s paper [1]. The phase space of such a flow decomposes into cells bounded by closed separatrices of regular fixed points and filled with periodic orbits, and an ergodic component in which orbits on one side of a fixed point visit its neighborhood more frequently than on the other. (See the figure, for example.) V. I. Arnold has shown that there exists a smooth closed curve transversal to the orbits of the ergodic component. The invariant measure and the flow naturally induce a smooth parameterization on the curve (this procedure was described in detail, e.g., in [3]). The first-return map is determined everywhere on the curve, except for a finite number of points that are the points of the last intersection of the stable separatrices with the curve. In the induced parameterization, this map is a circle rotation. The return time is a smooth function of the parameter everywhere Received by the editors June 14, 2004 and, in revised form, August 17, 2004. 2000 Mathematics Subject Classification. Primary 37E35, 37A25. The work was partially supported by the program “Leading Scientific Schools of Russian Federation”, project no. NSh-457.2003.01. c ©2004 American Mathematical Society Reverts to public domain 28 years from publication