Well-approximable angles and mixing for flows on 𝕋² with nonsingular fixed points

Abstract

Abstract. 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

Multiple mixing for a class of conservative surface flows

“…The use of a base interval exchange transformation of the form of T is similar to the one used by Chaika and Wright [2], that was in turn inspired by [26,9,21]. Observe that by a result of Kochergin (Theorem 2 in [15]), if follows that any smooth flow with (strongly) asymmetric singularities on the T 2 is mixing and so our example is optimal in terms of the genus of the surface. The return time above the points z 1 and z 2 has identical asymmetric logarithmic asymptotes whose global contribution is therefore asymmetric logarithmic.…”

“…Mixing of these flows was proved by Kochergin in the case of degenerate power like singularities (see exact description below) in [13] and by Khanin and Sinai in a particular case of non-degenerate asymmetric saddle type singularities (see exact description below) [27]. The study of the mixing properties of these flows has known a revival of interest since the beginning of the 2000's, with results such as the computation of the speed of mixing [4] or extension of the Kahnin-Sinai mixing result to include all irrational translation vectors [17] (see also [15], [16]), or advances in the study of Arnol'd and Kochergin flows in the general case where the Poincaré section return map is an interval exchange and not just a circular rotation [30,31,32].…”

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

“…The question about mixing of such flows, risen in the same paper [Arn91], was answered by Sinai and Khanin in [SK92], where it was proved that, under a generic diophantine condition on the rotation angle, suspension flows with asymmetric singularities over a rotation are strongly mixing (see also [Kha96]). The diophantine condition of [SK92] was weakened by Kochergin in a series of works ([Koc03b,Koc04a,Koc04b,Koc04c]).…”

Quantitative Mixing for Locally Hamiltonian Flows with Saddle Loops on Compact Surfaces

“…The question posed by Arnold was answered by Sinai and Khanin [25], who proved that, under a full-measure Diophantine condition on the rotation angle, the flow is mixing. This condition was weakened by Kochergin [12,13,14,15]. The presence of singularities in the roof function is necessary, as well as the asymmetry condition: in this setting, mixing does not occur for functions of bounded variation or, assuming a full-measure Diophantine condition on the rotation angle, for functions with symmetric logarithmic singularities; see the results by Kochergin in [8] and [11] respectively.…”