Absence of mixing in area-preserving flows on surfaces

Abstract

We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus g ≥ 2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. Definitions and main results1.1. Flows given by multi-valued Hamiltonians. Let us consider the following natural construction of area-preserving flows on surfaces. On a closed connected orientable surface S of genus g ≥ 1 with a fixed smooth area form ω, consider a smooth closed real-valued differential 1-form η. Let X be the vector field determined by η = i X ω = ω(η, ·) and consider the flow {ϕ t } t∈R on S associated to X. Since η is closed, the transformations ϕ t , t ∈ R, are areapreserving. The flow {ϕ t } t∈R is known as the multi-valued Hamiltonian flow associated to η. Indeed, the flow {ϕ t } t∈R is locally Hamiltonian; i.e., locally one can find coordinates (x, y) on S in which it is given by the solution to the equationsẋ = ∂H/∂y,ẏ = −∂H/∂x for some smooth real-valued Hamiltonian function H. A global Hamiltonian H cannot be in general be defined (see [NZ99, §1.3.4]), but one can think of {ϕ t } t∈R as globally given by a multi-valued Hamiltonian function.The study of flows given by multi-valued Hamiltonians was initiated by S. P. Novikov [Nov82] in connection with problems arising in solid-state physics i.e., the motion of an electron in a metal under the action of a magnetic field. The orbits of such flows arise also in pseudo-periodic topology, as hyperplane sections of periodic surfaces in T n (see e.g. Zorich [Zor99]).From the point of view of topological dynamics, a decomposition into minimal components (i.e., subsurfaces on which the flow is minimal) and periodic components on which all orbits are periodic (elliptic islands around a center and cylinders filled by periodic orbits) was proved independently by Maier

Multiple mixing for a class of conservative surface flows

“…The study of the mixing properties of surface flows has known a revival of interest since the beginning of the 2000s, with results such as the computation of the speed of mixing [6] or extensions of the Khanin-Sinai mixing result to include all irrational translation vectors [19,20] (see also [21]), or advances in the study of multi-valued Hamiltonian flows on surfaces in the general case where the Poincaré section return map is an IET and not just a circular rotation [3,[35][36][37].…”

Ergodic properties of infinite extensions of area-preserving flows

“…The definition (1.3) of symmetry appears often in the literature, for example in [22,34,40]. In this paper we need a more restrictive notion of symmetry: we give in Sect.…”

“…In Sect. 3.2 we formulate results on the growth of Birkhoff sums based on the work of the second author in [40]. The correction operator, which is crucial to define the correction χ in Theorem 1.2, is constructed in Sect.…”

“…The additional difficulty that we have to face to achieve tightness is the presence of logarithmic singularities. Here the assumption that the singularities are symmetric is crucial to exploit the cancellation mechanism introduced by the second author in [40] in order to show that locally Hamiltonian flows are typically not mixing.…”

“…A typical locally Hamiltonian flow (φ t ) t∈R on S with no saddle connection is (uniquely) ergodic, by a celebrated theorem by Masur and Veech [28,43]. Moreover, mixing properties of locally Hamiltonian flows have been investigated in [22,23,34,[38][39][40]. On the other hand, very little is understood in the case of noncompact extensions with the exception of the special case of g = 1 (see [8,10]) and the case where f vanishes on the set of fixed points of the flow (φ t ) t∈R (see [5,11,26]).…”