Ergodic properties of infinite extensions of area-preserving flows

Abstract

We consider volume-preserving flows (Φ f t ) t∈R on S × R, where S is a compact connected surface of genus g ≥ 2 and (Φwhere (φ t ) t∈R is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in C 2+ (S), then the following dynamical dichotomy holds: if there is a fixed point of (φ t ) t∈R on which f does not vanish, then (Φ f t ) t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ 0 t ) t∈R . The proof of this result exploits the reduction of (Φ f t ) t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φ t ) t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

“…We then use our result to show the existence of infinite extensions of such flows which are ergodic with respect to the natural infinite invariant measure. This result generalizes to higher genus a classical result by Krygin [42] in genus one and extends a previous result in higher genus by the authors (see [20], where we showed the existence of ergodic extensions in any genus, but only for flows with self-similar foliations) to a full measure set of flows.…”

“…One of the main results of this paper is that infinite ergodic extensions exist in any genus g ≥ 1 for a full measure set of (minimal) locally Hamiltonian flows with non-degenerate fixed points (with respect to the Katok fundamental class for each stratum, see § 2.1.1). More precisely, we are able to extend the result previously proved in [20] only for a measure zero class of self-similar IETs to a full measure set of locally Hamiltonian flows, by proving the following dichotomy for the dynamics of the extensions: Theorem 1.2 (Ergodic or reducible extensions of locally Hamiltonian flows). For a full measure set of locally Hamiltonian flows ψ R with non-degenerate saddles in U min , for any > 0, for any f in a infinite dimensional (finite codimension) subspace K ⊂ C 2+ (M ), we have the following dichotomy:…”

“…We considered the case of a locally Hamiltonian flow ψ R with non-degenerate saddles and a general observable f : M → R and, correspondingly, of a cocycle ϕ with logarithmic (symmetric) singularities in our previous joint work [20], where we showed the existence of ergodic extensions in any genus, but for a very restrictive class of locally Hamiltonian flows. More precisely, in [20] we could treat only the special (measure 8 Recall that an Arnold flow is the restriction to the minimal component of a locally Hamiltonian flow in genus one with one saddle and one center, see Figure 1(a).…”

Ratner’s property and mild mixing for smooth flows on surfaces

“…This will give (A). To obtain (B), we use again that the IETs under consideration are of bounded type -this time the phenomenon we observe is that after the points reach the neighborhood of some singularity at time M , they stay "far away" from all singularities for time interval of length εM and we may use estimates from the following result: Proposition 6.2 (Proposition 4.1 in [32], see also Proposition 3.1 in [9]). Let π = (π 0 , π 1 ) be an admissble pair of bijections from A to {1, .…”

“…cf. Remark 3.2 in[9]). One can check that if T is of bounded type, the estimate (18) from the above proposition holds and, furthermore, one can take as (n k ) k∈N the sequence (m k ) k∈N associated with the Marmi-Moussa-Yoccoz acceleration of the Rauzy induction.…”