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Mixing for smooth time-changes of general nilflows. (English) Zbl 1479.37022

Continuous-time dynamical systems can be roughly divided into hyperbolic, for which the speed of divergence of nearby orbits is exponential, parabolic, for which the divergence happens at a subexponential speed and flows with no divergence, which are called elliptic. Hyperbolic and elliptic flows have deep theories and the landscape is much more solid than that of parabolic flows. This paper aims at this a gap and it contributes in the understanding of typical properties of parabolic flows.
In particular, the authors try to answer to the following question: which ergodic and spectral properties are generic among smooth parabolic flow? To formalize a setting in which this question can be put in precise terms, they consider time changes of {completely irrational nilflows}. A {nilmanifold} is the quotient \(M=\Gamma\setminus G\) of a nilpotent Lie group \(G\) by a lattice \(\Gamma\) of step at least \(2\) (which excludes the possibility of \(M\) being a torus). One considers then a translation flow \(\phi=\{\phi\}_{t\in\mathbb{R}}\) on \(M\) which is uniquely ergodic. This means that the linear flow on the torus factor of \(M\) is completely irrational (see Theorem 2.2). A (smooth) time-change of a flow is a new flow, with the same orbits, travelled with the same orientation but with different speeds (and with smooth reparametrization function).
The mains result of the paper is that there exists a dense set \(\mathcal{P}\) of smooth (reparametrization) functions such that every time-change generated by a positive element in \(\mathcal{P}\) is either measurably trivial (and therefore the resulting flow is measurably conjugated with the original one) or it is mixing. It should be pointed out that in general there are some (cohomological) obstructions for a smooth time change to be measurably trivial, and the set of functions which generate non-trivial time changes (as in the statement of the main result) are generic.
I would like to remark that the paper is beautifully written with a very informative introduction which is nice to follow for non-experts on parabolic dynamics.

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37A25 Ergodicity, mixing, rates of mixing
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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