Quantitative weak mixing for interval exchange transformations

Twisted translation flows and effective weak mixing

“…The completion of the proof of weak mixing for typical IET's (see [AvF07], [AFS23]) is based on a non-linear, probabilistic elimination scheme.…”

“…In other terms, in the definition of the effective stable space the convergence takes place only in average and excursion of the cohomology class outside of any given neighborhood of the integer lattice are possible provided their frequency converges to zero in the large time limit. Despite this importance difference, all the basic properties of the stable space (in particular concerning its Hausdorff dimension) hold for the effective weak stable space, and the linear elimination procedure outlined in section 4.5 can be carried out (see [F22a]), as well as the non-linear, probabilistic elimination (see [AFS23]). By the effective Veech criterion (Theorem 4.23 and by the linear elimination argument we can then derive the following:…”

“…The above theorem implies the last equivalent property in Definition 2.20 for the space W 1 (M), hence it concludes (see also Exercise 2.21) the outline of the proof of polynomial weak mixing for typical translation flows stated in Theorem 2.22. For the proof of effective weak mixing for IET's, based on the effective Veech criterion and the probabilistic elimination procedure, we refer the reader to [AFS23].…”

Size of exceptional sets in weakly mixing systems

“…The rate of weak mixing on the typical non-rotation class of IETs is C N = o(N −α ) for some α > 0. The rate on the typical rotation class of IETs is C N = o(log −a N ) for some a > 0 (See [4]).…”