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Hölder regularity for the spectrum of translation flows. (Régularité Hölder pour le spectre des flots de translation.) (English. French summary) Zbl 1470.37051

J. Éc. Polytech., Math. 8, 279-310 (2021); corrigendum ibid. 9, 1513-1514 (2022).
Given a holomorphic one-form \(\omega\) on a compact orientable surface \(M\) the flow at unit speed along the leaves of the foliation \(\operatorname{Re}(\omega)=0\) defines the vertical flow \(t\mapsto h_t^{+}\) on \(M\), which preserves the area form induced by \(\omega\). A. Katok [Isr. J. Math. 35, 301–310 (1980; Zbl 0437.28009)] showed that this vertical flow is never mixing, and H. Masur [Ann. Math. (2) 115, 169–200 (1982; Zbl 0497.28012)] and W. A. Veech [Ann. Math. (2) 115, 201–242 (1982; Zbl 0486.28014)] showed independently that for almost every abelian differential (with respect to a natural volume measure on the moduli space of abelian differentials) the vertical flow is uniquely ergodic under some additional technical hypotheses. W. A. Veech [Am. J. Math. 106, 1331–1359 (1984; Zbl 0631.28006)] showed also that almost all translation flows, again under a technical hypothesis, are weak mixing. The latter result was later shown in full generality by A. Avila and G. Forni [Ann. Math. (2) 165, No. 2, 637–664 (2007; Zbl 1136.37003)]. Thus the spectrum of translation flows is almost surely continuous and always has a singular component. The authors [Isr. J. Math. 223, 205–259 (2018; Zbl 1386.37035); J. Anal. Math. 141, No. 1, 165–205 (2020; Zbl 1462.37025)] developed tools to find local asymptotics for the spectral measures of translation flows using uniform estimates for twisted Birkhoff integrals and Diophantine arguments. G. Forni [“Twisted translation flows and effective weak mixing”, Preprint, arXiv:1908.11040] found Hölder estimates for spectral measures for surfaces of any genus. Here many of these ideas, including a symbolic formulation of some of Forni’s approach are used to simplify and generalise the author’s earlier arguments, giving a Hölder property (and hence, for example, implications for quantitative rates of weak mixing) for a general class of random Markov compacta, giving in particular the result for almost all (with respect to any of a more general class of measures) translation flows on surfaces of any genus.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
37A25 Ergodicity, mixing, rates of mixing
37E35 Flows on surfaces
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References:

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