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UID:7114@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20190226T110000
DTEND;TZID=Europe/Paris:20190226T120000
DTSTAMP:20241120T203423Z
URL:https://www.i2m.univ-amu.fr/evenements/cohomological-equations-for-lin
 ear-involutions/
SUMMARY:Alexandra Skripchenko (Higher School of Economics\, Moscow\, Russia
 ): Cohomological equations for linear involutions
DESCRIPTION:Alexandra Skripchenko: The famous Roth's theorem about diophant
 ine approximation states that a given algebraic number may not have too ma
 ny rational number approximations\, that are “very good”. More precise
 ly\, Roth first defined a class of numbers that are not very easy to appro
 ximate by rationals (they are called Roth numbers) and then showed that al
 most all algebraic irrationals are of Roth type\, and that they form a set
  of a full measure which is invariant under the natural action of the modu
 lar group SL(2\,Z).\n\nIn addition to their interesting arithmetical prope
 rties\, Roth type irrationals appear in a study of the cohomological equat
 ion associated with a rotation {R}{a} : {R}{a}({x}) = {x} + {a} of the cir
 cle T={{R}}/{{Z}}: {a} is of Roth type if and only iff for all {r}\, {s} :
  {r} &gt\; {s} + 1 &gt\; 1 and for all functions Φ of class {C}{r} on T w
 ith zero mean there exists a unique function Ψ ∈ {C}{s}(T) with zero me
 an such that\n\nΨ−Ψ∘{R}{a} = Φ.\n\nIn 2005 Marmi\, Moussa and Yocco
 z established an analogue of Roth theorem for interval exchange transforma
 tions (IETs). In particular\, they defined the notion of Roth type IETs an
 d proved existence of the solution of cohomological equation for this clas
 s\; they also showed that IET of Roth type form a full measure set in the 
 parameter space of IETs.\n\nIn a fresh joint work with Erwan Lanneau and S
 tefano Marmi we get a certain generalization of this result for linear inv
 olutions that can be considered as a natural extension of IETs to non-orie
 ntable case.\nhttps://arxiv.org/abs/1908.09107\n\n&nbsp\;
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Alexandra_Skripchenko.jpg
CATEGORIES:Séminaire,Ernest
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