Cohomological equations for linear involutions

Alexandra Skripchenko
Higher School of Economics, Moscow, Russia

Date(s) : 26/02/2019   iCal
11 h 00 min - 12 h 00 min

The famous Roth’s theorem about diophantine approximation states that a given algebraic number may not have too many rational number approximations, that are “very good”. More precisely, Roth first defined a class of numbers that are not very easy to approximate by rationals (they are called Roth numbers) and then showed that almost 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 modular group SL(2,Z).

In addition to their interesting arithmetical properties, Roth type irrationals appear in a study of the cohomological equation associated with a rotation {R}{a} : {R}{a}({x}) = {x} + {a} of the circle T={{R}}/{{Z}}: {a} is of Roth type if and only iff for all {r}, {s} : {r} > {s} + 1 > 1 and for all functions Φ of class {C}{r} on T with zero mean there exists a unique function Ψ ∈ {C}{s}(T) with zero mean such that

Ψ−Ψ∘{R}{a} = Φ.

In 2005 Marmi, Moussa and Yoccoz established an analogue of Roth theorem for interval exchange transformations (IETs). In particular, they defined the notion of Roth type IETs and proved existence of the solution of cohomological equation for this class; they also showed that IET of Roth type form a full measure set in the parameter space of IETs.

In a fresh joint work with Erwan Lanneau and Stefano Marmi we get a certain generalization of this result for linear involutions that can be considered as a natural extension of IETs to non-orientable case.



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