Circle diffeomorphisms with several break points: symmetry and hyperbolicity of the renormalization operator
speaker: Alexey Teplinsky (University of Toronto)abstract: In our previous project, we showed that any two $C{2+\alpha}$-smooth circle diffeomorphisms, $\alpha>0$, each having a single break (i.e. a jump in the first derivative), with the same size of breaks and with the same irrational rotation number from certain class are $C1$-smoothly conjugate (Commun. Math. Phys., 320:2, 2013). In order to achieve this, in particular, we have constructed a renormalization operator in appropriate functional space and proved its uniform hyperbolicity. Our present aim is to expand this theory to the case of $N\ge1$ breaks. We introduce the renormalization procedure based on the S.Ferenczi's induction for interval exchanges and show that the consecutive renormalizations, which are arrays of $2N$ smooth functions acting on definite intervals, approach a $2N$-parameter family of (normalized) arrays of linear-fractional functions intertwined in $N$ commutation relations. We explicitly describe the time-reverse symmetry of the renormalization operator acting on that family and explain the essence of its $N$ contracting and $N$ expanding directions. We believe that this will eventually lead to the following rigidity statement: any two $C{2+\alpha}$-smooth, $\alpha>0$, circle diffeomorphisms with breaks with the same irrational rotation number from certain class (containing all numbers of bounded type), the same sizes of corresponding breaks and the same invariant measures of intervals between corresponding break points are $C{1}$-smoothly conjugate. This is work in progress. (Joint with K.Khanin.)
timetable:
Wed 12 Jun, 15:00 - 15:50, Aula Dini
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