On \(C^1\) rigidity of circle maps with a break point
speaker: Elio Mazzeo (University of Toronto)abstract: I will discuss the two main results that were obtained in the rigidity theory of circle maps with a break point, as part of my thesis. The first main result is a proof that \(C^1\) rigidity holds for circle maps with a break point for almost all irrational rotation numbers. In particular, the class of rotation numbers for which \(C^1\) rigidity is proven to hold are those rotation numbers \(\rho\) for which (for every other \(n\)) the partial quotient \(k_{n+1}\) of the continued fraction expansion of \(\rho\) does not grow larger than a certain exponentially large quantity in \(n\). In other words, \(\rho\) should not be too well approximated by rationals in order for \(C^1\) rigidity to hold.
The second main result has to do with the 3-parameter family of fractional linear transformation (FLT) pairs. An FLT-pair \(T\) is a circle homeomorphism that consists of two branches each of which is an FLT. An FLT-pair can be viewed as a Generalized Interval Exchange Transformation (G-IET) with two intervals. In this context, it is not a natural condition to assume that at each of the two connection points of the branches the left and right derivatives of \(T\) must agree. If one considers the family of FLT pairs such that the product of the jump ratios (the ratio of the left to right derivative) at each of the two `break points' is a fixed number \(c^2\neq1\), one obtains a \(3\)-parameter family. We proved that for this family \(C^1\) rigidity holds for all irrational rotation numbers without any restriction on the growth rate of \(k_{n+1}\).
timetable:
Wed 12 Jun, 16:30 - 16:55, Aula Dini
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