Super-statistics of Blaschke products
speaker: Christopher Penrose (Queen Mary University of London)abstract: Suppose we are given two \(C^2\) orientation-preserving circle maps A and B which both expand Lebesgue measure \(\lambda\) on the unit circle. Given an arbitrary composition \(C = C_l\circ\ldots\circ C_1\) (or word) with \(C_i\in \{A,B\}\) we obtain the unique invariant probability measure \(\lambda_C\) equivalent to \(\lambda\). Considering the special case of iteration alternating between blocks \(C_i = A\) of length m and blocks \(C_i = B\) of langth n (averaged over the \(m+n\) ending phases) we obtain the Birkhoff measure \[ \frac{1}{m+n}\{\sum_{i=1}^m \lambda_{A^i\circ B^n\circ A^{m-i}} + \sum_{j=1}^n \lambda_{B^j\circ A^m\circ B^{n-j}}\} .\] We are interested in the statement that as m and n tend to infinity with fixed ratio \(p:q\) (satisfying \(p+q = 1\)) this Birkhoff measure tends to the super-statistical limit \(p\lambda_A + q\lambda_B\). We examine this convergence in the special case of A and B degree two Blaschke products.
timetable:
Tue 15 Oct, 11:45 - 12:15, Aula Dini
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