Lisa Lorentzen

Affiliation: Norwegian University of Science and Technology


Title Of Talk: Convergence of random continued fractions

Abstract: It is amazing to see how willingly a continued fraction converges. It has therefore been a dream of mine to prove that "almost all continued fractions converge" in some sense. Now, a continued fraction can be regarded as a sequence of Möbius transformations, and thus, as a sequence of non-singular $(2\times 2)$ matrices. So if we define a random continued fraction $K(a_n/b_n)$ as a stochastic variable where the elements $(a_n,b_n)$ are picked independently from a given distribution $\mu$, we can adapt the theory of random products of matrices to prove convergence of $K(a_n/b_n)$ with probability 1. Of course, there are conditions on the measure, but it turns out that a finite expectation of the expression $\log (|a_1|+(1+|b_1|^2)/|a_1|)$ plus some extra very mild technical conditions suffices. I have talked on this topic on different occasions. The new thing in this talk is that we are interested in the speed of convergence that can be obtained with probability 1.

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