Lisa Lorentzen

Affiliation: Norwegian University of Science and Technology

Email: lisa@math.ntnu.no

Title Of Talk: Continued Fractions Converge with Probability 1

Abstract: An old dream of mine is to some day produce a paper titled ``Almost All Continued Fractions Converge", because that is what I believe to be true in some sense. Unfortunately, this is not what I have proved. But, by means of Furstenberg's deep convergence theorems in ergodic theory, I have proved: Let $\Phi$ be a probability distribution for pairs $(a,b)$ of complex numbers. Construct a random continued fraction $K(a_n/b_n)$ where the pairs $(a_n,b_n)$ are picked $\Phi$-randomly and independently. Then, under proper conditions on $\Phi$, the random continued fraction $K(a_n/b_n)$ converges with probability 1. And under very mild conditions on $\Phi$, $K(a_n/b_n)$ is restrained with probability 1.

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