Seminar 11.12.15 Constantine

Speaker: Dave Constantine (Wesleyan)

Title: Circle rotations and conditionally convergent series

Abstract: Take the harmonic series and assign to its terms a pattern of + and – signs by a fair coin flip. One can show that this process produces a convergent series almost surely, although there are (of course) sequences of flips which give divergence. Now, instead of a random process, let’s produce these signs deterministically. In particular, let $f$ be a function on $X$ taking values in $\{+1,-1\}$ and let $T:X\to X$. We can similarly investigate the convergence of $\sum f(T^nx)/n$.

In this talk I’ll discuss this problem for an irrational circle rotation and a function assigning +1 to the top half of the circle and -1 to the bottom half. I’ll show that while convergence is the almost sure behavior, there are irrational rotations which give divergence, that all of them are Liouville numbers, but that not all Liouville numbers give divergent series. We’ll also take a quick look at what seems to happen when the series converges.

This is joint work with Joanna Furno (IUPUI).