Håkan Hedenmalm: The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps

Abstract. This reports on joint work with A. Montes-Rodriguez. The original problem was to decide whether for two atomic inner functions,the linear span of the nonnegative integer powers of each are able to span

$H^\infty$ in the weak-star topology. This problem is a sharpening of a problem raised by Matheson and Stessin in 2005, who allowed products as well (so as to form the algebra). It was known that a condition was necessary (because points were not separated without it) but not whether it was sufficient.

In a 2011 Annals paper, Hedenmalm and Montes-R solved the corresponding problemfor L^\infty with all integer powers, not just nonnegative. Unexpectedly, the proofused the Birkhoff ergodic theorem in the case of an infinite invariant measure, associated with a certain Gauss-type map. Here, we return to the original problem and solve it completely. The methods require an extension of ergodic theory to spaces of distributions in place of the standard space L^1. Transfer operators are no longer contractive, so really novel methods were conceived: a new amalgam of ergodic theory and harmonic analysis.