Noninvariance of an approximation property for closed subsets of Riemann surfaces

Abstract:

A closed subset E of an open Riemann surface M is said to have the approximation property $\mathcal {a}$ if each continuous function on E which is analytic at all interior points of E can be approximaed uniformly on E by functions which are everywhere analytic on M. It is known that $\mathcal {a}$ is a topological invariant (i.e., preserved by homeomorphisms of the pair $(M,E)$) when M is of finite genus but not in general, not even for ${C^\infty }$ quasi-conformal automorphisms of M. The principal result of this paper is that $\mathcal {a}$ is not invariant even under a real-analytic isotopy of quasi-conformal automorphisms (of a certain M). M is constructed as the two-sheeted unbranched cover of the plane minus a certain discrete subset of the real axis, and the isotopy is induced by $(x + iy, t) \mapsto x + ity$, for $t > 0$; E can be taken to be that portion of M which lies over a horizontal strip.