Carathéodory Convergence and Harmonic Measure

Abstract

We give several new characterizations of Carathéodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

On the continuity of half-plane capacity with respect to Carathéodory convergence

“…By assumption, D \ F n → D \ F ∞ as n → ∞ in Carathéodory's sense. Then these domains have "arbitrarily good common interior approximations" by the implication (i) ⇒ (ii) in Theorem 2.3 of Binder, Rojas and Yampolsky [2]. Although this theorem assumes the simple connectivity, the implication needed here remains valid even if domains are not simply connected.…”

“…The merit of our method is that the hitting probability of the ABM to a hull F is given by the harmonic measure of H \ F . In fact, Binder, Rojas and Yampolsky [2] showed that the Carathédory convergence implies the weak convergence of harmonic measures under certain assumptions. Their result plays a central role in the proof of (1.3).…”

“…Propositions 2.3 and 2.4 hold for BMD and BMD half-plane capacity with obvious modifications. Thus, in order to prove Theorem 3.1, it suffices to show the following: Fn in terms of ABM;(ii) Use Theorem 3.1 of Binder, Rojas and Yampolsky[2] to show that hm D\Fn (ζ, •) → hm D\F∞ (ζ, •) weakly.…”

Computability in Harmonic Analysis

“…x at a single point x = x 0 ∈ Ω is sufficient to guarantee the existence of such an algorithm for any other point x in the domain Ω. The proof of this theorem is based on a new concept of harmonic approximation of domains, which builds on the ideas of another work of the subset of the authors [3]. We show that having an algorithm for approximating a domain harmonically is equivalent to being able to compute all of its harmonic measures.…”

“…It was proven in [3] that for simply connected planar domains the weak convergence of harmonic measures is equivalent to the classical Carathéodory convergence. In light of these results, it is not surprising that the condition of Theorem F is exactly the same as the condition for the existence of computable Riemann bijection in Computable Riemann Mapping Theorem (see [5]).…”