Riesz $\alpha$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces

TL;DR

This paper investigates the cyclicity threshold of functions in Dirichlet-type spaces, constructing counterexamples using Cantor sets and Riesz capacity to show cyclicity persistence fails at critical indices.

Contribution

It provides a novel construction of functions demonstrating the failure of cyclicity persistence at the critical index in Dirichlet-type spaces.

Findings

Constructed functions cyclic below the critical index but not at it.

Used generalized Cantor sets and Riesz capacity in the construction.

Showed cyclicity does not necessarily persist at the critical index.

Abstract

We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_{\alpha}$, $\alpha\in(0,1]$. Given a fixed $\alpha^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{\alpha^{*}}$ which is cyclic in $\mathcal{D}_{\alpha}$ for all $\alpha<\alpha^{*}$, but fails to be cyclic in $\mathcal{D}_{\alpha^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $\alpha^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $\alpha$-capacity.