Incomputability of Simply Connected Planar Continua

Article type: Research Article

Authors: Kihara, Takayuki^;

Affiliations: Mathematical institute, Tohoku University, Sendai, Miyagi, Japan. [email protected]

Note: [] This work was supported by Grant-in-Aid for JSPS fellows.

Abstract: Le Roux and Ziegler asked whether every simply connected compact nonempty planar $\Pi^0_1$ set always contains a computable point. In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a planar $\Pi^0_1$ dendroid without computable points. We also provide several pathological examples of tree-like $\Pi^0_1$ continua fulfilling certain global incomputability properties: there is a computable dendrite which does not *-include a $\Pi^0_1$ tree; there is a $\Pi^0_1$ dendrite which does not *-include a computable dendrite; there is a computable dendroid which does not *-include a $\Pi^0_1$ dendrite. Here, a continuum A *-includes a member of a class $\mathcal{P}$ of continua if, for every positive real ε, A includes a continuum $B \in \mathcal{P}$ such that the Hausdorff distance between A and B is smaller than ε.

Keywords: Computable Analysis, Type-two-theory of effectivity, Effectively closed sets

DOI: 10.3233/COM-12012

Journal: Computability, vol. 1, no. 2, pp. 131-152, 2012