Incomputability of Simply Connected Planar Continua

Le Roux and Ziegler asked whether every simply connected compact nonempty planar Π 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 Π 1 dendroid without computable points. We also provide several pathological examples of tree-like Π 1 continua fulfilling certain global incomputability properties: there is a computable dendrite which does not ∗-include a Π 1 tree; there is a Π 1 dendrite which does not ∗-include a computable dendrite; there is a computable dendroid which does not ∗-include a Π 1 dendrite. Here, a continuum A ∗-includes a member of a class P of continua if, for every positive real ε, A includes a continuum B ∈ P such that the Hausdorff distance between A and B is smaller than ε.