Supercritical Holes for the Doubling Map

For a map ({S : X to X}) and an open connected set (= a hole) ({H subset X}) we define ({mathcal{J}_H(S)}) to be the set of points in X whose S-orbit avoids H. We say that a hole H ₀ is supercritical if

1. for any hole H such that ({overline{H}_0 subset H}) the set ({mathcal{J}_H(S)}) is either empty or contains only fixed points of S;

1. for any hole H such that ({overline{H} subset H_0}) the Hausdorff dimension of ({mathcal{J}_H(S)}) is positive.

The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx =  2x mod 1.