Continuity of the Hausdorff Dimension for Invariant Subsets of Interval Maps

Let T : [0, 1] → [0, 1] be an expanding piecewise monotonic map, and consider the set R of all points, whose orbits omit a certain finite union of open intervals. It is shown that the Hausdorff dimension HD (R) depends continuously on small perturbations of the endpoints of these open intervals. A similar result for the topological pressure is also obtained. Furthermore it is shown that for every t ∈ [0, 1] there exists a closed, T -invariant Rt ⊆ [0, 1] with HD (Rt) = t. Finally it is proved that the Hausdorff dimension of the set of all points, whose orbit is not dense, is 1.