Large sets with small injective projections

Abstract

Let 1 , 2 , . . . be a countable collection of lines in R d . For any t ∈ [0, 1] we construct a compact set Γ ⊆ R d with Hausdorff dimension d − 1 + t which projects injectively into each i , such that the image of each projection has dimension t. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection E of disjoint, non-parallel k-planes in R d , for d ≥ k + 2, whose union is a small subset of R d , either in Hausdorff dimension or Lebesgue measure, while E itself has large dimension. As a second application, for any countable collection of vertical lines w i in the plane we construct a collection of nonvertical lines H, so that F , the union of lines in H, has positive Lebesgue measure, but each point of each line w i is contained in at most one h ∈ H and, for each w i , the Hausdorff dimension of F ∩ w i is zero.

Conformality for a robust class of non-conformal attractors

“…We conclude this subsection computing the Hausdorff dimension of the image of the whole boundary through the graph map. See [17] for examples of homeomorphisms between Cantor sets for which the Hausdorff dimension of the graph exceeds the maximal Hausdorff dimension of the factors. Proposition 5.12.…”