Dynamics of a higher dimensional analog of the trigonometric functions

We introduce a quasiregular analog F of the sine and cosine function such that, for a sufficiently large constant λ, the map x 7→ λF (x) is locally expanding. We show that the dynamics of this map define a representation of Rd, d ≥ 2, as a union of simple curves γ : [0,∞) → Rd which tend to ∞ and whose interiors γ∗ = γ((0,∞)) are disjoint such that the union of all γ∗ has Hausdorff dimension 1.