Author(s)

Ruotian Yang

Corresponding Author

Ruotian Yang

Abstract

We construct, for any dtarget > 0, a subset of a Euclidean space with Hausdorff dimension dtarget. The fractional part is realized by a linear, symmetric two-strip Smale horseshoe on [0, 1]2 with expansion λ > 2 (horizontal contraction 1/λ), C1+α-smoothed off the invariant set; in this model the invariant set has dimension D(λ) = 2ln2/lnλ, a continuous, strictly decreasing map with range (0, 2). The integer part follows from dimH (A × [0, 1]n ) = dimH (A) + n. We briefly recall the needed tools and give explicit examples.

Keywords

Hausdorff Dimension, Smale Horseshoe, Dynamical Systems, Con-Structive Proof, Fractal Geometry