Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

David Maps and Hausdorff Dimension

David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show: – Given α and β in [0, 2] , there exists a David map φ: C → C and a compact set Λ such that dimH Λ = α and dimH φ(Λ) = β . – There exists a David map φ: C → C such that...

Stability index for chaotically driven concave maps

We study skew product systems driven by a hyperbolic base map Ŝ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R+ like x → ĝ(θ) arctan(x), where θ ∈ Θ is a parameter driven by the base map. The fibrewise attractor is the graph of an upper semicontinuous function θ → φ̂∞(θ) ∈ R+. For many choices of ĝ, φ̂∞ has a residual set of zeros, b...

Hausdorff dimension of the multiplicative golden mean shift

Hausdorff Dimension for Fractals Invariant under the Multiplicative Integers

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences (xk) such that xkx2k = 0 for all k. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for mult...

Hausdorff Dimension of Metric Spaces and Lipschitz Maps onto Cubes

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can be always mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff...