PACKING DIMENSION OF MEASURES ON A RANDOM CANTOR SET

Translating the Cantor set by a random

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness” in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructiv...

On the Dimension of Deterministic and Random Cantor-like Sets

Is Quantum Space a Random Cantor Set with a Golden Mean Dimension at the Core?

In ref. [l], Mauldin et al. proved a theorem which at first sight may seem slightly paradoxical but we perceive as exceedingly interesting. This theorem states that the Hausdorff dimension dy’ of a randomly constructed Cantor set is d:’ = @, where $ = (d5 1)/2 is the G o Id en Mean. That such disordered indeterministic construction which actually epitomize dissonance should single out the Golde...

On Homeomorphic Product Measures on the Cantor Set

Let μ(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1 − r. It is a long-standing open problem to characterize those r and s such that μ(r) and μ(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending μ(r) to μ(s)). The (possibly) weaker property of μ(r) and μ(s) being contin...

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a param...