Abstract
We show that there exists 0<α0<1 (depending on the parameters) such that the fractal percolation is almost surely purely α-unrectifiable for all α>α0.
Highlights
Broman et al [4] showed that, in the case d = 2 and p ≥ pc, the set E can be decomposed as E = Ec ∪ Ed, where Ed is the totally disconnected part of E and Ec consists of non-trivial connected components of E
DimH Ec < dimH Ed = dimH E and there exists 0 < β < 1, depending on the parameters, such that Ec is an uncountable union of non-trivial β-Hölder curves
Due to the definition of unrectifiability, one has to show that, almost surely, the intersection of the fractal percolation set with all α-Hölder curves has zero measure
Summary
We describe the heuristic idea behind the proof of our main result concerning the α-unrectifiability of fractal percolation. Using α-Hölder continuity and Jensen’s inequality on the right hand side, one obtains (1 + r)q|γ(a) − γ(b)| ≤ C |ai − ai+1|α ≤ CLq(L−q|a − b|)α If this is true for large q, we have that 1 + r ≤ L1−α, that is, the exponential growth rate of the broken line approximation is controlled by the Hölder exponent. The idea for the existence of r is as follows: If the intersection of a curve with the fractal percolation set has positive measure, the density point theorem implies that, at small scales, there cannot be big gaps, that is, parts of the curve whose intersection with the set is empty. In order to have many successive scales with holes, high probability for the existence of a hole is needed This is obtained by decreasing the size of a hole, which, in turn, makes r smaller. Due to the definition of unrectifiability, one has to show that, almost surely, the intersection of the fractal percolation set with all α-Hölder curves has zero measure
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