Correlated fractal percolation and the Palis conjecture

TL;DR

This paper proves that for correlated fractal percolation, the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions exceeds one, confirming a strong version of the Palis conjecture.

Contribution

The authors develop a new condition to analyze correlated fractal percolation and prove the Palis conjecture holds in this setting, extending previous results.

Findings

The algebraic difference contains an interval almost surely if and only if the sum of Hausdorff dimensions exceeds one.

A new condition for correlated fractal percolation is introduced and shown to imply previous conditions.

The strong version of the Palis conjecture is confirmed for correlated fractal percolation.

Abstract

Let F1 and F2 be independent copies of correlated fractal percolation, with Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will contain an interval? The well known Palis conjecture states that `generically' this should be true. Recent work by Kuijvenhoven and the first author (arXiv:0811.0525) on random Cantor sets can not answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of (arXiv:0811.0525) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.