Differences of random Cantor sets and lower spectral radii

Details Differences of random Cantor sets and lower spectral radii Differences of random Cantor sets and lower spectral radii

2008-11-04

arxiv.org/abs/0811.0525v1

Mathematics

Probability

26 pages, 8 figures

Scientific paper

We investigate the question under which conditions the algebraic difference between two independent random Cantor sets $C_1$ and $C_2$ almost surely contains an interval, and when not. The natural condition is whether the sum $d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that \emph{generically} it should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities $(p_0,p_1)$ the interior of the region where the Palis conjecture does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>\sqrt{2}$ and $p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of $2\times 2$ matrices.

Affiliated with

Also associated with

No associations

Rating

Differences of random Cantor sets and lower spectral radii does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Differences of random Cantor sets and lower spectral radii, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Differences of random Cantor sets and lower spectral radii will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-184364