Differences of random Cantor sets and lower spectral radii
| dc.creator | Dekking, F. Michel | |
| dc.creator | Kuijvenhoven, Bram | |
| dc.date | 2008-11-04 | |
| dc.date.accessioned | 2026-07-07T10:15:25Z | |
| dc.date.available | 2026-07-07T10:15:25Z | |
| dc.description | 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. | |
| dc.description | 26 pages, 8 figures | |
| dc.identifier | https://arxiv.org/abs/0811.0525 | |
| dc.identifier | http://arxiv.org/abs/0811.0525 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/173167 | |
| dc.subject | Probability | |
| dc.subject | 60D05 | |
| dc.title | Differences of random Cantor sets and lower spectral radii | |
| dc.type | text |