On the fine structure of stationary measures in systems which contract-on-average
| dc.creator | Nicol, Matthew | |
| dc.creator | Sidorov, Nikita | |
| dc.creator | Broomhead, David | |
| dc.date | 2000-05-22 | |
| dc.date.accessioned | 2026-07-07T04:35:25Z | |
| dc.date.available | 2026-07-07T04:35:25Z | |
| dc.description | Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a corollary show that the measure will be singular if the modulus of the entropy $\sum_i p_i \log p_i$ is less than $d$ times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of $\mathbb{R}$. | |
| dc.description | Final version; 14 pages in Latex | |
| dc.identifier | https://arxiv.org/abs/math/0005211 | |
| dc.identifier | http://arxiv.org/abs/math/0005211 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59248 | |
| dc.subject | Probability | |
| dc.subject | Dynamical Systems | |
| dc.subject | 60J05; 28D20, 20M20 | |
| dc.title | On the fine structure of stationary measures in systems which contract-on-average | |
| dc.type | text |