Abstract
We consider linear iterated function systems with a random multiplicative error on the real line. Our system is {x → di + λiYx. where di ∈ ℝ and λi > 0 are fixed and Y > 0 is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors y1, y2,..., distributed as Y, independent of everything else. Let h be the entropy of the process, and let χ =double struck E sign[log(λY)] be the Lyapunov exponent. Assuming that χ < 0, we obtain a family of conditional measures νy on the line, parametrized by y = (y1, y2, . . . ), the sequence of errors. Our main result is that if h > |χ|. then νy is absolutely continuous with respect to the Lebesgue measure for almost every y. We also prove that if h < |χ|, then the measure νy is singular and has dimension h/|χ| almost every y. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia. motivated by probabilistic number theory.
| Original language | English |
|---|---|
| Pages (from-to) | 739-756 |
| Number of pages | 18 |
| Journal | Journal of the London Mathematical Society |
| Volume | 74 |
| Issue number | 3 |
| DOIs | |
| State | Published - Dec 2006 |
| Externally published | Yes |
Bibliographical note
Funding Information:The research of the first author was partially supported by NSF grants #DMS-0104073 and #DMS-0244479. Part of this work was completed while he was visiting Microsoft Research. The research of the third author was partially supported by NSF grants #DMS-0099814 and #DMS-0355187. The research of the second author was partially supported by OTKA Foundation grant #T42496. The collaboration of the second and third authors was supported by NSF-MTA-OTKA grant #77.