Absolute continuity for random iterated function systems with overlaps

Yuval Peres, K. Simon, Boris Solomyak

Research output: Contribution to journalArticle

16 Citations (Scopus)

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 languageEnglish
Pages (from-to)739-756
Number of pages18
JournalJournal of the London Mathematical Society
Volume74
Issue number3
DOIs
Publication statusPublished - Dec 2006

Fingerprint

Random Systems
Absolute Continuity
Iterated Function System
Absolutely Continuous
Overlap
Ergodic Processes
Random Sets
Perturbed System
Continuous Distributions
Number theory
Stationary Process
Lebesgue Measure
Real Line
Identically distributed
Lyapunov Exponent
Multiplicative
Random variable
Linear Systems
Entropy
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Absolute continuity for random iterated function systems with overlaps. / Peres, Yuval; Simon, K.; Solomyak, Boris.

In: Journal of the London Mathematical Society, Vol. 74, No. 3, 12.2006, p. 739-756.

Research output: Contribution to journalArticle

@article{2ee4ea060d244f34b369b0ab45fad7d4,
title = "Absolute continuity for random iterated function systems with overlaps",
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.",
author = "Yuval Peres and K. Simon and Boris Solomyak",
year = "2006",
month = "12",
doi = "10.1112/S0024610706023258",
language = "English",
volume = "74",
pages = "739--756",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Oxford University Press",
number = "3",

}

TY - JOUR

T1 - Absolute continuity for random iterated function systems with overlaps

AU - Peres, Yuval

AU - Simon, K.

AU - Solomyak, Boris

PY - 2006/12

Y1 - 2006/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33846080538&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846080538&partnerID=8YFLogxK

U2 - 10.1112/S0024610706023258

DO - 10.1112/S0024610706023258

M3 - Article

AN - SCOPUS:33846080538

VL - 74

SP - 739

EP - 756

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

ER -