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Stability of iterated function systems on the circle
Author(s) -
Szarek Tomasz,
Zdunik Anna
Publication year - 2016
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdw013
Subject(s) - mathematics , iterated function , iterated function system , equicontinuity , function (biology) , operator (biology) , orbit (dynamics) , pure mathematics , stability (learning theory) , fixed point , mathematical analysis , attractor , engineering , biology , aerospace engineering , biochemistry , chemistry , repressor , evolutionary biology , machine learning , gene , transcription factor , computer science
We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron–Frobenius operator is shown to satisfy the e‐property, that is, for any continuous function its iterates are equicontinuous. The Strong Law of Large Numbers for trajectories starting from an arbitrary point for such function systems is also proved.