Embeddability of homeomorphisms of the circle in set-valued iteration groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:433

Article

Zdun, Marek Cezary¹ 

[1] Pedag Univ, Inst Math, PL-30084 Krakow, Poland

关键词: Iteration groups;    Flows;    Set-valued functions;    Functional equations limit sets;    Fractional iterates;    Homeomorphisms of the circle;   

DOI  :  10.1016/j.jmaa.2015.08.052

来源: Elsevier

【 摘 要 】

Let F: S-1 -> S-1 be a homeomorphism without periodic points. It is known that F is embeddable in a continuous iteration group if and only if F is minimal. We deal with F which is not minimal. In this case, F satisfying some additional assumptions can be embedded but only in a nonmeasurable iteration groups. There are infinitely many such nonmeasurable groups. We propose here a new approach to the problem of embeddability. For a given homeomorphism F without periodic points we construct some substitute of an iteration group, namely the unique special set-valued iteration group {F-t: S-1 -> cc[S-1], t is an element of R}, which is regular in a sense and in which F can be embedded i.e. F(x) is an element of F-1 (x). We also determine a maximal countable and dense subgroup T subset of R such that {F-t: S-1 -> cc[S-1], t is an element of T} has a continuous selection {f(t): S-1 -> S-1, t is an element of T} being the best regular embedding of F. If there exists a nonmeasurable embedding {f(t): S-1 -> S-1, t is an element of R} of F, then there exists an additive function gamma:R -> T such that f(t)(z) is an element of F-gamma(t)(z),t is an element of R. We determine a unique maximal subgroup T with this property. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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