On set-valued iteration groups generated by commuting functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:398

Article

Zdun, Marek Cezary

关键词: Iteration groups;    Multi-valued functions;    Functional equations;    Set-valued iteration semigroups;    Embedding;   

DOI  :  10.1016/j.jmaa.2012.09.016

来源: Elsevier

【 摘 要 】

Let I be an open interval and f , g:l -> I be commuting homeomorphisms such that f < id, g < id and inf{n, m is an element of N, f(n) < g(m)} =: s is not an element of Q. Definer f(+)(t) := sup{f(n) o g(-m) : n, m is an element of N, n - sm < t), f_(t) := inf{f(n) o g(-m) : n, m is an element of N, n - sm > t} and F-t (x) := [f_(t)(x), f(+)(t) (x)], x E I. We show that the family (F-t :I -> cc(I), t is an element of R) is an iteration group in the sense of set-valued functions such that f (x) is an element of F-1 (x) and g(x) E F-s (x). If f and g are embeddable in an iteration group of continuous functions (f(r) : t is an element of R), then there exists an additive function gamma such that f(t) (x) is an element of F-gamma(t)(X), t is an element of R. Several other properties of set-valued iteration groups (F-t : t is an element of R} as well as iteration groups {f_(t) : t is an element of R} and {f(+)(t) : t is an element of R) are given. (C) 2012 Elsevier Inc. All rights reserved.

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