Aequationes mathematicae

, Volume 87, Issue 3, pp 201–245 | Cite as

Recent results on iteration theory: iteration groups and semigroups in the real case

Open Access
Article

Abstract

In this survey paper we present some recent results in the iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embedddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in \({{\mathbb{R}}^n}\) space; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers–Ulam stability of the translation equation. Most of the results presented here was obtained by the means of functional equations. We indicate the relations between the iteration theory and functional equations.

Mathematics Subject Classification (2000)

39B12 39B82 39B22 37C15 37E10 37E30 26A18 26E25 54C25 54H15 57S05 

Keywords

Iteration groups and semigroups flows iterative root embedding conjugacy functional equations stability set-valued functions mean-type mapping nonmonotonicity height 

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Authors and Affiliations

  1. 1.Institute of MathematicsPedagogical UniversityKrakówPoland

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