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General construction of non-dense disjoint iteration groups on the circle

Krzysztof Ciepliński — 2005

Czechoslovak Mathematical Journal

Let = { F v 𝕊 1 𝕊 1 , v V } be a disjoint iteration group on the unit circle 𝕊 1 , that is a family of homeomorphisms such that F v 1 F v 2 = F v 1 + v 2 for v 1 , v 2 V and each F v either is the identity mapping or has no fixed point ( ( V , + ) is a 2 -divisible nontrivial Abelian group). Denote by L the set of all cluster points of { F v ( z ) , v V } for z 𝕊 1 . In this paper we give a general construction of disjoint iteration groups for which L 𝕊 1 .

The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński — 2004

Czechoslovak Mathematical Journal

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle 𝕊 1 , that is, families = { F v 𝕊 1 𝕊 1 v V } of homeomorphisms such that F v 1 F v 2 = F v 1 + v 2 , v 1 , v 2 V , and each F v either is the identity mapping or has no fixed point ( ( V , + ) is an arbitrary 2 -divisible nontrivial (i.e., c a r d V > 1 ) abelian group).

On conjugacy equation in dimension one

Krzysztof CieplińskiZbigniew Leśniak — 2013

Banach Center Publications

In this paper, recent results on the existence and uniqueness of (continuous and homeomorphic) solutions φ of the equation φ ∘ f = g ∘ φ (f and g are given self-maps of an interval or the circle) are surveyed. Some applications of these results as well as the outcomes concerning systems of such equations are also presented.

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