GENERAL CONSTRUCTION OF NON-DENSE DISJOINT ITERATION GROUPS ON THE CIRCLE


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Abstract

Let ℱ = {Fv: 𝕊1 → 𝕊1, vV} be a disjoint iteration group on the unit circle 𝕊1, that is a family of homeomorphisms such that Fv1 O Fv2 = Fv1+v2 for v1, v2V and each Fv 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 {Fv(z), vV} for z ∈ 𝕊1. In this paper we give a general construction of disjoint iteration groups for which θ ≠ L ≠ 𝕊1.

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