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The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${𝕊}^1$, that is, families $ℱ=\{F^v{𝕊}^1⟶{𝕊}^{1}v\in V\}$ of homeomorphisms such that $$F^{v_1}\circ F^{v_2}=F^{v_1+v_2},v_1,v_2\in V,$$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathrm{c}ardV>1$) abelian group).

Let $ℱ=\{F^v{𝕊}^1\to {𝕊}^1,v\in V\}$ be a disjoint iteration group on the unit circle ${𝕊}^1$, that is a family of homeomorphisms such that $F^{v_1}\circ F^{v_2}=F^{v_1+v_2}$ for $v_1$, $v_2\in V$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is a $2$-divisible nontrivial Abelian group). Denote by $L_{ℱ}$ the set of all cluster points of $\{F^v\left(z\right)$, $v\in V\}$ for $z\in {𝕊}^1$. In this paper we give a general construction of disjoint iteration groups for which $\varnothing \ne L_{ℱ}\ne {𝕊}^1$.

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.