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 $\varnothing \ne L_{ℱ}\ne {𝕊}^1$.

We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of...