Let ϕ be an analytic function defined on the unit disk$D$, with ϕ($D$)⊂$D$, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_{$n$}/λ^{$n$}converges uniformly on compact subsets of$D$to a function σ analytic in$D$which satisfies$σ$°φ=λ$σ$. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space$H$_{$p$}, the sequence Φ_{$n$}/λ^{$n$}converges to σ in the norm of$H$_{$p$}. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

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We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in $${\mathbb{R}^{3}}$$ , we show that they can be transformed with a$C$^{$m$}-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

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