Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let $f{̃}_{ϵ}$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $Fix\left(f{̃}_{ϵ}\right)=\varnothing$ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving homeomorphisms...

Soit $f$ un homéomorphisme du plan qui préserve l’orientation et qui a un point périodique $z^{*}$ de période $q\ge 2$. Nous montrons qu’il existe un point fixe $z$ tel que le nombre d’enlacement de $z^{*}$ et $z$ ne soit pas nul. En d’autres termes, le nombre de rotation de l’orbite de $z^{*}$ dans l’anneau ${ℝ}^2\setminus \left\{z\right\}$ est un élément non nul de $ℝ/\mathbb{Z}$. Ceci donne une réponse positive à une question posée par John Franks.

Two dynamical deformation theories are presented – one for surface homeomorphisms, called pruning, and another for graph endomorphisms, called kneading – both giving conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. The theories are related to each other and to Thurston’s classification of surface homeomorphisms up to isotopy.