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Existence of fixed points linked with a periodic orbit of a homeomorphism of the plane. (Existence de points fixes enlacés à une orbite périodique d’un homéomorphisme du plan.) (French) Zbl 0782.55001
Recall first the following definition: Let \(f:\mathbb{R}^ 2\to\mathbb{R}^ 2\) be a homeomorphism of the plane, \(x_ 0\) a fixed point of \(f\), \(x\) a periodic point of period \(n\) and \({\mathcal O}=\{x,f(x),\dots, f^{n- 1}(x)\}\) that orbit. Let \(c\) be an arc in \(\mathbb{R}^ 2-\text{fix}(f)\) joining \(x\) to \(f(x)\) and \(\gamma_ c\) the closed curve obtained by concatenation of the arcs \(c\), \(f(c),\dots,f^{n-1}(c)\). Let now \(\omega(x_ 0,\gamma_ c)\) be the index of \(x_ 0\) with respect to \(\gamma_ c\). Then there is a well-defined integer \(\ell\in \{0,1,\dots,n-1\}\) such that \(\omega(x_ 0,\gamma_ c)-\ell\) is a multiple of \(n\). It is called the linking number \(Lk(x_ 0,{\mathcal O})\).
In this paper, the authors prove the following Theorem: Let \(f:\mathbb{R}^ 2\to \mathbb{R}^ 2\) be an orientation preserving homeomorphism such that \(f- \text{Id}\) is \(k\)-Lipschitz, with \(k\in[0,1]\). Then for every periodic orbit \({\mathcal O}\) of \(f\), there is a fixed point \(x_ 0\) such that \(\omega(x_ 0,\gamma_ c)\) is nonzero.
This gives a partial answer to a question by J. Franks. The proof is short and elementary (it uses only basic plane geometry). As the authors say, the theorem is a special case of a result of M. Handel in which the same conclusion holds with a weaker hypothesis, requiring that the homeomorphism of the plane be isotopic to the identity relative to its fixed point set (which is assumed to be finite), the special case here being that the straight line homotopy gives the desired homotopy. Handel’s result is discussed in the final version of the paper [J. Franks, Invent. Math. 108, 403-418 (1992; Zbl 0766.53037)]. (This is cited as a preprint in the paper under review).

MSC:
55M20 Fixed points and coincidences in algebraic topology
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[1] Kolev, C. R. Acad. Sci. 310 pp 831– (1990)
[2] Brown, Pacific J. Math. 143 pp none– (1990) · Zbl 0728.55001 · doi:10.2140/pjm.1990.143.37
[3] Gambaudo, Math. Proc. Camb. Phil Soc. 108 pp 307– (1990)
[4] Guaschi, Fixed points and linking with periodic orbits of surface difleomorphisms and a generalisation of Brouwer’s lemma (1990)
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