Posted by: matheuscmss | July 4, 2019

“Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant”

Patrice Le Calvez and Jean-Christophe Yoccoz showed in 1997 that there are no minimal homemorphisms on the infinite annulus \mathbb{R}/\mathbb{Z}\times\mathbb{R}.

Their beautiful paper was motivated by the quest of finding minimal homeomorphisms on punctured spheres \mathbb{S}^2\setminus\{p_1,\dots,p_k\}. More concretely, the non-existence of such homeomorphism was previously known when k=0 (as an easy application of the features of Lefschetz indices), k=1 (thanks to the works of Brouwer and Guillou), and k\geq 3 (thanks to the work of Handel), so that the main result in Jean-Christophe and Patrice paper ensures the non-existence of minimal homeomorphisms in the remaining (harder) case of k=2.

A key step in Jean-Christophe and Patrice proof of their theorem above is to establish the following result about the sequence of Lefschetz indices i(f^k,z) of iterates f^k of a local homeomorphism f of the plane at a fixed point z of f: if z is not a sink nor a source, then there are integers q, r\geq 1 such that

i(f^k,z) = \left\{\begin{array}{cc} 1-rq & \textrm{ if }k\in q\mathbb{Z} \\ 1 & \textrm{ otherwise } \end{array}\right.

As it turns out, Jean-Christophe and Patrice planned a sequel to this paper with the idea of extending their techniques to compute the sequences of Lefschetz indices of periodic points of f belonging to any given Jordan domain U with K=\bigcap\limits_{n\in\mathbb{Z}} f^{-k}(U) is compact.

In fact, this plan was already known when the review of Jean-Christophe and Patrice paper came out (see here), and, as Patrice told me, some arguments from this promised subsequent work were used in the literature as a sort of folklore.

Nevertheless, a final version of this preprint was never released, and, even worse, some portions of the literature were invoking some arguments from a version of the preprint which was available only to Jean-Christophe (but not to Patrice).

Of course, this situation became slightly problematic when Jean-Christophe passed away, but fortunately Patrice and I were able to locate the final version of the preprint in Jean-Christophe’s mathematical archives. (Here, the word “final” means that all mathematical arguments are present, but the preprint has no abstract, introduction, or other “cosmetic” details.)

After doing some editing (to correct minor typos, add better figures [with the aid of Aline Cerqueira], etc.), Patrice and I are happy to announce that the folklore preprint by Jean-Christophe and Patrice (entitled “Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant“) is finally publicly available here. We hope that you will enjoy reading this text (written in French)!

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