Ensemble oscillant d’un homéomorphisme de Brouwer, homéomorphismes de Reeb. (Oscillation set of a Brouwer homeomorphism, Reeb homeomorphisms).

*(French)*Zbl 1026.37033This interesting paper is devoted to the study of Brouwer’s homeomorphisms, that is, fixed-point-free orientation-preserving homeomorphisms of the plane. Beginning with the plane translation theorem, the authors introduce a new conjugacy invariant which they name oscillation set as an undertaking to give a precise description of the way the translations are glued together.

Since H. Nakayama’s example [Houston J. Math. 21, 569-572 (1995; Zbl 0857.54040)] shows that the singular set is not an invariant which permits a total classification of the homeomorphisms of Brouwer by conjugation, the idea is to bring up this new invariant in order to obtain a finer classification. If \(h\) is a homeomorphism of Brouwer, a couple \((M,N)\) of points of the plane which do not belong to the same orbit of \(h\) is said to be singular if, for every neighborhood \(U\) of \(M\) and for every neighborhood \(V\) of \(N\), there is a positive integer \(n\) such that \(h^{n}(U)\cap V\) is a nonempty set.

The definition of an oscillation set is based in the following notion: the origin of an arc \(c\) is said to be indifferent to a point \(N\) if there exists another arc \(\hat{c}\) with the same origin that \(c\) and a neighborhood \(V\) of \(N\) such that, for every nonnegative integer, \(h^{n}(\hat{c})\) miss \(V\). Then a singular couple \((M,N)\) is an oscillation if every arc with origin at \(M\) and such that \(M\) is not indifferent to \(N\), enjoys the property that the family \((h^{n}(c))_{n\geq 0}\) is not equilocally connected in \(N\). The oscillation set is defined as the set of all oscillation couples of \(h\) and it is denoted by \(\text{Osc} (h)\).

The authors use the oscillation set in several ways. For instance, it is shown that there exist uncountable homeomorphisms of Reeb (that is, homeomorphisms that preserve each leaf of that Reeb foliation) which are pairwise nonconjugated, and it is also proved that there is a flow of Reeb \((h^{t})_{t\in \mathbb{R}}\) such that the homeomorphisms \(h^{1}\) and \(h^{2}\) are not conjugated. Even though the oscillation set is more advantageously than the singular set, the oscillation set does not provide either a total classification of Bouwer’s homeomorphisms. However an interesting case where it characterizes the dynamic is the following: there is only one class of conjugation of Brouwer’s homeomorphisms \(h\) which satisfies the following properties: (1) the set of singular couples of \(h\) is the product of two real lines, and (2) the oscillation set is empty.

Since H. Nakayama’s example [Houston J. Math. 21, 569-572 (1995; Zbl 0857.54040)] shows that the singular set is not an invariant which permits a total classification of the homeomorphisms of Brouwer by conjugation, the idea is to bring up this new invariant in order to obtain a finer classification. If \(h\) is a homeomorphism of Brouwer, a couple \((M,N)\) of points of the plane which do not belong to the same orbit of \(h\) is said to be singular if, for every neighborhood \(U\) of \(M\) and for every neighborhood \(V\) of \(N\), there is a positive integer \(n\) such that \(h^{n}(U)\cap V\) is a nonempty set.

The definition of an oscillation set is based in the following notion: the origin of an arc \(c\) is said to be indifferent to a point \(N\) if there exists another arc \(\hat{c}\) with the same origin that \(c\) and a neighborhood \(V\) of \(N\) such that, for every nonnegative integer, \(h^{n}(\hat{c})\) miss \(V\). Then a singular couple \((M,N)\) is an oscillation if every arc with origin at \(M\) and such that \(M\) is not indifferent to \(N\), enjoys the property that the family \((h^{n}(c))_{n\geq 0}\) is not equilocally connected in \(N\). The oscillation set is defined as the set of all oscillation couples of \(h\) and it is denoted by \(\text{Osc} (h)\).

The authors use the oscillation set in several ways. For instance, it is shown that there exist uncountable homeomorphisms of Reeb (that is, homeomorphisms that preserve each leaf of that Reeb foliation) which are pairwise nonconjugated, and it is also proved that there is a flow of Reeb \((h^{t})_{t\in \mathbb{R}}\) such that the homeomorphisms \(h^{1}\) and \(h^{2}\) are not conjugated. Even though the oscillation set is more advantageously than the singular set, the oscillation set does not provide either a total classification of Bouwer’s homeomorphisms. However an interesting case where it characterizes the dynamic is the following: there is only one class of conjugation of Brouwer’s homeomorphisms \(h\) which satisfies the following properties: (1) the set of singular couples of \(h\) is the product of two real lines, and (2) the oscillation set is empty.

Reviewer: Manuel Sanchis (Castelló)