A rigidity theorem for transverse dynamics of real analytic foliations of codimension one. (English) Zbl 0831.57017

Nakai, Isao

Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 327-343 (1994).

The author proves the following Theorem: “Let \((M_i^n,{\mathcal F}_i)\), \(i=1, 2\), be a real analytic and orientable foliation of \(n\)- manifolds of codimension 1 and \(h: (M_1^n,{\mathcal F}_1)\to (M_2^n,{\mathcal F}_2)\) a foliation preserving homeomorphism. Assume that all leaves of \({\mathcal F}_1\) are dense and that there exists a leaf of \({\mathcal F}_1\) with holonomy group \(=1, \mathbb{Z}\). Then \(h\) is transversely real analytic.” The proof is based on a rigidity theorem for pseudogroups. As a natural interpretation of the work by E. Ghys and T. Tsuboi [Ann. Inst. Fourier 38, No. 1, 215-244 (1988; Zbl 0633.58018)], the author applies this theorem to prove a topological rigidity of the Godbillon-Vey class of real analytic foliations of codimension one.
For the entire collection see [Zbl 0797.00019].

Reviewer: J.L.Viviente (Zaragoza)

MSC:

57R30

Foliations in differential topology; geometric theory

Keywords:

real analytic and orientable foliations; codimension 1; foliation preserving homeomorphism; transversely real analytic; topological rigidity of the Godbillon-Vey class of real analytic foliations