Un point fixe commun pour des difféomorphismes commutants de \(S^ 2\). (A common fixed point for commutating diffeomorphisms of \(S^ 2\).). (French) Zbl 0689.57019

Bonatti, Christian

Ann. Math. (2) 129, No. 1, 61-69 (1989).

The author proves that a finite set of commuting diffeomorphisms of the 2-sphere have a common fixed point, provided that they are sufficiently \(C^ 1\)-close to the identity. More precisely, there is a neighborhood U of the identity in \(Diff^ 1(S^ 2)\) such that, whenever the elements of \(\{f_ 1,f_ 2,...,f_ n\}\subset U\) commute, then there is a point of \(S^ 2\) that is fixed by \(f_ i\) \(1\leq i\leq n\). This gives an affirmative answer to a question of H. Rosenberg [Sém. Bourbaki 1972/1973, Exposé No.434, Lect. Notes Math. 383, 294-306 (1974; Zbl 0342.57013)].

Reviewer: L.Conlon

MSC:

57R30	Foliations in differential topology; geometric theory
37-XX	Dynamical systems and ergodic theory
57R50	Differential topological aspects of diffeomorphisms

Keywords:

finite set of commuting diffeomorphisms of the 2-sphere; common fixed point

Citations:

Zbl 0342.57013