## 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

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

Zbl 0342.57013
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