Fixed point sets of isotopies on surfaces

Francois Beguin (1) , Sylvain Crovisier (2) , Frederic Le Roux (3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LMO - Laboratoire de Mathématiques d'Orsay
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

We consider a self-homeomorphism h of some surface S. A subset F of the fixed point set of h is said to be unlinked if there is an isotopy from the identity to h that fixes every point of F. With Le Calvez' transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the two dimensional sphere that preserves the orientation and pointwise fix some given closed connected set F.

Domaines: Systèmes dynamiques [math.DS]

Francois Beguin, Sylvain Crovisier, Frederic Le Roux. Fixed point sets of isotopies on surfaces. Journal of the European Mathematical Society, 2020, 22 (6), pp.1971-2046. ⟨10.4171/JEMS/960⟩. ⟨hal-02367499⟩