On the realisability of Smale orders

Ioannis Iakovoglou (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We give an answer to a generalised form of Smale's problem in dimension 2 and 3 (Problem 6.6 in [S. Smale, Bull. Amer. Math. Soc. 73 (1967)]) concerning the realisability of Smale orders. More specifically, we are considering the question of whether a partial order on a finite set is realisable as the Smale order of a structurally stable diffeomorphism or flow acting on a closed manifold. We classify the orders that are realisable by 1) an Omega-stable diffeomorphism acting on a closed surface, 2) an Anosov flow on a closed 3-manifold, and 3) a stable diffeomorphism with trivial attractors and repellers acting on a closed surface.

Mots clés: Smale orders; structurally stable diffeomorphisms and flows

Domaines: Mathématiques [math]

Ioannis Iakovoglou. On the realisability of Smale orders. Fundamenta Mathematicae, 2022, 258 (3), pp.291-334. ⟨10.4064/fm178-4-2022⟩. ⟨hal-03894930⟩