On the realisability of Smale orders
Volume 258 / 2022
Fundamenta Mathematicae 258 (2022), 291-334
MSC: Primary 37D20; Secondary 37D05, 37E30, 37E15.
DOI: 10.4064/fm178-4-2022
Published online: 25 June 2022
Abstract
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.
