Continuous flows generate few homeomorphisms

Abstract

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.