GEOMETRY OF MINIMAL FLOWS DRAFT (DRAFT) WILLIAM BASENER Abstract. Our main result is that a minimal flow on a compact manifold is either topologically conjugate to a Riemannian flow or every parametrization of ϕ is nowhere equicontinuous, defined as follows. A flow is Riemannian if given any points x, y ∈ M, the value of d(ϕt (x),ϕt (y)) is independent of t ∈ R . A flow is nowhere equicontinuous if there exists an ǫ> 0 such that given any point x ∈ M and any δ> 0, there exists a point y ∈ N δ (x) and time t ∈ R such that d(ϕt (x),ϕt (y)) >ǫ. Equivalently, a flow ϕt is nowhere equicontinuous if there exists an ε> 0 such that given any x the set Eε(x)= {y ∈ M | d(ϕt (y),ϕt (x)) >ε for some t ∈ R } is open and dense in M. Our results depend heavily on the work of Lopez and Candel. 1. Introduction Throughout this paper, let M denote a closed Riemannian n-dimensional man- ifold. A flow ϕ : M × R → M is minimal if every orbit O(x)= {ϕ t (x) | t ∈ R } is dense in M . Our main theorem, Theorem 1, provides a dichotomy for minimal flows. Either the flow is topologically conjugate to a Riemannian flow, in which case the distance between points is preserved by the flow, or the flow is nowhere equicontinuous, in which case the flow tends move points apart. The irrational flows on tori fall into the first category and horocylce flows on unit tangent bundles of surfaces of constant negative curvature fall into the second category. In both of these cases the flow is defined using the geometry of the manifold. This observa- tion, together with our main theorem, suggests that perhaps every minimal flow is related to some geometry on the ambient manifold. THEOREM 1. A minimal flow ϕ on a compact Riemannian manifold M is ei- ther topologically conjugate to a Riemannian flow or every parametrization of ϕ is nowhere equicontinuous. Our primary tool for proving Theorem 1 is Theorem 3, proven by Lopez and Candel in [LC1]. A flow is Riemannian if given any points x, y ∈ M , the value of d(ϕ t (x),ϕ t (y)) is independent of t ∈ R . A flow is nowhere equicontinuous if there exists an ǫ> 0 such that given any point x ∈ M and any δ> 0, there exists a point y ∈ N δ (x) and time t ∈ R such that d(ϕ t (x),ϕ t (y)) >ǫ. Proposition 1 gives an alternative formulation of nowhere equicontinuous. In [G], Gottschalk conjectured wether there exists a minimal flow on S 3 . By Theorem 1, a minimal flow on S 3 must either be Riemannian or nowhere equicon- tinuous. The Riemannian flows on S 3 are well-known, and are not minimal. Hence, we get the following corollary. COROLLARY 1. If ϕ is a minimal flow on S 3 then ϕ is nowhere equicontinuous. 1