Fiberwise divergent orbits of projective flows with exactly two minimal sets

Abstract

Let ’t be a nonsingular flow on a 3-dimensional manifold M. Denote by P : PX ! M the projectivized bundle of the quotient bundle of TM by the line bundle tangent to’t. The derivative of’t induces a flow t on PX, called the projective flow of ’t. In this paper, we consider the dynamical properties of t restricted to 1 P (M) for a minimal set M of’t, under the condition that the restriction of t to 1 P (M) has exactly two minimal sets N1 and N2. If’t has no dominated splitting over M, we find two types of orbits of t in the domain between N1 and N2: one is “bounded below” and the other is “bounded above”. As an application we prove that, if’t is further assumed to be almost periodic on the minimal set, there is a dense orbit in that domain.