Chapter 45 - Minimal flows

Abstract

This chapter discusses the concepts and various problems related to minimal flows. A topological dynamical system is a continuous group action φ: X ×T →X written φt(x), where T is a group and X is a compact Hausdorff space. The most common cases are where T = ℝ, called a continuous flow, and T = ℤ, called a discrete flow. In this chapter, a flow means a continuous flow and discrete flows are referred to as maps, homeomorphisms, and diffeomorphisms. Flows arise most naturally as the set of solutions to a system of differential equations. A compact invariant subset X ⊂M is said to be a minimal set if it is a minimal compact invariant set under containment. Explicitly, X is a minimal set if the only compact invariant subsets of X are itself and the empty set. The simplest examples of minimal flows are the trivial flow on a single point and a flow on a circle without fixed points. The notion of Minimal flows on 3-manifolds and transitive Anosov flows is also explored. Asymptotic properties of flows are also elaborated.