Chapter 1. Hyperbolic chain recurrent sets The object of these lectures is to study qualitative^ the time evolution of smooth dynamical Systems. We will limit our attention to smooth flows on compact manifolds and their discrete time analogues, smootli diffeomorphisms. In fact, we will place further limita- tions on the Systems considered and the aim of this chapter is to describe and motivate these restrictions. Á qualitative analysis of the time evolution of a dynamical System should include a description of the long run or asymptotic behavior of points of the manifold Ì under the System, i.e., a description of the behavior of ftx as / tends to ±°° (or fnx as ç ±» for a diffeomorphism). Ideally we should have this description for all ÷ EM. It is clear that in such a description a special role is played by the periodic orbits and stationary points for flows and the periodic points for a diffeomorphism. It turns out that the set of points satisfying a kind of recurrence weaker than periodicity is crucial to the de- scription of long run behavior. (1.1) DEFINITION. If /: Ì —• Ì is a diffeomorphism, then ÷ EM is called chain re- current if for any e 0 there exist points xx = x, x2, ... , xn_x, xn = ÷ (n depends on e) such that d(fxif xi+ ë ) e for 1 / ç, where d( , ) is a metric on M, For a flow ft, ÷ Å Ì is chain recurrent if for any e 0 there exist points x^ = = x, x2, ... , xn = ÷ and real numbers t(i) 1 such that d(ft^xit xi+l) e, for 1 / n. In either case the set of chain recurrent points is called the chain recurrent set and will be denoted by R or R(/). It is an easy exercise to show that the chain recurrent set R is invariant under the flow or diffeomorphism and is closed and hence compact since we are assuming Ì to be compact. One might think of R as the points which come within e of being periodic for every e 0. The importance of chain recurrence for the description of asymptotic behavior of orbits of the system is shown in the following theorem of Conley. (1.2) THEOREM [C]. Jfft is á continuous flow on M, there exists á continuous func- tion g: Ì R such that (1) ffx* R(/f), g(ftx) g(fsx) when t s. (2) Ifx,y Å R(/r) then g(x) = g(y) if and only if for e 0 there exist points x1 = x, x2 · · ·' x n = x n + ñ · · ·' x 2n = x ' n ^ an ^ rea ^ num bers t(i) 0, 1 / 2«, such that d(fmxit x.+ 1 ) e, Ê é 2n. The obvious analogue of this theorem for diffeomorphisms is also valid. 1
Previous Page Next Page