On Conley's Fundamental Theorem of Dynamical Systems

a rX iv:mat h /9184v1[mat h.DS]2Sep2On Conley’s Fundamental Theorem of Dynamical Systems M.R.Razvan (To the memory of C.Conley)Abstract In this paper,we generalize Conley’s fundamental theorem of dynamical systems in Conley index theory.We also conclude the existence of regular index filtration for every Morse decomposition.11Introduction Charles Conley is mostly known for his fundamental theorem of dynamical systems and his homotopy index theory [C].In the latter,he proved that every continuous flow on a compact metric space admits a Lyapunov function which is strictly decreasing along the orbits offthe chain recurrent set.This result has been developed by Franks for homeo-morphisms [F]and Hurley for noncompact metric spaces [H1]-[H4].In the former,Conley defined a homotopy invariant for any isolated invariant set for a continuous flow.This invariant gives some invaluable information about the behavior of the isolated invariant set.This paper concerns a combination of these two masterpieces.Indeed we generalize Conley’s fundamental theorem of dynamical systems in Conley index theory.We alsoconclude the existence of regular index filtration for every Morse decomposition.The generalized Morse inequalities is a consequence of the existence of regular index filtration.2Conley index theoryWe start this section with some basic definitions.Letϕt be a continuousflow on a metricspace X.An isolated invariant set is a subset S⊂X which is the maximal invariantset in a compact neighborhood of itself.Such a neighborhood is called an isolatingneighborhood.A Morse decomposition for S is a collection{M i}n i=1where each M i is anisolated invariant subset of S and for all x∈S− n i=1M i there exist i,j∈{1,···,n} such that i>j,α(x)∈M i andω(x)∈M j.A pair(A,A∗)of subsets of S is calledan attractor-repeller pair if{A,A∗}is a Morse decomposition for S i.e.α(x)∈A∗andω(x)∈A for every x∈S−(A∪A∗).Let S be an isolated invariant set with an isolating neighborhood V and a Morsedecomposition{M i}n i=1.In[S],it is proved that ifϕ[0,+∞)(x)⊂V,thenω(x)⊂M i forsome1≤i≤n.Similarly ifϕ(−∞,0](x)⊂V,thenα(x)⊂M i for some1≤i≤n.Nowfor j=0,···,n,we defineI+j=I+j(V)={x∈V|ϕ[0,∞)(x)⊂V,ω(x)⊂M j+1∪···∪M n}I−j=I−j(V)={x∈V|ϕ(−∞,0](x)⊂V,α(x)⊂M1∪···∪M j}S∗j={x∈S|ω(x)⊂M j+1∪···∪M n},S j={x∈S|α(x)⊂M1∪···∪M j}. Moreover if(A,A∗)is an attractor-repeller pair for S,we set={x∈V|ϕ[0,∞)(x)⊂V,ω(x)⊂A∗},I−A={x∈V|ϕ(−∞,0](x)⊂V,α(x)⊂A} I+A∗In[S]and[CZ],it is proved that I+j and I−j are compact and(I+j,I−j)is an attractor-repeller pair for S.This fact allows us to prove our results for an attractor-repeller pairand then extend them to every Morse decomposition.In order to define the concept of index pair,we follow[RS]and[S].Given a compactpair(N,L)with L⊂N⊂X,we define the induced semi-flow on N/L byϕt♯:N/L−→N/L,ϕt♯(x)= ϕt(x)ifϕ[0,t](x)⊂N−L[L]otherwise.In[RS]it is proved thatϕt♯is continuous if and only ifi)L is positively invariant relative to N,i.e.x∈L,t≥0,ϕ[0,t](x)⊂N⇒ϕ[0,t](x)⊂L.ii)Every orbit which exits N goes through Lfirst,i.e.x∈N,ϕ[0,∞)(x)⊂N⇒∃t≥0withϕ[0,t](x)⊂N,ϕt(x)∈L,or equivalently if x∈N−L then there is a t>0such thatϕ[0,t](x)⊂N.Definition.An index pair for an isolated invariant set S⊂X is a compact pair(N,L) in X such thatProposition2.2.An index pair(N,L)is regular provided thatϕ[0,t](x)⊂N−L.If B is a compact subset of N/L−I+A∗and U is a neighborhood of[L]∪I−A,then there exists T∈R+such thatϕ[T,+∞)♯(B)⊂U.Proof.We may assume that U is a compact neighborhood of[L]∪I−A with U∩I−A=∅.Now suppose that there are x n∈B and t n−→∞such thatϕt n♯(x n)∈U.Since B iscompact,we may choose x n’s so that x n−→x∈B.It is easy to see thatϕ[0,+∞)(x)⊂N−L.Since B∩I+A∗=∅,we haveω(x)⊂A,hence there is a t∈R+such thatϕ[t,+∞)(x)∈◦U.Since x n−→x,there are t n∈R+such that t n−→+∞,ϕ[t,t n]♯(x n)⊂◦U and t n>t for every n∈N.It follows that there exists t′n∈[t,t n]such thatϕ[t,t′n]⊂U,ϕt′n(x n)∈∂U and t′n−t−→+∞.(The latter is true sinceϕ[t,+∞)(x n)⊂◦U and x n−→x.)Thereforethe sequenceϕt′n(x n)has a limit point y∈∂U such thatϕ(−∞,0](y)⊂U∩(N−L)and y∈ω(B)⊂S.Thusα(y)⊂A which follows that y∈A.This contradicts y∈∂U.Theorem3.2.Let S be an isolated invariant set with an attractor-repeller pair(A,A∗) and an index pair(N,L).There exists a continuous function g:N/L−→[0,1]such that(i)g−1(0)=[L]∪I−A and g−1(1)=I+A∗.(ii)g(ϕt(x))<g(x)for every x∈[L]∪I−A ∪I+A∗and t∈R+.Proof.Letρ:N/L−→[0,1]be a continuous function withρ−1(0)=[L]∪I−Aandρ−1(1)=I+A∗.We define h:N/L−→[0,1]by h(x)=supt≥0ρ(ϕt♯(x)).It is not hard tosee that h−1(0)=[L]∪I−A ,h−1(1)=I+A∗.We show that h is upper semi-continuous.For every x∈N/L andǫ>0there is a t∈R+such thatρ(ϕt♯(x))>h(x)−ǫ.Now there is a neighborhood U of x such thatϕt♯(y)>h(x)−ǫfor every y∈U.Therefore h(y)>h(x)−ǫfor every y∈U which proves the upper semi-continuity of h.As a result, h is continuous in h−1(1).Now suppose that x∈h−1(1)=I+A∗andǫ<1−h(x).If we set B=ρ−1[0,h(x)+ǫ]and U=ρ−1[0,h(x)+ǫ)in the above lemma,we obtain a T∈R+ withρ(ϕt♯(y))<h(x)+ǫfor every y∈B and t≥T.Now by continuity ofϕ♯,there exists an open set V⊂N/L such that x∈V andρ(ϕt♯(y))<h(x)+ǫfor every t∈[0,T]and y∈V,Therefore h(y)<h(x)−ǫfor every t∈[0,T]and y∈U∩V which shows that h is lower semi-continuous in x.Now it is not hand to check that g:= +∞0e−t f(ϕt♯(x))dt is the desired function[S].Theorem3.3.Let S be an isolated invariant set with an index pair(N,L)and a Morse decomposition{M i}n i=1.There is a continuous function g:N/L−→[0,N+1]such that (i)g−1(0)=[L]and g(M i)=i for every1≤i≤n.(ii)If x∈N−L− n i=1M i and t>0,then g(ϕt♯(x))<g(x).Proof.Consider the attractor-repeller pairs(I j,I∗j)for0≤j≤n.By Theorem3.2, there are continuous functions g i:N/L−→[0,1]with g−1i(0)=[L]∪I−j,g−1i(1)=I+j and g j(ϕt(x))<g j(x)for every x∈[L]∪I−j∪I+j.Now g:=g0+···+g n is the desired function.Corollary3.4.Every Morse decomposition admits a regular indexfiltration. Proof.Let f be the above Lyapunov function.If we set N k:=π−1(f−1[0,k+1/2])for 0≤k≤n and N n:=N,then by Proposition2.2,(N k,N k−1)is a regular index pair for M k,for every1≤k≤n.Definition.Letϕt be a continuousflow on a compact metric space X.Anǫ-chain forϕt is a sequence x0,···,x n in X and t1,···,t n in[1,+∞)such that d(ϕt i(x i−1))<ǫ.A point x∈X is called chain recurrent if for everyǫ>0,there is anǫ-chain x=x0,···,x n=x. The set of all chain recurrent points forϕt is denoted by R(ϕt).It is not hard to Check that R(ϕt)is a closed invariant subset of X containing the non-wandering setΩ(ϕt).In[C]and[Ro],it has been shown that R(ϕt|R(ϕt))=R(ϕt) and R(ϕt)= (A∪A∗)where the intersection is taken over all attractor-repeller pairs (A,A∗)in X.It is also known that the number of all attractor-repeller pairs in a compact metric space is at most countable.Theorem3.5.Let S be an isolated invariant set with an index pair(N,L).Then there is a continuous function g:N/L−→[0,1]such that(i)g−1(0)=[L]and g(ϕt(x))≤g(x)for every x∈N/L and t≥0.(ii)if x∈N−L−R(ϕt|I)and t≥,then g(ϕt♯(x))<g(x).Proof.Let{(A i,A∗i)}∞i=1be the sequence of all attractor-repeller pairs in S including (∅,S)and(S,∅).Now by the Theorem3.2,there are continuous functions g i:N/L−→[0,1]such that g−1i(0)=[L]∪I−A i ,g−1i=I+A∗iand g i(ϕt(x))<g i(x)for every t∈R+andx∈[L]∪I+A∗i ∪I−A i.Now g= ∞i=12−i g i is the desired function.The above result can be considered as a generalization of Conley’s Fundamental The-orem of Dynamical Systems.A similar result for maps and semi-flows can be obtained by following[FR],[H3],[H4]and[Sz].Acknowledgment.The author would like to thank Institute for studies in theoretical Physics and Mathematics,IPM,for supporting this research.References[C]Conley,C.,Isolated invariant set and the Morse index,CBMS Notes,38(1978),AMS Providence,RI.[CZ]Conley,C.and Zehnder,E.,Morse type index forflows and periodic solutions for Hamiltonian systems,Comm.Pure App.Math.,37(1984),207–253.[F]Franks,J.,A variation on the Poincare-Birkhofftheorem,Contemporary Math.81(1988),111–117.[FR]Franks,J.,and Richeson,D.,Shift equivalence and Conley index,Trans.Amer.AMath.Soc.352(2000),3305–3332.[H1]Hurley,M.,Chain recurrence and attraction in noncompact spaces,Ergodic Theory Dynamival Systems,11(1991),709–729.[H2]Hurley,M.,Noncompact chain recurrence and attraction,Proc.Amer.Math.Soc.,115(1992),1139-1148.[H3]Hurley,M.,Chain recurrence,semiflows and gradients,J.Dynamics.Diff.Equatuions,7(1995),437–459.[H4]Hurley,M.,Lyapunov functions and attractors in arbitrary metric spaces,Proc.Amer.Math.Soc.,126(1998),245–256.[Ro]Robinson,D.,Dynamical Systems,CRC Press,1995.[RS]Robbin,J.and Salamon,D.,Dynamical systems,shape theory and the Conley index,Ergodic Theory Dynamical Systems,8*(1998),375–393.[S]Salamon,D.,Connected simple systems and the conley index for isolated in-variant sets.Trans.Amer.Math.Soc.,22(1985),113–140.[Sz]Szymczak,A.,The Conley index for discrete semidynamical systems,Topology Appl.66(1995),215–240.Institute for Studies in Theoretical Physics and MathematicsP.O.Box:19395−5746,Tehran,IRAN.e-mail:razvan@karun.ipm.ac.ir,Fax:009821-2290648。