Chaos Of Invertible Dynamical Systems

The idea of chaos in connection with a map was introduced by Li Tien Yien and his teacher James A. Yorke, after that there are various definitions of what it means for a map to be chaotic and there is a series of papers on Topological Dynamics and Ergodic Theory about chaos. Following Topological Dynamics, Linear Dynamics is also a rapidly evolving branch of functional analysis, which was probably born in 1982 with the Ph. D. thesis of C. Kitai, Toronto University. It has become rather popular because of the efforts of many mathematicians, such as G. Godefroy, Joel H. Shapiro, Karl-G. Grosse-Erdmann, F. Bayart, E. Matheron and A. Peris.Except for tuple operators that is first studied by N. Feldman, J. W. Robbin and N. H. Kuiper have made a topologically conjugate classification for only one operator on finite dimensional space. So for the eigenvalues |λ|≠1, the operators of Jordan blocks are not chaotic. S. Rolewicz prove that a Jordan block is not supercyclic when its eigenvalues |λ|= 1, and following a easy discussion it is not chaotic too. So the definition of chaos is valid only in infinite dimensional topological linear space and hence in this paper the Banach spaces or Hilbert spaces considered are separable infinite dimension.In Chapter 1, the author of this paper introduces the histories, the background and today’s situation about Dynamical Systems, also the preliminaries are ready in Chapter 2. The distributional chaos of invertible bounded linear operator on Banach space is studied in Chapter 3. F. Martinez-Gimenez first showed that uniform DC1 is a topological uniform conjugate invariant for bounded linear operators on Banach spaces, but the author of this paper gives a generally conclusion:Theorem 3.2.1 Let X, Y be Banach spaces, T ∈ L(X), S ∈L(Y). If T and S are topologically conjugate, then T is DC1 or DC2 if and only if S is DC1 or DC2.And the author also obtains a necessary condition for DC2:Theorem 3.2.2 Let T ∈L(B),suppose that {x ∈B; lim ||Tnx||= 0} is a dense subset of B. Then T is DC2 imply T is absolutely Cesaro unbounded.After that, the author investigates the properties of weighted shift operators on Lp(N)Theorem 3.2.3 On Lp(N), 1 ≤ p < +∞, there exists a strongly mixing backward weighted shift operator, but it is not DC2Theorem 3.2.4 There exists an invertible bilateral forward weighted shift operator T ∈LP(Z), 1 ≤ p < +∞ such that T is DC1, but T-1 is not DC1.At the end of Chapter 3, the author of this paper gives some properties of the adjoint multiplier determined by Cowen-Douglas functions on Hardy space and gives some chaos about λI + T the scalars perturbation of an operator.Theorem 3.3.3 Let φ(z) ∈H∞(D)) be a m-folder analytic function, Mφ be the multiplication by φ on H2(D). For any given z0 ∈ D, if the rooter function hz0 of φ at z0 is an outer function, then φ is a Cowen-Douglas function, that is, the adjoint multiplier Mφ*∈Bm(φ(D))Corollary 3.3.1 Let φ∈H∞(D) be a m-folder analytic function. For any given z0 ∈D, if the rooter function hz0 of φ at z0 is invertible on the Banach algebra H∞(D), then φ is a Cowen-Douglas function. Especially, for any n ∈ N, if a, b are both non-zero complex number, then a + bzn ∈H∞(ID) is a Cowen-Douglas function.Theorem 3.3.4 If φ∈ H∞(D) is a Cowen-Douglas function, Me is the multiplication by φ on H2 (D), then the following assertions are equivalent.(1) Mφ* is Devaney chaotic;(2) Mφ* is distributionally chaotic;(3) is strongly mixing;(4) Mφ* is Li-Yorke chaotic;(5) Mφ* is hypercyclic;(6) φ(D)∩T ≠ 0Corollary 3.3.2 If φ∈H∞(D) is a invertible Cowen-Douglas function in the Banach algebra H∞(D), Me is the multiplication by φ on H2(D), then the adjoint multiplier Mφ* is Devaney chaotic or distributional chaotic or strong mixing or Li-Yorke chaotic or hypercyclic if and only if Mφ*-1 is.Lemma 3.3.2 If T ∈ L(H) is a normal operator, then SLY(T) = (?).Lemma 3.3.3 There is a quasinilpotent compact operator T ∈L(H) such that SLy(T) = T is closed and SLy(T*) = (?), where T = (?)D,D = {z ∈ C, |z| < 1}.Lemma 3.3.4 Let T be backward shift operator on L2(N), T(x1, x2,… ) = (x2, x3,… Then SLy(T) = SDc(T) = Soy(T) = SH(T) = 2D \ {0} and SLy(2T) = SDc(2T) = SDv(2T) = SH(2T) = 3D. Hence SLy(T) and SLy(2T) are open sets.Corollary 3.3.3 Let T be the backward shift operator on L2(N), T(x1,x2,…) = (x2, x3,… ) . For any given n ∈ N, any given 0 ≠λ∈C and any given 0 ≠ a ∈ C, if λ + aTn is an invertible bounded linear operator, then λ+aTn is strongly mixing or Devaney chaotic or distributionally chaotic or Li-Yorke chaotic if and only if (λ + aTn)-1Theorem 3.3.5 There is T ∈L(H) such that SLy(T) is neither open nor closed.In Chapter 4, because there is enough methods to do with operator theory, this paper is studying on separable Hilbert space and finally get the following beautiful results:Lemma 4.1.4 Let T be an invertible bounded linear operator on the separable Hilbert space H over C, let A(|Tk|) be the complex C* algebra generated by |Tk| and 1 and let H|Tk| = A(|TK|)ζki be a sequence of non-zero A(|Tk|)-invariant subspace, there is a decomposition H = (?)iH|Tk|i, ζki∈H,i, k ∈ N. Given a proper permutation of HTki and H|T(k+1)|j, we get T*Hi|Tk|=Hi|T(k+1)andT-1Hi|Tk|=Hi|T(k+1)|Following the decomposition the author gives a theoretical foundation for Lebesgue operator:Theorem 4.1.1 Let T be an invertible bounded linear operator on the separable Hilbert space H over C, there is ζ∈ H such that A(|T|)ζ = H. For any given n ∈ N, let A(|Tn|) be the complex C* algebra generated by |Tn| and 1 and let ζn be a A(|Tn|)-cyclic vector such that A(|Tn|)ζn = H. Then:(1) For any given ζn,there is a unique positive linear functional ∫ f(z)dμ|Tn|(z) =< f(|Tn|)ζn,ζn>, (?)f ∈L2(a(|Tn|)). σ(|Tn|)(2) For any given ζn, μ|Tn|, there is a unique finite positive complete Borel measure such that L2(σ(|Tn|)) is isomorphic to H.Theorem 4.1.2 Let T be an invertible bounded linear operator on the separable Hilbert space H over C and let A(|T|) be the complex algebra generated by |T| and 1 There is v(|T|-1) = σ(|T-1|) and we get that |T|-1 and |T-1| are unitary equivalent by the unitary operator Fxx*H., moreover the unitary operator Fxx*H. is induced by an al-most everywhere non-zero function √(φ|T|), where √(φ|T|)∈L∞(σ(|T|),μ|T|) . That is, d μ|T-1| = |φ|T||dμ|T|-1.Corollary 4.1.1 Let T be an invertible bounded linear operator on the separable Hilbert space H over C and let A(|T|) be the complex algebra generated by |T| and 1. There is σ(|T|) = σ(|T*|) and we get that |T| and |T*| are unitary equivalent by the unitary operator Fxx* H., more over the unitary operator Fxx* H. is induced by an almost everywhere non-zero function √(|Φ|T||), where √(Φ|T|)∈(σ(|T|),μ|T|). That is, dμ|T| = [Φ|T||d μ|T|This is a based analysis for the following results:Theorem 4.2.1 Let T be a Lebesgue operator on the separable Hilbert space H over C, then T is Li-Yorke chaotic if and only if T*-~ is.Corollary 4.2.1 Let T be a Lebesgue operator on the separable Hilbert space H over C, then T is DC1 (or DC2 or DC3) if and only if T*-1 is.Theorem 4.2.2 There is an invertible bounded linear operator T on the separable Hilbert space H over C, T is Lebesgue operator but not is a normal operator.Corollary 4.2.2 There is an invertible bounded linear operator T on the separable Hilbert space H over C, T is a Lebesgue operator and also is a positive operator.In Chapter V, at the begin the author of this paper gives some example on Hilbert space to confirm the theory of the last chapter that the inverse of an invertible dynamic is not chaotic, moreover gives a global conclusion of chaos for scalars perturbation of an operator.Example 5.1.1 Let {en}N∈N be a orthonormal basis of L2(N) and let Sω be a backward shift operator on/32(N) with weight sequence ω = (ωn}n>1 such that Sω(e0) = 0, Sω(en) = ωnen-1,where 0 <|ωn| < M < +∞, (?)n > 0.(1)" If |λ| = 1, then λI + Sω is Li-Yorke chaotic, but λI + Sω and (λI + Sω*)-1 are not.(2) Let (λI + Sω)n = Un|(λI + Sω)n| be the polar decomposition of (λI + Sω)n, {Un}n=1∞ is not a constant sequence.Example 5.1.2 There is an invertible bounded linear operator I+ Kε on H =L2 (N) such that I + Kε is Li-Yorke chaotic, but (I + Kε)-1,(I + Kε)*-1 and (I + Kε)* are not.Corollary 5.1.1 There is an invertible bounded linear operator I+Kε on H = L2(N) such that I + Kε is distributionally chaotic, but (I + Kε)-1,(I + Kε)*-1 and (I + Ks)* are not.Theorem 5.1.2 There is T ∈L(H) such that SLy(T) = Sot(T) = F is an open arc oft = {|λ| = 1;λ∈ C}, and for (?)λ∈ F, we get that (λ + T)*,(λ + T)*-1 and (λ + T)-1 are not Li-Yorke chaotic.Finally the author of this paper investigates some properties of self-mapping on noncompact finite dimension space:Theorem 5.2.1 There is a self-homeomorphism.f on D={z∈C；|z|<1),such that.f is Li-Yorke chaotie,but.f-1 is not.Theorem 5.2.2 Let D={z∈C；|z|<1).,there is a self.homeomorphism f on D ang f is Li-Yorke chaotic.there is a self.homeomorphism h on C such that f and h are topolofivally conjugative.But h and h-1 are not Li-Yorke chaotic.