In this paper we study Li-Yorke chaos in linear operator on Banach space, in addition to  establishing some basic properties of Li-Yorke chaos and explanation when the operator be Li-Yorke chaos or not. We also prove...............

A. Ahmed, Buthainah A.

AL-Zubaidi, Mohanad Nafea Jaafar

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  98 Some Properties of Li-Yorke Chaos Mohanad Nafea Jaafar AL-Zubaidi1*  and   Buthainah A. A. Ahmed2*  1.Mathematics department/ College of Education for pure science Ibn Al –Haitham /University of Baghdad 2.Mathematics department /College of science/ University of Baghdad  Abstract    In this paper we study Li-Yorke chaos in linear operator on Banach space, in addition to  establishing some basic properties of Li-Yorke chaos and explanation when the operator be Li-Yorke chaos or not. We also prove the following the theorem, if 𝜒𝑇(𝒟) ∩ 𝜒𝑇(ℂ\?̅?) ≠ ∅, where  𝒟 is the interior of the unit circle, and ℂ\?̅?  is the exterior of the unit circle then T satisfied Li-Yorke Chaos Criterion. Key words: Li – Yorke,Choas,Irregular vectors,Li –Yorke chaos criterion.  1. Introduction       The dynamics of linear operators have been widely studied in the last few years. Several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, mixing and weakly mixing properties, and frequent hypercyclicity, among others. In the paper, we are mainly interested with the notion of Li-Yorke chaos. Let (X, 𝑇) be Banach space  and 𝑇 continuous operator from X to itself .The definition of Li-Yorke chaos is based on ideas in [8]. A pair of points {𝑥, 𝑦} X is said to be a Li-Yorke pair if one has simultaneously         lim𝑛 inf‖𝑇𝑛𝑥 − 𝑇𝑛𝑦‖ = 0 and lim𝑛 sup‖𝑇𝑛𝑥 − 𝑇𝑛𝑦‖) > 0.         A set S  X is called scrambled if any pair of distinct points {𝑥, 𝑦}  S is a Li- Yorke pair. Finally, a system     (X, 𝑇) is called chaotic in the sense of Li and Yorke if X contains an uncountable scrambled set, and definition a vector 𝑥 X is said to be irregular for 𝑇 if 𝑙𝑖𝑚 infn‖ 𝑇n𝑥‖= 0 and 𝑙𝑖𝑚 supn‖ 𝑇n𝑥‖= ∞ [7]. While [3] give an equivalent definition of irregular vector, that is            A vector x is said to be irregular vector of T if there are two sequences kn and ln increasing to ∞ such that lim𝑛 𝑇𝑘𝑛𝑥 = 0 and lim𝑛 𝑇𝑙𝑛𝑥=∞.      We will recall some properties of Li-York Chaos that needed later.  1.2. Theorem:[7]        Let T: X → X is an operator. The following assertions are equivalent: (i) T is Li-Yorke chaotic. (ii) T admits a Li-Yorke pair. (iii) T admits an irregular vector. 2. Main Result Now, we will give our main results 2.1. Proposition:           Li-Yorke Chaos is preserved under conjuacy. Proof:     let 𝑇: X            →   X be conjugate to 𝑆:Y             →   Y via 𝜑: X             →   Y , then 𝑥X , yY and  y=𝜑(𝑥), and suppose 𝑆 is Li-Yorke chaos, by theorem above  , 𝑆 admits an irregular  vector. Let a vector y is an irregular for 𝑆, there are two  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  99 sequences 𝑛𝑘 and 𝑙𝑘 increasing to ∞ such that 𝑆𝑛𝑘𝑦                →    0 mean that 𝑆𝑛𝑘𝜑(𝑥)                →    0 mean that 𝜑𝑇𝑛𝑘(𝑥)                →    0, we have  𝑇𝑛𝑘(𝑥)                →    0 . And ‖𝑆𝑙𝑘𝑦‖                →    ∞ mean that ‖𝑆𝑙𝑘𝜑(𝑥)‖                →    ∞ mean that ‖𝜑𝑇𝑙𝑘(𝑥)‖                →    ∞   we have ‖𝑇𝑙𝑘(𝑥)‖                →    ∞. Then 𝑇 is Li-Yorke chaos. 2.2. Proposition:          If at least one of 𝑇1 or 𝑇2 has irregular vectors then 𝑇1  𝑇2 has irregular vectors. Proof.        Let x be an irregular vector of 𝑇1. Thus there is a sequence kn such that   𝑇1𝑘𝑛𝑥                →    0. This means that (𝑇1𝑘𝑛𝑥𝑇2𝑘𝑛𝑦 )                 →    0 which means that (𝑇1  𝑇2)𝑘𝑛 (x  y)                 →    0. In the same time, there is a sequence ln such that ‖𝑇1𝑙𝑛𝑥‖                 →    ∞ which means that  ‖𝑇1𝑙𝑛𝑥‖ ‖𝑇2𝑙𝑛𝑦‖                →    ∞ , then ‖(𝑇1𝑇2)𝑙𝑛(𝑥𝑦)‖                →    ∞.       In following theorem is improved atheorem that found in [1], which proved that if the sum of operators is Li-Yorke chaos then at least one of operator is Li-Yorke while we prove the following:  2.3. Theorem:         𝑇1 and 𝑇2 are operator on Banach space if 𝑇1 has irregular vector and there is a sequence 𝑘𝑛such that 𝑇1𝑘𝑛𝑥                →    0, where x irregular vector and 𝑇2𝑘𝑛𝑦                →    0 if and only if  𝑇1⨁ 𝑇2 has irregular vector. Proof:         Let 𝑥 be an irregular  vector of 𝑇1. thus are sequence 𝑘𝑛 such that 𝑇1𝑘𝑛𝑥 → 0 and 𝑇2𝑘𝑛𝑦 → 0.We have 𝑇1𝑘𝑛𝑥⨁𝑇2𝑘𝑛𝑦                →    0, that mean (𝑇1⨁𝑇2)𝑘𝑛(𝑥⨁𝑦)                →    0.       In the same time, there is a sequence‖𝑇1ℓ𝑛𝑥‖                →    ∞, we have ‖𝑇1ℓ𝑛𝑥‖2+ ‖𝑇2ℓ𝑛𝑦‖2                 →    ∞, means that ‖𝑇1ℓ𝑛𝑥⨁𝑇2ℓ𝑛𝑦‖2                 →    ∞, thus ‖(𝑇1⨁𝑇2)ℓ𝑛(𝑥⨁𝑦)‖2                 →    ∞. Then 𝑇1⨁𝑇2 has irregular vector. The conversely is similarity.     2.4. Theorem:           If 𝑇 B(H), and 𝑇 is Li-Yorke, then 𝑇∗ has no eigenvectors. Proof:       Suppose that 𝑇 is Li-Yorke chaos and 𝑇∗𝑣 =𝑣 when 𝑣 ≠ 0. If A vector 𝑥 ∈H is irregular for 𝑇 then lim sup‖𝑇𝑛𝑥, 𝑣 >‖ = lim sup ‖< 𝑥, 𝑇𝑛∗𝑣 >‖ = lim sup‖< 𝑥, 𝑛𝑣 >‖ = lim sup‖𝑛̅̅ ̅ < 𝑥, 𝑣 >‖ =  and  lim inf‖𝑇𝑛𝑥, 𝑣 >‖ = liminf ‖< 𝑥, 𝑇𝑛∗𝑣 >‖ = lim inf‖< 𝑥, 𝑛𝑣 >‖ = lim inf‖𝑛̅̅ ̅ < 𝑥, 𝑣 >‖ = 0 If || < 1 or < 𝑥, 𝑣 = 0 then the set  is bounded and if || ≥ 1 and < 𝑥, 𝑣 ≠ 0 then the last set is bounded below, then 𝑇 is not Li-Yorke chaos. 2.5. Corollary:           If X is finite dimensional, then 𝑇 has not Li-Yorke chaos in X. Proof:     Suppose 𝑇 is Li-Yorke Chaos in X. since X is finite dimensional, hense 𝑇∗ has eigenvalues a contradiction.  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  100       The following theorem described the relation between irregular vectors commuting operators.   2.6. Theorem:           Let S and T be commuting operators on X. then the set of all irregular vectors for S is T-invariant. Proof:         Let M be the set of all irregular vectors for S, xM to prove TxM, by definition of irregular vector , there exist two sequence 𝑘𝑛and 𝑙𝑛 increasing to  such that 𝑆𝑘𝑛𝑥                →    0 and ‖𝑆𝑙𝑛𝑥‖                →    ∞. By induction we can show that         𝑆𝑘𝑛𝑇𝑥 = 𝑇𝑆𝑘𝑛𝑥. Then 𝑆𝑘𝑛𝑇𝑥 = 𝑇𝑆𝑘𝑛𝑥                →    0 and‖𝑆𝑙𝑛𝑇𝑥‖ = ‖𝑇𝑆𝑙𝑛𝑥‖ =‖𝑇‖‖𝑆𝑙𝑛𝑥‖                →    ∞. 2.7. Theorem:          Let  𝑇 ∈ 𝐵(𝑋) , if there exist sequence 𝑘𝑛increasing to  such that lim𝑛→∞‖𝑇𝑘𝑛‖ = 0 and ‖𝑇‖ > 1 Then T is Li-Yorke chaos. Proof:      Let R=‖𝑇‖  1. Let {𝜀𝑘}𝑘=1∞  is a sequence of positive numbers decreasing to zero. First of all, fix 𝑁1 ∈ ℕ (for example, set  𝑁1 = 2). Then there is 𝑥1 such that ‖𝑥1‖ = 1 and     lim𝑛→∞‖𝑇𝑘𝑛𝑥1‖ = 0 and  sup‖𝑇𝑖𝑥1‖ = ∞ , i=1,…,𝑁1. So we can choose a positive integer M1 such that ‖𝑇𝑘𝑛𝑥‖ < 𝜀1 for any n M1. For convenience. Then ‖𝑇𝑁1𝑥1‖ ≥ 1. Now we will construct a sequence of points {𝑥𝑘}𝑘=1∞  associated with two sequences of integers {𝑁𝑘}𝑘=1∞ and  {𝑀𝑘}𝑘=1∞  such that for every k2,   1) ‖𝑥𝑘‖ = 𝑅−𝑀𝑘−1 . 2−𝑘𝜀𝑘−1; 2) ‖𝑇𝑖𝑥𝑘‖ ≥ 1, i=1,…,𝑁𝑘; 3) ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ < 𝜀𝑘𝑘𝑗=1 , for any  𝑘𝑛 ≥ 𝑀𝑘. Select there is 𝑥2 such that   ‖𝑥2‖ = 𝑅−𝑀1 . 2−2𝜀1 and                   lim𝑛→∞‖𝑇𝑘𝑛𝑥2‖ = 0 and  sup‖𝑇𝑁2𝑥2‖ ≥ 1 So we can choose 𝑀2such that ‖𝑇𝑘𝑛𝑥1‖ + ‖𝑇𝑘𝑛𝑥2‖ < 𝜀2 for any n≥ 𝑀2. Continue in this manner. If we have obtained  {𝑥𝑘}𝑘=1∞ , {𝑁𝑘}𝑘=1∞ and {𝑀𝑘}𝑘=1∞  such that for each k=2,…, m.   1)  ‖𝑥𝑘‖ = 𝑅−𝑀𝑘−1 . 2−𝑘𝜀𝑘−1; 2) 𝑠𝑢𝑝𝑛 ‖𝑇𝑖𝑥𝑘‖ ≥ 1, i=1,…,𝑁𝑘; 3) ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ < 𝜀𝑘𝑘𝑗=1 , for any n≥ 𝑀𝑘. Select there is 𝑥𝑚+1 such that   ‖𝑥𝑚+1‖ = 𝑅−𝑀𝑚 . 2−(𝑚+1)𝜀𝑚 and              lim𝑛→∞‖𝑇𝑘𝑛𝑥𝑚+1‖ = 0 and  sup‖𝑇𝑁𝑚+1𝑥𝑚+1‖ ≥ 1 . So we can choose 𝑀𝑚+1such that ∑ ‖𝑇𝑘𝑛𝑥𝑗‖𝑚+1𝑗=1 < 𝜀𝑚+1 for any n≥ 𝑀𝑚+1. If we have obtained  {𝑥𝑘}𝑘=1∞ , {𝑁𝑘}𝑘=1∞  and {𝑀𝑘}𝑘=1∞  such that for each k=2,…,m,   1)  ∑ ‖𝑥𝑘‖∞𝑘=1  is finite. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  101 2) For each p,  ‖𝑇𝑖𝑥𝑘‖ < 2−𝑘𝜀𝑘−1 , for any 𝑘 > 𝑝 and any 1  i  𝑀𝑝. Hence, ∑ ‖𝑇𝑖𝑥𝑘‖ <∞𝑘=𝑝+1 ∑ 2−𝑘𝜀𝑘−1  <∞𝑘=𝑝+1 𝜀𝑝for any 1  i  𝑀𝑝  3) For each k, ‖𝑇𝑖𝑥𝑘‖ ≥ 1, 𝑖 = 𝑁𝑘′ , … , 𝑁𝑘. 4) 𝑀𝑘 > 𝑁𝑘 > 𝑀𝑘−1 for each k. 5) ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ < 𝜀𝑘−1𝑘−1𝑗=1 , for n=1,… , 𝑁𝑘. 6) For each p, ∑ ‖𝑇𝑘𝑛𝑥𝑘‖ < 𝜀𝑝∞𝑘=𝑝+1 , for n=1,… , 𝑁𝑝. Let 2 = {0, 1}ℕ be a symbolic space with two symbols. According to condition (1), we can define a map    f: 2→ X as                                     𝑓(𝜉) = ∑ 𝜉𝑘𝑥𝑘∞𝑘=1  For every element 𝜉 = (𝜉1, 𝜉2, … ) ∈  2. Obviously one can get an uncountable subset D∈ 2 such that for any two distinct points  𝜉, 𝜉′ 𝐷, 𝜉 and 𝜉′ have infinite coordinates that are different and infinite coordinates that are equivalent. Then                     ‖𝑓(𝜉) − 𝑓(𝜉′)‖ = ‖∑ (𝜉𝑘 − 𝜉𝑘′ )𝑥𝑘∞𝑘=1 ‖. Set 𝜃 = (𝜃1, 𝜃2, … ) =(𝜉1 − 𝜉1′ , 𝜉2 − 𝜉2′ ,…).Then‖𝑓(𝜉) − 𝑓(𝜉′)‖ = ‖∑ (𝜃𝑘)𝑥𝑘∞𝑘=1 ‖. Note that the possible values of 𝜉𝑘 − 𝜉𝑘′ are only 0,-1, or 1, and 𝜃 has infinite coordinates being zero and infinite coordinates bring nonzero. Now we will prove that { 𝑓(𝜉), 𝑓(𝜉′)} is a Li-Yorke chaotic pair . Let 𝑧 = ∑ 𝜃𝑘𝑥𝑘∞𝑘=1 . Suppose {𝑘𝑞}𝑞=1∞ is the infinite subsequence such that the 𝑘𝑞th coordinate of 𝜃 is nonzero (1 or -1) and {𝑘𝑟}𝑟=1∞  is the infinite subsequence such that the 𝑘𝑟th coordinate of 𝜃 is zero. By (5),(6) and (2), for n=1,… , 𝑁𝑘𝑞, ‖𝑇𝑘𝑛𝑧‖ ≥ ‖𝑇𝑘𝑛(𝜃𝑘𝑞𝑥𝑘𝑞)‖ − ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ − ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ > 1 − 𝜀𝑘𝑞−1 − 𝜀𝑘𝑞∞𝑗=𝑘𝑞+1𝑘𝑞−1𝑗=1. Since {𝜀𝑘}𝑘=1∞  decrease to zero, then lim 𝑠𝑢𝑝𝑛→∞‖𝑇𝑘𝑛(𝑓(𝜉)) − 𝑇𝑘𝑛(𝑓(𝜉′))‖ = lim 𝑠𝑢𝑝𝑛→∞‖𝑇𝑘𝑛(𝑧)‖ ≥ lim 𝑠𝑢𝑝𝑞→∞‖𝑇𝑁𝑘𝑞(𝑧)‖ ≥ 1 On the other hand, ‖𝑇𝑘𝑛𝑧‖ ≤ ‖𝑇𝑘𝑛(𝜃𝑘𝑟𝑥𝑘𝑟)‖ − ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ − ∑ ‖𝑇𝑘𝑛𝑥𝑗‖ < 𝜀𝑘𝑟−1 − 𝜀𝑘𝑟∞𝑗=𝑘𝑟+1𝑘𝑟−1𝑗=1 . lim 𝑖𝑛𝑓𝑛→∞‖𝑇𝑘𝑛(𝑓(𝜉)) − 𝑇𝑘𝑛(𝑓(𝜉′))‖ = lim 𝑖𝑛𝑓𝑛→∞‖𝑇𝑘𝑛(𝑧)‖ ≤ lim 𝑖𝑛𝑓𝑞→∞‖𝑇𝑁𝑘𝑞(𝑧)‖ ≤ 0 Therefore, { 𝑓(𝜉), 𝑓(𝜉′)} is a Li-Yorke chaotic pair for any distinct { 𝜉, 𝜉′}D , then T is Li-Yorke chaos.   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  102 3. The criterion for Li-Yorke chaos             The following criterion for Li-Yorke was introduced in [7]. Some definitions and theorems on Li-Yorke Chaos Criterion. 3.1. Definition[7]          An operator T: XX satisfies the Li-Yorke Chaos Criterion (LYCC) if there exist an increasing sequence of integers(nk)k and a subset 𝑋0⊂X such that (a) limk→∞ 𝑇nk𝑥 = 0 , 𝑥 ∈ 𝑋0, (b) sup   ‖𝑇n|Y‖ = ∞ , where Y:=span(𝑋0)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅  and 𝑇n|Y denotes the restriction operator of 𝑇𝑛to Y.  3.2. Definition[5]          Given an arbitrary operator 𝑇 ∈ 𝐵(𝑋) on a complex Banach space 𝑋 and any 𝑋 and any closed set Fℂ , the glocal spectral subset 𝜒𝑇(𝐹) consists of all 𝑥 ∈ 𝑋 for which there exists an analytic function 𝑓: ℂ\F → X with the property that (𝑇 − 𝐼)𝑓() = 𝑥   for all  ∈  ℂ\F.  3.3. Theorem[5, P.225]            Let 𝑇 ∈ 𝐵(𝑋) be an operator on a Banach space 𝑋 and let 𝑓:𝑈 → ℂ be an analytic function on neighbourhood 𝑈 of 𝜎(𝑇). Then 𝑋𝑓(𝑇)(𝐹) = 𝜒𝜑(𝑇)(𝐹) = 𝜒𝑇(𝜑−1𝐹) For every closed subset F of ℂ.  3.4. Theorem:            Suppose that the operator TB(X) on the Banach space X. if 𝜒𝑇(𝒟) ∩ 𝜒𝑇(ℂ\𝒟) ≠ ∅, and then T satisfied Li-Yorke Chaos Criterion. Proof:      To show that T satisfies Li-Yorke chaos, we will prove that every vector x ∈ 𝜒𝑇(𝒟) ∩ 𝜒𝑇(ℂ\?̅?) is irregular vector for T.      If 𝑥 ∈ 𝜒𝑇(𝑘) for some compact k⊂  𝒟, then there exists resolvent function f: ℂ\𝑘 →  𝜒 such that                    (𝑇 − 𝐼)𝑓() = 𝑥 for all  ℂ\𝑘. Choose 0p1 so that kB(0,p).       Let  and  respectively denote the positively oriented circles {: || = 𝑝} and {: || = ‖𝑇‖ + 1} respectively, then By [6,P.140]               𝑇𝑛𝑥 =12𝑖∫ 𝑛𝑔()𝑑           for every n0 and hence by [2,P. 205]              𝑇𝑛𝑥 =−12𝑖∫ 𝑛𝑔()𝑑=−12𝑖∫ 𝑛𝑔()𝑑         for every n0      In particular, for every  𝑥 ∈ 𝜒𝑇(𝒟), it follows that 𝑇𝑛𝑥 → 0 as  𝑛 → ∞. Thus the first condition, in the irregular vector for T is satisfied.      Now for the second condition, if 𝑥 ∈ 𝜒𝑇(𝑘) for some compact k⊂  ℂ\?̅?, then there exists resolvent function  𝑔: ℂ\𝑘 →  𝜒 such that (𝑇 − 𝐼)𝑔() = 𝑥 for all  ℂ\𝑘.      Choose 1𝑝1𝑝2, so that k is contained in the annulus {:𝑝1 < 𝑝2}. let 1and 2be the inner and outer boundaries of the annulus respectively, each with counterclockwise orientation, and let  = 1- 2.      If 𝑎 ∈ 𝑘, then a is the inside of  and hence n(, 𝑎) = 1. Thus n(, 𝑘) = 1.       If 𝑎 ∈ ?̅?, then a is the outside of  and hence n(, 𝑎) = 0. Thus n(, ?̅?) = 0.      Define             𝑇𝑛𝑥 =12𝑖∫ 𝑛𝑔()𝑑        for every n0      In particular, for every 𝑥 ∈ 𝜒𝑇(ℂ\?̅?), it follows that 𝑇𝑛𝑥 → ∞ as 𝑛 → ∞.  Thus the second condition, in the irregular vector for T is satisfied. 3.5. Corollary:           Suppose that 𝑇 ∈ 𝐵(𝑋) and 𝜑 analytic in a neighborhood of 𝜎(𝑇). If there exists open sets U, Vℂ so that each of the subspace 𝜒𝑇(𝑈) ∩ 𝜒𝑇(𝑉) ≠ 𝜑 , then 𝜑(𝑇) is Li-Yorke Chaos Criterion if 𝜑 separates U and V is the sense that 𝜑(𝑈)𝒟 and 𝜑(𝑉)ℂ\?̅?.   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.1, 2016  103 Proof:    Since 𝜑 is analytic in a neighborhood of 𝜎(𝑇) then theorem (3.3). 𝜒𝜑(𝑇)(𝐹) = 𝜒𝑇(𝜑−1𝐹) For every closed subset F of ℂ. If H is an open subset of ℂ, let                                         𝜒𝜑(𝑇)(𝐻) =∪ {𝜒𝜑(𝑇)(𝐹): 𝐹 𝑖𝑠 𝑎 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐻} =∪ {𝜒𝑇(𝜑−1(𝐹)): 𝐹 𝑖𝑠 𝑎 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐻} = 𝜒𝑇(𝜑−1(𝐻))       For every open subset H of  ℂ. Since 𝜑(𝑈)𝒟 and 𝜑(𝑉)ℂ\?̅? it follows that 𝑈 ⊆ 𝜑−1(𝒟) and  𝑉 ⊆𝜑−1(ℂ\?̅?). Hence 𝜒𝑇(𝑈) ⊆ 𝜒𝑇(𝜑−1(𝒟)) = 𝜒𝜑(𝑇)(𝒟) 𝜒𝑇(𝑉) ⊆ 𝜒𝑇(𝜑−1(ℂ\?̅?)) = 𝜒𝜑(𝑇)(ℂ\?̅?) Thus                           𝜒𝑇(𝑈) ⊆ 𝜒𝜑(𝑇)(𝒟) and 𝜒𝑇(𝑉) ⊆ 𝜒𝜑(𝑇)(ℂ\?̅?)       Since 𝜒𝑇(𝑈) ∩ 𝜒𝑇(𝑉) ≠ 𝜑 then𝜒𝑇(𝒟) ∩ 𝜒𝑇(ℂ\?̅?) ≠ 𝜑. Thus 𝜑(𝑇) is Li-Yorke Chaos Criterion.   References [1] B. Beauzamy. Introduction to Operator Theory and Invariant Subspaces.1990.Americam Mathematical Society.    22(1) [2] Conway, J.B.(1985).A course in functional analysis. Springer-Verlage,Berlin Heidelberg,New York. [3] Prajitura,G.T.(2009).Irregular vectors of Hilbert space operators. J. Math. Anal. Appl.354: 689–697. [4] Banks,J.,Brook,J.,Cairns,G., Davis,G. andStacey,P.(1992)."On Devaney ’s definition  of chaos". American Mthematic Monthly. 99: 332-334.  [5] Laursen, K.B. and Neumann, M.M.(2000).Introduction to local spectral theory. London Math. Soc. Monographs New Series, Clarendon Press, Oxford Science publication.  [6] Sehechter, M.(1971).principles of function analysis. Academic Press, Inc.New  york and London. [7] Bermudez,T., Bonilla,A., Martinez-Gimenez, F. and Peris, A.(2010). Li-Yorke and distributionally chaotic operater.J.math.Anal.Appl.373:83-93.  [8] Li,T.Y. and Yorke,J.A.(1975).Period three implies chaos. Amer.Math. Monthly 82: 985–992. 

Some Properties of Li-Yorke Chaos

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Some Properties of Li-Yorke Chaos

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