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DATE DUE DUE DUE DATE DUE 1/98 chlRCMDmpGS—p.“ Entropy Zero Systems and Morse—Smale Systems By Wei Wang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT Entropy Zero Systems and Morse—Smale Systems By Wei Wang In this thesis, we prove that if f is a real analytic diffeomorphism on a two di- mensional compact Riemannian manifold, the non—wandering set of f is finite and f satisfies locally normalized condition, then, f can be 0' (r > O) approximated by a Morse—Smale diffeomorphism. ACKNOWLEDGMENTS I am deeply indebted to my advisor Professor Sheldon Newhouse who provided me with invaluable help, guidance and encouragement both in my personnel life and in academic research. I would like to thank the members of my dissertation committee, Professor T.Y. Li, W. Sledd, C. Wei], Z.F. Zhou, for their serving on my committee, their kindness and their time. Finally, I wish to express my thanks to my parents and my wife for their continuing love, encouragement and unsparing support. iii 4 5 TABLE OF CONTENTS Introduction and Main Result Preliminary Elementary Cycles Advanced Cycles 3.1 When p 6 F3 ............................... 3.2 When p 6 F3 ............................... 3.3 When 1) e F; ............................... 3.4 Removing advanced cycles ........................ No Cycles Conclusion and Remarks BIBLIOGRAPHY iv 21 22 23 26 37 40 49 50 CHAPTER 0 Introduction and Main Result Let M be a compact Riemannian manifold. f be a C" (r > 0) diffeomorphism on M. One of the important quantities to describe the complexity of the structure of a system is topological entropy, which tells roughly how many different orbits f has. A formal definition of entropy is as follows. h(f) = lim lim sup “*0 n—ioo log s(n, c) n where h( f ) be the topological entropy of f, 3(n, e) = maxECM{cardinality of E} and E is a set such that if x, y E E then d(f":r, fky) > e for some k E [0, n). In other words, h( f ) is the asymptotic growth rate of the number of finite length orbits known with precision 15 as the length goes to infinity. Topological entropy is a topological invariant and has very nice properties: 1. (Katok—Newhouse—Yomdin Theorem)[5] [8] [13] If the dimension of M is two, then h( f ) is continuous with respect to f in the C°° case. 2. (Bowen)[2] h( f) = h(le), where Q is the non—wandering set of f, Q = {ml for any neighborhood U, U|m|>0 f'"(U) n U 74 (0}. 3. If the dimension of M is two, then the set of systems with zero entropy is a closed set. 4. The systems with finite non-wandering set have zero entropy. 5. The system whose non-wandering set contains a horse shoe has positive entropy. A natural question is what we can say about the systems that have zero entropy. A Morse-Smale system is a system with following properties: (1). its non— wandering set is finite and hyperbolic; (2). all the intersections between stable man- ifolds and unstable manifolds are transversal. Morse-Smale systems are very important systems. Their structures are well un- derstood. A interesting question is that whether we can use Morse—Smale systems to approximate systems with zero entropies? If the dimension of the manifold M is greater than three, the answer to that question is no. Dankner [3] constructed a counterexample example in the early 19805. If the dimension of M is two, Newhouse conjectures that it is true. Conjecture (Newhouse) 0n the two dimensional compact manifold, the difi'eomor— phisms with zero entropies are on the boundary of the set of Morse—Smale systems. In this direction, the earliest important work was done by Newhouse and Palis [10] in early 19705. They developed the breaking cycle technique and proved that any diffeomorphism with hyperbolic non-wandering set can be approximated by Q-stable diffeomorphism. About fifteen years later, Malta and Pacifico [6] generalized this result by relaxing the condition from hyperbolic non-wandering to hyperbolic limit set. As a corollary of their results, we see that, in dimension two, a diffeomorphism with finite hyperbolic non-wandering set (or limit set) is on the boundary of Morse- Smale system set. Therefore, the next question is that if we drop the hyperbolic condition, which is a very strong condition, will the same result hold? Problem (Newhouse) Can any difieomorphism on the two dimensional compact manifold with finite non-wandering set be approximated by a Morse—Smale difi’eomor- phism .9 Definition (locally normalized condition) A diffeomorphism f is called to have locally normalized condition at a fixed point 0, if it satisfies the following conditions: (1). in a small neighborhood of o, f can be embedded in an analytic vector field with no elliptic sectors; (2). there is an invariant analytic curve through 0. In this paper, we prove the following theorem Theorem Let M be a two dimensional compact manifold, f be a real analytic difieo- morphism on M, the no-wandering set of f 9( f) be finite and f satisfy the locally normalized condition. Then, f can be approximated in C' (r > 0) by a Morse-Smale difieomorphism. In chapter 1, we review some basic definitions and facts. Chapter 2—5 is the proof of above theorem. In Chapter 2, we work with elementary cycles. We will break all the elementary cycles without causing Q—explosion. Chapter 3 is about advanced cycles. We will study the local structures around fixed points and remove the advanced cycles without causing Q-explosion. Systems that have no cycles are discussed in chapter 4. Chapter 5 concludes this paper. CHAPTER 1 Preliminary In this Chapter, we review some basic definitions and facts. Throughout this paper, we let M be a two dimensional compact manifold, f be an analytic diffeomorphism on M. (2( f ) be the non-wandering set of f, i. e. the set of points with the property that for every neighborhood U such that Ulml>0 f’"(U) D U 96 (b. When 0( f) is finite, 9( f) = per( f ), per( f ) is the set of periodic points of f. Let p E 9(f), we denote the set of points a: such that d(f"a:, f"p) -—> 0(n —> 00), where n 6 2‘“, by W‘(p). W" (p) is called the stable set of f at p. The stable set of f '1 at p is called the unstable set of f at p, denoted by W"(p). If Tp(M) can be split as a direct sum E; 6E; so that T, f (E;) = E}? and Tp f (E3) = E}‘p, where E; = {22 such that ITpfvl Z Alvl} and E; = {U such that |Tpfv| _<_ A'llvl} for some constant A > 1, then, p is called a hyperbolic point. Clearly, if p 6 per( f ) and the two eigenvalues of f at p are not on the unit circle, then p is hyperbolic. If one of its two eigenvalues lies inside of the unit circle, while the other one lies outside of the unit circle, then, p is called a saddle. If both eigenvalues lie inside of the unit circle or outside of the unit circle, then, p is called a node. If one of its two eigenvalues lies on the unit circle, the other one is not on the unit circle, then p is called a semi-hyperbolic periodic point. Let p €per( f ), N (p) be a neighborhood of p. Let p be a saddle of a diffeomorphism f, N (13) be a neighborhood of {2. If, in N (p), f is topologically equivalent to f in N (p), then, we call p a topological saddle. Similarly, we can define topological node and topological saddle-node. A semi-hyperbolic point can be a saddle-node or a topological saddle-node or a topological saddle or a topological node. By the Invariant Manifold Theorem, when p is a saddle, the stable set W’(p) and unstable set W“ (p) are both one dimensional manifolds. A diffeomorphism f is called a Morse-Smale diffeomorphism, if the following pr0p- erties are satisfied: 1. {2( f ) is finite and hyperbolic, 2. the intersections between W“(p) and W3(q) for any points p q in 9( f) are transversal. We say that a curve P = {($(t), y(t)),t 6 [0,6]} enters p under f, if 1- ($(0),y(0)) = (0,0), 2. limHo y(t)/z(t) or limt_,o:c(t)/y(t) exists, ll ”:3 A O "1 3. P is invariant under f and lim,Hoe f"(:c(t),y(t)) limnsoo f ’"($(t), y(t)) = p) for t E (0, 6)- CHAPTER 2 Elementary Cycles In this chapter, we consider the diffeomorphisms whose nonwandering sets consist of finite fixed points and that has only elementary cycles. We will break the elementary cycles without causing Q—explosion. In our following discussion, we only consider orientation preserving diffeomor- phism. Let 9( f ) be the set of all non-wandering points of f in Q( f) which have at least one non-empty hyperbolic sector, one stable separatrix and one unstable separatrix, such that one of which must be a separatrix of this hyperbolic sector. We denote a hyperbolic sector of p by HS (p) and a non-hyperbolic sector by NHS (p) Definition Let p, q E O( f ), S (p) and S (q) be a sector of p and q respectively, and an unstable separatrix S§(p) C S (p) and a stable separatrix S§(q) C S (q) Then, we call that sector S (q) follows sector S (p), if following two properties are satisfied: A A 8 1- S"(P) as (q) 75 0, where 58(4) = 53(4) - {q}, 9‘09) = Ungo f"(52‘(p)), 5‘01) = Unzo f ‘"(S§(q)) A 2. Let z E S"(p)flSs(q), x 6 53(1)), y E S§(q), let D be an arbitrary small neighborhood of curve segment 7 = [z 2] U [z y]. Then, S (p)flD aé (b and S(q) n D 75 0, where D is a connected component of D \ 'y. 6 We denote that sector of q, S (q), follows sector of p, HS (p), by S (p) >— S (q) or S (q) -< S (p) We also have the corresponding notations when the sectors are hyperbolic and non-hyperbolic sectors. For any p, q 6 52(f), if there is a series of points p = p0,p1,p2, . . . ,pk = q in Q(f) such that W“(p,-)nW"(p,-+1) aé 0, i = 0,1,...,k — 1, we call that p > q. If there is a series points p1,p2, - - - ,p, in 0(f) such that p1 > p2 > > p,, we call it a chain, denoted by C[p1,p2, . . . ,pz]. Definition A chain [p0, p1, . . . , pk], p, E O( f ), is called an elementary chain, if A S“(p,-_1) (183(1),) 3£ 0fori=1,2,. . . , k (2.1) and S'"(p.-)CHS(p,) fori=1,2,...,k-—1ando=uors (2.2) In this case, we call that S"(p,) are in this chain, for o = u or s. We denote the elementary chain by C = [HS (p0), HS (p1), . . . , HS(pk)]. Definition If (2.1) and (2.2) are both true for all i E Z”, where p,- = p,- mod(k+1), then the chain is called an elementary cycle. All the other cycles are called advanced cycles, and we call that 5" (p,) is in this cycle, for o = u or s. We denote the elementary cycle by A = [HS(po),HS(P1), ° ° ' a HS(pk)a HS(p0)] In the following, we list some simple lemmas which will be used later in this section. Lemma 2.0.1 Let p E 6( f ) and be a fixed point of f, HS (p) be a hyperbolic sector A determined by S’(p) and S"(p), V be a neighborhood of p, let q E VflS’(p), y E VflHS (p) and J = [q, y] be a C" (r > 0) curve segment. Then, given 5 > 0, there exists no 6 N such that ifn > no, f"J is e-close to S“(p). D Definition Let p E 8(f), q E Q(f), HS(p) be determined by S"(p) and Ss(p), we say that S“(p) is directly accumulated by S“(q) through HS (p), if there exists a are [93.31] C 5“(Q) flHSOD) and x 6 5’09)- We say that an open set V meeting S"(p) (o = u or s) at x, x E S"(p), if V = V’ 0 HS (p), where V’ is a neighborhood of x in M. Lemma 2.0.2 Let p, q be fixed points of f in 8( f ), HS (p) and HS (q) be determined by S"(p) and S"(q) respectively, (a = s, u), HS(p) > HS (q), let V be an open set meeting S‘(p) at x, x E S‘(p)nHS(p), U be an open set meeting S"(q) at y, y E S“(q) nHS(q). Then, there exists an integer m > 0 such that f'"(V) H U 9e 0. Proof: Let J be a segment in V that is transversal with S" (p) at x. By Lemma 2.0.1, there exists n1 such that f"1(J) is a -close to S“(p). Let 2 6 S“(p) nS"(q) and N’ be a neighborhood of z in M, such that N = N’ 0 HS (p) 96 0. Then, there exists an integer m1 such that fm‘J ON 76 0, let fmlJ nN = J’. We know that there exists n2 such that fn’z E ITS—(i1), since HS (p) > HS (q), we have an open disk D, as in the definition at page 6, such that z E D, fn’z E D and HS (p) n D, HS(q) n D both are contained in the same connected component D of D \ 7. We claim that f"’(J’) flfianm) aé 0 To prove the claim, If S“(p) = Ss(q), the claim is clear; if S“(p) # S“(q) and S“(p) directly accumulates on Su(q), then 3 G’ C G such that f"’(G') C HS (q) 0 D, since f”2(z) E DflHS(q). Note that HS(q)flD is open, so f"2(J)flDflHS(q) 95 0; if S" (p) 7e S " (q) and S‘(q) is directly accumulates on S S(p), we use the similar argument to get the claim. For the situation when f is orientation reversing, we can follow the similar process as above to prove the claim. Now, in HS (g), from Lemma 2.0.1. we have 521(1) C U f‘"(U) n20 therefore 2 e U f-"(U) n20 so f"’(J')fl(U f’"(U)) 75 0 n20 this means that there exists m > 0 such that f'"(J’) nU 5i (0, thus f'"(V) n U 76 0, the lemma is proved. [:1 Definition Let {qj} be fixed points of f in 9( f ), HS (qj) be determined by S”(q,-), (o = s, u) and j = 0, 1, 2, . . . , h. Then, an elementary cycle A = [HS(QO), HS(QI), ° ° ' v HS(qk)i HS(QDH (k 2 0) is called a simple cycle if S“(q,-) = S’(q,+1) and HS(q,) > HS (q,-+1), where i: 0,1,1"? (Ii = (Ii mod(k+1)- Corollary 2.0.3 Let A = [HS(qo),HS(q1), . . . ,HS(qk),HS(qo)], k 2 0, be a simple cycle, then, S"(q,-) C 9(f), where S"(q.-) C HS(qi), 0 S i g k, a = s, u. Proof: It is enough to prove that S’(qo) C 9( f). For any x E S‘(q0), let V be an open set meeting S‘(qo) at x. To prove x 6 Q( f), we only have to prove that there is a 10 m > 0 such that f’"(V) D V at 0. Since, S“(qk) = S‘(qo), there exists n; such that f ‘"=x E HS(qk) and f ‘"’x e Un>0 f “"V. Let U be an Open set such that it meets S“(qk) at f‘”=x and U C UnZO f""(V). We claim that there is an integer l > 0 such that f‘(V)flU#0 therefore, f'"(V)n(U f"“(V)) 76 0 n>0 it follows that there is a m > 0 such that f'"(V) f) V 75 0. Now, we prove the claim. Let x1 6 S“(q1)flm, V1 be an open set meeting S“(q1) at x1. Since, HS((I2) > HS (ql), by Lemma 2.0.2, there is a m1 > 0 such that fm1(V)flV1 ¢ 0. Since, S"(q1) = S’(q2), there is a n3, > 0 such that fnzlxl E S’(q2) flHS(q2). Let U1 be an open set contained in fnlel and meeting S"(q2) at f"=1x1, then, f"“+"=1(V) HUI 75 0. We repeat above process by starting with U1 instead of V1, we find an integer l > 0 such that f‘(V) flU 7£ (D. This proves the corollary. [3 Definition An elementary chain C = [HS(qo),HS(q1), ' - -,HS(qk)] is called an im- proper chain if HS(QO) >' HS(QI) > * HS(CIk) If in above definition qo = qk, then it is called an improper cycle. Corollary 2.0.4 Let q,- be fixed points off fori = 0, 1,2, . . .,k and A = [HS(Qo),HS(q1), ' - - . HS(qk), HS(qo)l be an improper cycle. If there are two points q,, q,- on this cycle such that S“(q,~) fl S‘(qj) 75 (l) and S“(qj) is directly accumulated by S"(q,-) or S”(q,~) is directly 11 accumulated by S’(qJ-), then S"(q,-) fl 38(Qj) C 52(f). Proof: It is enough to prove this corollary when S“(qj) is directly accumulated by S "((11). Let x E S’(qj)flS"(q,-)flHS(q,-), y E S’(q,-)flS’(q,-), then there exists n,- > 0 such that f‘"J' (x) 6 S“(q,-) 0 HS (q,). Let V be an open set meeting S’(q,-) at x and V C m, U be an Open set meeting S“(q,-) at f‘"i (x) and U C mmf‘"i(V). By using the similar argument as in the proof of Corollary 2.0.3 and Lemma 2.0.2, we find a m > 0 such that f'"(V) 0 U ¢ 0, it follows fm+"i(V) n V 79 (0, this proves the corollary. C]. Proposition 2.0.5 If dim(M) = 2, f E Diff'(M), Q(f) is finite, then there are no improper cycles. Proof: Let us suppose, by the way of contradiction, that there are improper cycles and A = [HS(qo),HS(q1),...,HS'(qk),HS(qo)] be one of them. That means that S“(q.~)flS‘(q.-+1) 75 0 (0 S i < k) and S“(qk) fl S‘(qo) 96 (0 and HS(q,+1) 4 HS(q,), HS((Io) * 115(le- If there exists a q,- such that S “(q,-) does not coincides with S ’(q,+1). Since HS (q,+1) follows HS (q,-), then, S"(q,+1)is either directly accumulated by S"(q,-) or S’(q,-) is accumulated by S’(q,-+1). By Corollary 2.0.4, we know that S’(q,-) fl S“(q.-+1) C 9( f ). However, the set {S ’(q,) 0 S“(q,-)} is infinite. It contradicts with the assumption that 9( f) is finite. So we only have to consider the case when S“(q,-) = S‘(q,-+1), 0 _<_ i < k and S“(qk) = S’(qo), it means that A is just a simple cycle. By Corollary 2.0.3, we know that in this case S’(q.-) C 9( f). This makes f2( f) infinite. So there are no improper cycles. Cl Definition Let q, p E 0( f), S (p) and S (g) be two sectors, we say that S"(q) vis- its S(p) if there exists a sequence {xn | n = 1,2,...} C S“(q)flS(p), such that 12 1iInn—mo xn = 1’- Definition Let q,p 6 9( f ), S“(q) is said to die at p through NHS(p), if for any A x E S“(q), the w-limit set of x is {p}. Let p, g be any two points in 0( f ), we call that p is equivalent to q, if there exist a series of points in (2( f ) p =p0)pl:p21' ' °apk—lapk : q = (Ianla ° - -,QI—1,QI= p such that for any p,- and q,, the following are satisfied: Wu(pi)nW8(pi-i-l) ¢ 0) i: 0,1)' "3k —1 Wu(qj)nws(qj+l) :Ié 0: j: 0311"'al _1 Notice that this relation between points in 0( f ) is a equivalence relation, therefore, there is a classification of f2( f ) according to this equivalence relation. Let the equiv- alence classes are {7,}, i = 1,2, . . .,m, then, 9(f) = U? 7,. Remark: 1. The set {71, 72, . . . , 7",} of equivalence classes is naturally partially ordered by 7,- 5 71-, if there exist p.- 6 7,- and q,- 6 7,- such that W“(qj) n W’(p,~) 915 Q). 2. All the cycles (elementary cycles and advanced cycles) in Q( f ) are contained in 7,- fori=1,2,...,m. 3. We say that an equivalence class 7,- is trivial, if 7,- only contains a fixed point. Let E be the set of all fixed points in 7 whose unstable separatrix does not cross any stable separatrix of fixed points in 7. 13 Definition For any p E 7,- 0 E.( f ), we call that S“ (p) is free if S"(p) C U W’m) 7j<7i or S“(p) dies at (‘1' through a NHS(q'), where W‘(7j) = UqE’Y, S’(q). We also have a corresponding definition by interchanging u and 3. Lemma 2.0.6 Let q2, q be two fixed points off, ql be afixed point in 9(f), Sg(q1) C m, o = s, u, S"(q1) is directly accumulated by S“ (q) through HS (ql) and S“(q1) = S"(q2), then, S"(q) will visit S(qo), where S ((12) is a sector of q2 such that S (Q2) -< HS (€11)- Proof: Since S"(q1) is directly accumulated by S“(q) through HS(ql). By the definition, there exists a point x E mmSu(ql) and y 6 HS (ql) flS’ (ql) such that [x, y] C HS (ql). Let J = [x, y]. By Lemma 2.0.1, we know that there exists no 6 N such that for any n > no f"J 6 -close S“(q1). Since S“(q1) = S’(q2), hence f"J 5 -close 58(42)- Let No he a 6-neighborhood of go, then El m > 0 such that f "‘J {1 No # 0. Choose x5 6 fmJ 0N5, let 6 ——-> 0, then we get a sequence {xo} such that x5 6 S"(q) and xo —+ go. The lemma is proved. C] The 2—dimensional curve [‘1 is called to cross 2-dimensional curve F2, if [‘1 fl [‘2 75 0 and 3 a neighborhood V of x 6 1‘1 flI‘2 such that N1 flf‘o 76 (b, No 0 F2 7E (b, where N1 and N2 are two different connected components of N \ I‘l. Lemma 2.0.7 Let q, go be fixed points of f, ql be a point in 9(f) and S"(q1) C HS(ql), o = s, u, ifS'“(q) crosses S‘(q1) and S“(q1) crosses S‘(q2), then S“(q) crosses Ss(<12)- D 14 In the following discussion, we denote by 7 a non-trivial equivalence class in {7,}, i = 1, 2, . . . ,m, that contains an elementary cycle. Corollary 2.0.8 There exist points qo,q1 E 7 H E( f ) such that S“(qo)flS‘(ql) 75 (0 and S“(qo) does not cross any Ss(p) for allp E 7. Proof: Since 7 is non-trivial and contains an elementary cycle, we have q, p E 7 such that S“(Q)flS’(p) at 0 Suppose, by the way of contradiction, that for any go 6 7, if S“(qo) flS‘(q1) 95 0 for some ql 6 7, then, S“(qo) must cross S’(q1). Then, by the definition of 7 and Lemma 2.0.7, we have that S"(qo) crosses S‘(qo). By Corollary 2.0.4, this implies S"(qo) nS’(qo) C 0( f). This contradicts the assumption that 9( f ) is finite. This proves the corollary. C] By above Corollary 2.0.8, we know that E 76 (0, if 9( f ) is finite. For any p E 7, we denote A(p) = {q E 7 such that S"(p) flS’(q) 76 0}. Lemma 2.0.9 Suppose Q(f) is finite, then there exist points q E EflO(f) andp E 7 such that one of the following properties is satisfied: 1- S"(t1)9‘é 53(1)) and S"(q) 053(1)) 75 0 2. S“(q) = S’(p) and there exists a hyperbolic sector of p, HSz(p), such that a unstable separtrix S", (p) is free, where 5;" (p) C HSo(p) and S: (p) C HSo(p). 15 3. S“(q) = S’ (p) and there exists a non-hyperbolic sector of p, NHS (p), such that NHS (p) -< HS (q), where S:(q) C HS (q) and S§(p) C NHS (p) Proof: Suppose there are no q E E n O( f ), p E 7 such that S“(q) 055(1)) 75 0 and S“(<1)9'é S"(19) that means that, for any two fixed points p and q in 7, if S“(q) fl S’(p) 75 (l) and S"(q) does not cross S’(p) then S“(q) = S’(p). Let po 6 7 n E. If El pl 6 A(po) such that S“(po) == S’(p1) and either (ii) or (iii) is satisfied by changing po to q and p1 to p, then we are done. If there is no such p1 in A(po), then we can choose a p1 in A(po) such that S"(po) = $8091) and HS(pl) '< HS(po) where S§(po) C HS(po) and S§(p1) C HS(pl). We claim that pl 6 E. In fact, if p1 ¢ 3 then El p 6 7, such that S"(p1) crosses S‘(p), where S:(p1) C m. By Lemma 2.0.7, we get that S“(po) crosses S’(p), which contradicts the fact that po 6 2. Based on point p1, we do the same thing as we just did based on po and repeat this process n times (or we may have already got (ii) or (iii) and hence, finished the proof), then we will get an elementary chain C = [HS(po),HS(p1), ' ° - , HS(pn)l where HS(pk) 4 HS(pk-1) and S"(pk_1) = S’(pk), k = 1,2,...,n. Because 7 is finite, n can not go to infinite. So, either we stop at some step by getting a point p,- such that (ii) or (iii) is satisfied if changing pJ-_1 to q and p,- to p, or we get a 16 elementary cycle A = [HS(po),HS(p1), - ~ - , HS(po)] Note that A actually is a simple cycle. By Corollary 2.0.3, we know that in this case, S“(pk) C 9( f), for k = 0, 1,. ..,n. This contradicts the assumption that 9( f) is finite. The lemma is proved. Cl Proposition 2.0.10 Suppose that 9( f ) be finite, then there exist ql E 7 and go 6 S such that S"(qo) 058011) 74 0 Moreover, I. If S"(qo) = S‘(q1). Then, either S“(q1) is free or there exists a point q2 6 7 such that S“(q1) = S’(q2), HS(Qg) 74 HS(ql) and S’(q2) C NHS(q2), where S"(q1) C NHS(q1), for o = u or s. 2. If S“(qo) 75 S’(q1). Then, either HS(ql) < HS(qo), S“(q1) is accumulated by S"(qo) and S“(q1) is free or S“(qo) dies at ql through the NHS(q1) or there exists q2 such that S“(q1)flS‘(q2) 7E Q) and S“(qo) dies at q2 through NHS(q2). Proof: By Lemma 2.0.9, we only have consider the case when 3 q E E and q1 in 7 such that S“(r1) US’(ql) at (0 and SW61)?é 53011) Now, there are only two possible situations between S“(q) and S"(q1). The first case is that there is a hyperbolic sector HS (ql) determined by S“(q1) and S‘(q1) through 17 which S“(q1) is directly accumulated by S“(q); The second case is that S “(q1) dies at ql through a non-hyperbolic sector NHS(q1), where 8;?(q1) C NW). If second case occurs, it is just a part of (ii) of the proposition, then we are done. Now, let us consider the first situation. If in this case, S“(q1) is free, then it is a part of (ii) of proposition, we finish. So, we only have to consider the case when S“(q1) is not free. We claim that in this case, ql 6 B. As a matter of fact, if ql ¢ E, that is to say that there exists a p E 7 such that S“(q1) crosses S‘(p). By Lemma 2.0.7, S“(q) will cross 8‘ (p), this contradicts the fact that q E 5. Let q2 E 7 such that S“(q1)flS‘(q2) 9'5 (D. We have to consider following six possible situations. Case 1. S"(q1) aé S’(q2), HS(Qz) -< HS(ql) and S“(q1) dies at (12 through NHS(q2), where S§(q2) C m and S§(q,-) C m, where o = u, s and i = 1, 2. Since S“(q1) does not cross S‘(q2), S’(q1) is accumulated by S‘(q2) through HS(ql); However, by the assumption, we know that S"(q1) is accumulated by S"(q) through HS(ql) and HS(ql) is determined by S“(q1) and S’(q1). So, S“(q) crosses S’(q2) in the hyperbolic sector HS(ql), this contradicts the fact that q E 3. Hence, this case can not occur. Case 2. S"(q1) 3A S‘(q2), HS(Qg) 74 HS(ql), S"(q1) directly accumulates on S“(q2) through HS(qo), where Sg(q,-) C m, where o = s, u and i = 1. 2. Since S“(q1) does not cross S‘(q2), S‘ ((12) will directly accumulate on S‘(q1). By using the same argument as in (1), we get that S“(q) crosses S“(q2), this contradicts with the fact that q E E. This means this case can not occur. Case 3. S“(q1) 9E S‘(q2), HS(Qz) K HS(ql), S“(q1) dies at go through a non- hyperbolic sector NHS(q2) of go, where S: (Q2) C NHS(q2) and S: (q,-) C HS (q,-), where o=s, uandi=l, 2. Since S“(q1) is directly accumulated by S“(q) through HS (ql), S“(q) will visit NHS(q2), therefore, S“(q) will die at qo. This proves the proposition in this case. 18 Case 4- S"(€h) = 53(92), HS(42) 74 H5011), Where 5:011) C HS((II), 5:012) C H5012)- Let S“(q2) be the unstable sepratrix of go such that 5:012) C HS(Q2) If there exists a hyperbolic sector HSo(q2) of go such that 5:012) C m and 3:012) mm = 0 (23) Let 52"(q2) be the unstable sepratrix of qo such that 52"(42) C m Then, since, HS (qo) 74 HS (ql), S;"(q2) is free. If there is no hyperbolic sector of q2 satisfying (3), then, there exists a non- hyperbolic sector NHS(q2) of go such that NHS(q2) -< HS (ql). Since S“(q1) is directly accumulated by S"(q) and S“(q1) = S’(q2), S“(q) will visit NHS(q2). So S“(q) will die at q; through NHS(q2) This proves the pr0position in Case 4. Case 5. S“(q1) 7e S’(q2), HS (q2) < HS (ql) and S“(q1) accumulates directly on S"(q2), where S"(q2) C m, o = s, u. We know that if S“(q2) is free then we are done. Now, suppose S"(q2) is not free, Then, S ’ (q2) must not cross any unstable separatrix of fixed point in 7. In fact, if there exists a po such that S"(q2) cross 8’ (po), then, S’(q2) will be directly accumulated by S’(po) through HS (go). So, S“(q1) crosses S’(po). This contradicts the assumption that S“(q1) does not cross any stable separatrix in 7. Now, we start with go as we did with ql, repeating our analysis from Case 1 to Case 6. This procedure must be stopped at one of two situations (a) and (b) listed below or else we get contradiction. 19 (a). we get a fixed point q,- and q,+1 in 7 such that S“(q,-) does not cross any unstable separatrix in 7. S“(q,-) = S’(q,-+1) and either S‘(q,-+1) is free or there exists q,-+2 such that S“(q,-+1) = S‘(q,~+2) and HS(q,+2) does not follow HS(q,+1), q,-+2 has a non-hyperbolic sector NHS(q,-+2) such that NHS(q,-+2) 4 HS(q,+1) and S;‘(q,-+2) C W. (b). we get two fixed points q, and q,-+1 in 7 such that either HS (q,-+1) follows HS(qg) and S“(q,-+1) is accumulated by S“(q,-) and S“(q,-+1) is free or S"(q,) dies at Q,“ through a non-hyperbolic sector NHS(q,-+1) or there exist q,-+2 such that S“(q,-) 0 S’ (q.-+1) ¢ 0 and S"(q,-) dies at q,+2 through the non-hyperbolic sector NHS(q,-+2). If (a) or (b) happens, then, let go = q,-,q1 = q,-+1,q2 = q,-+2, we prove the propo- sition. If our above procedure is not stopped at (a) or (b), then, we get a improper chain. Since the number of fixed points in 7 is finite, we actually get a improper cycle A = [HS (q), HS (ql), - . ~ , HS (q)] By Proposition 2.0.5, we know that it is impossible. Case 6. S"(q1) = S’(q2) and HS(qo) < HS(ql). If S“(q2) is free, then we are done. If S“(q2) is not free, it must not cross any stable separatrix in 7. As a matter of fact, if there exists a p E 7 such that S"(q2) crosses S’(p), then S’(q2) is directly accumulated at S‘(p) through HS (qo). Since S"(q2) = S’(q1) and S“(q) accumulated on S“(q1), S“(q) will visit HS(qo), hence S“(q) must intersect with S‘ (p). It contradicts with the assumption that Su (q) does not cross any stable separatrix. Now, we base on go to repeat our process as we did from Case 1 to Case 6. By using the same argument as in Case 5, we then finish the proof of this proposition. D Theorem 2.0.11 Suppose that 9( f) is finite, then, f can be approximated in Diff' (M) by a diffeomorphism g which has no elementary cycles and 0(g) is finite. Proof: We take on {7,}, for i = 1,2, . ..,m, a simple ordering compatible with _<_, so 20 that 71 S 72 S 3 7",. Suppose that for 1 S j < l 7,- either is trivial or does not contain elementary cycles. Let p,q E 7, given by Proposition 2.0.10. We can perform an arbitrary small Cr perturbation of f such that 7, remains same, while, S"(p) fl S‘(q) = (0. Moreover, there is no new intersections with Su (p) and such that each free unstable separatrix S"(p’) of p’ E 71-, for 1 S j g l remains free. (see Theorem B in [10]). Continue this process, we can achieve an arbitrary small 0’ perturbation of f which has the same non-wandering set as f and has one more free unstable separatrix. Finally, we will obtain a diffeomorphism g, 0' close to f, such that (2(g) = 0( f) and g has no elementary cycles. This proves the theorem. Cl CHAPTER 3 Advanced Cycles In this chapter, we consider the diffeomorphism that has advanced cycles, we will remove all the advanced cycles without causing Q-explosion. Let p be a fixed point of f, N be a neighborhood of p, f be analytic in N. We denote by J; (f) the first jet of f at p. In the following sections, we will discuss the local structures of f in a small neighborhood of p, when JFK f) has different Jordan forms. Suppose 9( f) is finite. Let 3:1 1 Fft = p E Q(f) such that J;(f) has Jordan form 0 :tl l 0 F; = p 6 Q(f) such that J;(f) has Jordan form 0 :l:1 e21rai 0 F3 = p 6 Q(f) such that J1](f) has Jordan form 0 e—21rai where (1 ¢ Z Let A1 = [po,p1,...,pk] be an advanced cycle in Q(f), S"(p,~), (a = u,s), be 21 22 separatrices that are in A1, that is S“(p.-) 058(1):“) 75 0, for i = 0,1,. ..,k — 1, and S“(pk) flS‘(po) 95 0. We denote by 91 the set 91 = {p 6 A such that S"(p) and S’(p) are not in a same sector } In the following discussion, we always assume that the diffeomorphism is orienta- tion preserving and satisfies locally normalized condition unless we explicitly specify others. 3.1 When p E Fft Without lose generality, we only consider the case when p E F 1+ . Theorem 3.1.1 (1) Let 0 be a singular point of a real analytic vector field (x cos 27m + ysin 27m + fix, y), —x sin 27m + y cos 21m + $(x, y)) with i5(x, y) and $(x, y) are the series with degree greater than 1. Define h as a transformation ~ (£5.31) —+ ($,y+§‘l(f—2’§%) Since a 5! Z, h is well defined and is analytic. Moreover, it is invertible, its inverse is ($.31) —+ (my + X(x, y)) 24 where X(x, y) sin 2m + #70:, y + X(x, y)) = 0 Consider h f h'l, we have hfh‘l = hf(x,y + X(x.y)) ..—.. h(x cos 27rd + ysin 27m + T(% y), —x sin 27rd + y cos 27m + (b133, y)) = (x cos 27m + y sin 27m, —x sin 27m + y cos 27m + (1501831)) where X(x, y) sin 2m + 213(23, :1 + X(x, y)) = 0 fire, 31) ¢*(x, y) = X(x, 31) cos 2m + 65(x, y + x(x, y)) = 0 and (Me, :1) = X(x, 31) ~ ¢(x cos 2m: + sin(27ra)x"‘ (x, y), —x sin 21m + cos(27roz)x*(x, y) sin 27m where x*($, y) = y + X(x, ii)- Let g = h f h_1. We can directly check that ¢(x, y) is as desired. This proves this lemma. E] Proposition 3.2.2 Let f be analytic in a neighborhood of a fixed point p, and 1 e21rai 0 J. (f) = 0 e—21rai 25 where 0: ¢ Z. Then, there is no separatrix in N. Proof: Without lose generality, we let p = 0(0, 0) be a fixed point of f. Then, by Lemma 3.2.1, f is analytically conjugate with f: (x, y) ——> (x cos 27m: + ysin 27m, —x sin 27rd: + ycos 27rd + ¢(x, y)) where ¢(x, y) is analytic and ¢(x, y) = o(\/x2 + yg). Suppose, by the way of contradiction, that there is a separatrix I‘ = {(x,y) such that A(x, y) = 0}. By the definition of separatrix, we know that either go géOOr Moofib. _(,00) Let us first consider the case when g—ylo 96 0. Then, I‘ can be written as (00) = h(x). By Corollary 3.0.14, h(x) is analytic. Suppose its Taylor expansion is = its Since fI‘ = r, we have that — sin(21ra)x + cos(27ra)u(x) + ¢(x, to» = u(cos(27ra)x + sin(27ra)u(x)) or ¢(x, to» = u(cos(27ra)x + sin(27ra)u(x)) + sin(27ra)x — cos(27ra)u(x) (3.1) Let ¢(x,u(x))= 2”,, b,- x. Since ¢(x, y) = o(\/x2 + ya), we have that bo = 0 and bl = 0. Therefore, equation (5) becomes 00 2: b,- ~'x —— —Z( a,-( cos (21ra)x + sin(27roz)u(x))’ + sin(27ra)x — cos(27ra) Z a,x‘ i=2 i=1 i=1 26 by comparing the coefficients of first order term in both sides of above equation, we have alx cos 27ch: + afx sin 27m + x sin 27m — xal cos 27m = 0 that is (a2 + 1) sin 27m = 0 since (1 ¢ Z, sin 27m 75 0, therefore, we get a contradiction, This means that I‘ does not exist. 76 0, we use the similar argument as above to prove this Q For the case when 61: (0’0) pr0position. [3 Corollary 3.2.3 F3ne( f) = (D, i.e. points in F3 can not appear in the advanced cycles. [:1 3.3 When p E F; In this section, we only consider the case when p E F}, For the case when p 6 F2“, the argument is similar. Proposition 3.3.1 Suppose f be an analytic diffeomorphism with O = (0 0) as its fixed point in an open neighborhood N(O) of 0 and F = {(x y)|f(x y) = 0} be a 0' curve entering 0. If the first jet J2)( f) of f at 0 is E, the unit 2-dimensional matrix and I‘ is invariant under f, then there exits a C" diffeomorphism h and a C" diffeomorphism ~ g = ($.31) —> (x + @(x, 21), 310+ we, y))) (3-2) (0.0) — such that f o h = h o g and h(O) = 0, where C(04)) 2 133(00) = 0 and 33.5 a~ : By (0.0) 27 Proof: By the definition of invariant curve entering a fixed point, we know that (1)030 (”3mm 3 ( 0) 75 0. By Implicit Function Theorem, f (x, y) = 0 has an unique in equation f (x y)- — 0 eitherQ-i 6:” 75 0. Without loss generality, suppose tha solution y = h(x) and u(0) = 0, h(x) E 0’. So I‘ = {(x,u(x)) | x e I}, I is a appropriate interval. Since the first jet J5( f ) of f is E, f has the following form: (MI) —> (e + We, 31), y + We. 31)) where (x,y — u(x)), then h‘1 is (x,y) ———> (x,y + u(x)) and h is a Cr diffeomorphism. So hefeh’1(x,y) = hef($,y+#($)) = h(x + $003.31 + M33», 31 + #(1') + We. 31 + M33)» = (x+ (x + 6506,31), y(1+ {5(23, y)) and em, 0) = 112(0, 0) = 0. 29 - _ 6F _ 60 0, a o, _ Slnce, 00(0) _ Eek:0 _ 1+ #[FO — m0) 1gp yzo _ 1, thus 123(0, 0) = 00(0) — 1 = 0 and also, we have 6” 8 6 :93 =5:- +.—: 00:1 ‘7‘ (0.0) 1’ (0.0) y (0.0) 695 690 _ = _ = 0 By (0.0) By (0.0) The proposition is proved. [:1 Definition Let f be a C’ (r > m) function defined on a neighborhood N = (xo — A xo +A) of xo. If f(‘)(xo) = 0 for i = 0, - - -,m— 1 and f(m)(xo) aé 0, then xo is called a zero point of f with multiplicity m. If m = 1, we call xo a simple zero point of f. Corollary 3.3.2 Let f be a C" (r > m) function on N = (xo — A, xo + A), xo be its zero point with multiplicity m, then f(x) = (x — xo)"‘(x) (3.4) where (x) is a continues function in N and (xo) # 0. Cl Lemma 3.3.3 Let f be a C" (r > 771) function on N = (xo — A, xo + A), xo be its zero point with multiplicity m. Let f be a 0' function such that d(f,f) <8 where d( , ) is the 0’" topology in Cm(N, R), then f has at most m zero points in a neighborhood of xo. Proof: 30 Since f(m)(xo) 74 0, without lose generality, we suppose that f(m)(xo) > k > 0. By the continuity of f, there exist a 6 > 0 such that f’"(x) > k > 0 for all x E I = (xo — 6 xo + 6). Because d(f, f) < e, we have fim) (x) > W” — e > k — e > 0 (3.5) for all x E I. Now suppose f(x) has m + j zero points in I (j > 1). By Roll Theorem, we know that f(1)(x) has m + j — 1 zero points in I, f”) (x) has m + j — 2 zero points in I, and so on. After m steps, we get that f(m) (x) has j zero points in I. This is a contradiction. The lemma is proved. CI Lemma 3.3.4 Let f be a C" (r > m) function on N = (—A A) and x = 0 be its zero point with multiplicity m, then for any 6 6 (0 A), A E (0 A), there exits a C" function f such that ~ 1- d(f, f) < 5 2. f has exactly m different zero points in the interval I = [0 6) Proof: By Corollary 3.3.2, f (x) = xm(x), where (x) is a continuous function and (0) 79 0. Without lose generality, we suppose that (0) > 0, then there exist a positive number Am_1 such that 0 < Am_1 < 6 < A and (x) > 0 for all x E (0 Am_1). Choose a number 17",-1 in the interval (0 Am-1), we have f(Tlm—l) = 77::—1¢(77m~1) > 0 31 So we can choose a number fim._1 satisfying 0 < Bm_1 < 17m_1(17m_1) such that for any |am_1| < ,Bm_1 we have (Jim—177,113+ "m—IQOIm—l) : 7771::i (am—1 + 77m-lq)(7lm—1)) > 773::l(—16m—1 + 77m—1(p(77m—1)) > 0 Define a function f1 (x) = am_1xm_1 + x'"(x) = xm‘1(am_1 + x(x)) when x is small enough, the sign of function f1 (x) is determined by the sign of am_1. Choose am_1 < 0, we have that there exists a positive number Am_2 such that 0 < Am_2 < 17m_1 and f1 (x) < 0 for all x 6 (0 Am_2). We select 17m_2 E (0, Am_2), then f1(77m—2) = am—ln$:% + n2_2¢(nm—2) < 0 Choose ,6m_2 such that f1(77m—1) m—2 111-2 0 < [Hm—2 < then, for any number am_2 satisfying Iam_2| < fim_2, we have m- - f - arm—2171..-; + f1(771) = 77173—3 (am—2 + 497732—12 m-2 < 77773:; (lam-2 + Il;(7%31_)_) m-2 < 0 32 Define function f2($) = am—2IBm—2 + am_1xm"l + xm(X) Choose am_2 > 0, when x is small enough, say, |x| < Am__3, we have f2($) = mm'zmm—z + arm—1x + 3322(3) > 0 Choose 17m_3 E (0, Am-3) then f2(17m_3) > 0. Continue this process, we get a function fm_1(x) = alx + (12x2 + - - - + ozm_1x"’—l + x'"(x) where lail = (_1)2i+1,i : 1’._.,m_ 1 lai] < 161') i and a sequence 0<17m 0, fm-1(T]m_1) < 0 If m lS Odd fm_1(17m) < 0, fm_1(nm_1) > 0 if m is even Let Ila?) = fm—1($) = 01517 + 02232 + ° - - + (:1,,,.1:1:"’_1 + xm(x) Claim f(m,_1) > 0, fine.) < 0 for 1' = 0,1,- u, ['32:] Proof of the claim. 33 Let fl = max{5m—1,' "131} ~ “7721—1) = O117721—1 + ° ' ° + 021—1773;: + ' ' ' + am_1n§?:1‘+ ”3-1207290 = a11721_1+--- + 02917735] + f21—1(7721—1) Since f2,_1(n2,-_1) < 0 and [01021—1 +1 - ~+a2,-_117§f:]| < fiAm_1, we can choose 5 small enough such that fiAm._1 < I f2,-_1(n2,-_1)| for all i, then ~ “7721—1) < flAm—l + f2i—1(772i—1) < 0 ~ By using the same argument, we can prove that f (172,-) > 0. The claim is proved. From the above discussion, we know that f (x) has at least one zero point in each interval (17,“, 17,-) i = 1, - - - ,m — 1. Since 0 is a zero point of f, f(x) has at least m distinct zero points in the interval [0, 6). By Lemma 3.3.3, we know that f has at most m zero points. So f has exactly m distinct zero points in the interval [0, 6). Now we finish the proof of this lemma by showing d( f, f) < 5. Since alx + 02x2 + - - - + ozm_1x"“1 + x'"(x) WI A H v II = (111: + (121:2 + - - - + am_lx"‘“ + f(x) then flea) = flee) + 1.1... +---+ (m —1)(m — 2) -~(m — k)0"“"“‘0m-1 34 thus [f(k)($) — f(k)($)l = Imak + . . . + (m _1)(m _ 2) , , , (m _ k)$m-k—lam_l < Kfi where K is a constant and k = 1, ~ - - , m. ~ Let [3 = e/ K , we have that d( f, f) < e. This finishes the proof of lemma. [:1 Lemma 3.3.5 Let f be the function defined in Lemma 5.13, then all the zero points Z ( f ) of f are simple zero points. Proof: Suppose f has a zero point xo 6 Z ( f ) which is not simple. That means that it has a multiplicity greater than 1, say, m > 1. By Lemma 3.3.4, we know that for any 5 > 0 and 6 > 0, there exists a C" function, say, 7, such that in the interval (xo — 6, xo + 6), we have that d(f, f) < e. f has m distinct zero points in the interval (xo — 6/2, xo + 6/2). Since Z ( f ) is finite, we can choose 6 small enough such that 20') no. - i 6 21 $0+—2') :{$0} Define a function ~ (x) xEN\(xo—6,xo+6) g(x) = 7(3) 17 6 ($0 — %, $0 + %) h(x) (LEO—(5, $0+0)\($0— %, Tod-g) where h(x) is a 0' function which makes g(x) to be C" function on N and d(f, g) < e forxe (xo—6, xo+6)\(xo— %, xo+g). We get that d(9,f) S 61(91):) +d(f~, f) < 5 35 but g has m + k — 1 > m zero points in a small neighborhood of 0, it contradicts Lemma 3.3.4. This lemma is proved. D Proposition 3.3.6 Let g be the C" diffeomorphism (0:, y) —> (a: + 9003,31), y(1+ 10(10, y))) where 0, 6 > 0, there exists a C" difi‘eomorphism y with the following properties: 1. d(g,§) < e 2. The fixed point set F ix(§) E N of y is finite, 3. F my) is semi-hyperbolic or hyperbolic, The hyperbolic manifolds of semi- hyperbolic fixed points are contained in x axis. Proof: Consider function F (x) = 1). Let a... = %F(m>(0), then F(x) = amxm + 0(x'") this means that x = 0 is a zero point of F (x) with multiplicity m. By Lemma 3.3.4, we have a function m—l G(x) = 2 tax" + F(x) k=l 36 such that d(F, G) < e and G (x) has m distinct zero points in (0,6). We denote them by0=$o<$1<$2< °'°<.’Em_1 <6. Now, let 3 =2} 0"ka + (:0 + We, 11), 110+ «h(x, y))) then 9’ is a C" diffeomorphism and d(g,g) < 5. Moreover, g has 111 different fixed points in N. They are P0 = (010)1P1 : ($110)1P2 = ($210)1°°'1Pm—l = (mm—110) Note that these are all the fixed points g has in N. We claim that all these fixed points are hyperbolic or semi-hyperbolic. That is to say that the first jets J P (g) of g at each fixed points 13,-, for i- — 0,1, ,m - 1, have 0 the Jordan form , where |A| 7t 1 and o = 0 or 1. 0 fl Proof of the claim: 1+ g; g2 31(9) = (x 0) *1 (x..0) O 1 + wt“) 0) E 2122' = 1 + 812 (Ibo) 6y ($130) 0 1 + l/l($i, 0) By Lemma 5.13, glam“ 76 0. Let A- -— 1+ w .0), fl = 1 + w(x,-,0), thus J,},(§) has A 0 the Jordan normal form , where o = 0 or 1. This finishes the proof of the Oh 37 claim and the of proof the proposition. [:1 3.4 Removing advanced cycles Lemma 3.4.1 Let f has only advanced cycles, then, f can be approximated by an analytic difieomorphism f~ such that the following are satisfied: 1. f has only advanced cycles, ~ 3. For any p, q in 9(f), if S"(p)flS’(q) 75 (0, then, S“(p) and S‘(q) intersect transversely. Proof: Let p, q 6 CU) such that S“(p) flS’(q) at (0. Let x E S"(P) nS‘(q) and N“(p) be a fundamental neighborhood for Su (p) that contains x. In N “(p), we make a small perturbation, as in [10], such that S" (p) intersects S ’(q) transversely at x. We denote by f the new diffeomorphism. Since f has only advanced cycles, this operation does not change the non-wandering set of f. This proves this lemma. [:1 Theorem 3.4.2 Let f be an analytic diffeomorphism and Q( f ) consist of finite fixed points. If f only has advanced cycles, then, f can be approximated in C’ (r > 0) by a diffeomorphism with no cycle and finite non-wandering set. Proof: This theorem can be proved more clearly, if we work with vector fields instead of directly with map by using the techniques developed by Poincare-Bendixson. That proof will appear else where. 38 By Lemma 3.4.1, we can suppose that all the intersections between stable sep- aratrices and unstable separatrices are transversal. So, small perturbations do not destroy these connections. Since f only has advanced cycles, 7,, for i = 1, 2,. . . , m, only contains advanced cycles. Suppose that 7, is trivial for 1 5 i < j. Let A = [qo,q1, . . . , p, . . . ,p,,] be a advanced cycle in 7,, where p 6 Ann, or p E A 092. If p 6 F13. By Proposition 3.1.2, p is a topological saddle-node and the stable separatrix S 3 (p) and unstable separatrix Su (p) which are in cycle A are not in a same sector. Choose a small neighborhood N (p) of p such that N (p) fl 9( f ) = 0. In N (p), we make a small perturbation to split topological saddle-node p into a topological saddle p1 and a topological node p2. This perturbation does not break old separatrix connections and no new separatrix connection is created. So, Q-explosion does not occur. While, cycle A = [qo,q1, . . . ,p, . . . ,pk] becomes A’ = [qo,q1, . . . ,p1,p2, . . . ,p,,] which no longer is a cycle, because p2 is a topological node. We denote by f1 the new diffeomorphism. Then, f1 has less advanced cycles than f and 9(f1) remains finite. If p 6 F2. Let N (p) be the neighborhood of p as above, Su (p) be an unstable separatrix in A. Since A is an advanced cycle, the separatrix of p which is in the same sector as S“(p) must be free, we denote it by S" (p) By Proposition 3.3.1 and Proposition 3.3.6, we can choose a coordinate system in N (p) such that S“(p) is a part of x-axis. We then make a small perturbation in N (p) so that p is split into several hyperbolic or semi-hyperbolic points {p,p1,p2. . . . ,p,}, all these points are in x-axis and the two of the hyperbolic separatrices of each point are part of x-axis. We denote by S3(p) the image of S" (p) under this perturbation and denote by S“'(p)(9£ the image of S “ (p) under this perturbation) the separatrix of p which is also in a same sector of p as Sv"(p). Then, S‘" (p) is also a part of x-axis. Notice that S‘ (p) remains free and no seapratrix connection is broken. No new cycle is created. 39 If {p, p1, p2, . . . , p1} contains a node, then we are done. We find a diffeomorphism f1 which is close to f and has less advanced cycles than f and its non-wandering set remains finite. If {p, p1, p2, . . . , p1} does not contains node. We choose a point y E S“, (p) and let N " (p) be a fundamental neighborhood for S", (p) that contains y. Since S3(p) is free, we can make a perturbation in N “ (p) to break x-axis at y, as in [10]. Therefore, the cycle A is broken. This operation does not make Q-explosion. We denote by f1 the new diffeomorphism. Then, we have that f1 has less one advanced cycle than f and (2(f1) remains finite. This proves this theorem. [:1 As a corollary of the proof of above theorem, we have Corollary 3.4.3 Let f be an real analytic diffeomorphism, 9( f ) be finite and f sat- isfy locally normalized condition, let P be a fixed point of f. Then, there is a neigh- borhood of P, N (p), and a difi’eomorphism 9 such that g is 1: close to f, (2(g) is finite, g only has hyperbolic or semi-hyperbolic fixed points in N (p) and g has no cycles in N (P) CHAPTER 4 N o Cycles In this chapter, we discuss the diffeomorphism that has no cycles. We will approxi- mate it by a Morse—Smale diffeomorphism. Let f be a diffeomorphism on M, its non-wandering set 0( f ) be finite. If for any chain C[p1,p2, - - - , p;] in 0( f), p,- 7é p,- for i 76 j, we call that f satisfies no cycle condition or f has no cycles. Corollary 4.0.4 If f has no advanced cycles and elementary cycles, then f satisfies no cycle condition. Cl Lemma 4.0.5 Suppose that f 6 Diff’ (M), (r > 0), and 9( f ) is finite, then U W"(p) = U W309) = M peflm p601!) Proof: We first prove that Upeflm W“(p) = M. It suffices to prove that M C Unemf) W"(p). Since 9(f) is finite, we can let {2(f) = {0(p1),0(p2),---,0(p,,)}, where 0(p,) denotes the periodic orbit of periodic point p.- and 0(1),) 00(pj) = (b for i ¢ j. Let N,- be a neighborhood of 0(p,) such that Nian = Q) for i 31$ j. N = Uf=1N,-. Let 40 41 x be any point in M — 9( f ), then x is not a periodic point, it follows that { f ""x} must be an infinite sequence. Since M — N is closed, hence it is compact, and (M — N) 09(f) = (b, {f‘mx} can not stay in M - N for all m > 0, thus there exists an integer K > 0 such that f"mx E N for m > K. Note that N,- an = 0 for i aé j, there must be a large integer L 2 K > 0 such that when m > L, f ""x are contained in one particular Nk, it follows that a—limit set a(x) of x is contained in M, 0 9( f ), that is a(x) C N], nfl(f), hence, x E W"(0(pk)). So, M C UpeQU) W“(p). To prove UPGQU)W’(P) = M, we substitute f for f"1 and repeat the above argument. Ci Let 01 = {10 E 9(f) such that W"(P) no} = {p}} 01 consists of all the sinks of f on M 92 = {p E 0(f) — {21 such that W“(p) flu, 79 (i and W“(P) (WW) - 01) = 0} Ole = {p E 9(f) - U 9,- such that W"(p)fl U Q, 75 (b i 0. Since x E W’(0(q)), the w-limit set of x w(x) E 0(q), it follows 42 that f"x —-> g 6 0(q), (n —-) 00). So a E Wu—(p), thus, Wm 0(q) 96 0. Suppose W F] 0(q) 51E 0. Let U be a small neighborhood of 0(q) such that U 09( f) = {0(q)}. Suppose, by the way of contradiction, that W“(p) fl W’(0(q)) = 0, then, there is no x in W“(p) flU such that f"x E U for all n > 0. Since W0 0(q) 75 (b, W"(p) flU must be an infinite set. Consider set D = f(W"(p) 0U) flU—W“(p) nU. Since D090) = (0, W"(p)nD is a finite set. Let W"(p) 0 D = {x1,x2, . . . ,xk} and n,- be the largest number such that f“‘x, E U, then there are at most total 2le n,- points in W“(p)flU, it contradicts the fact that W"(p) (1 U is infinite. this proves the lemma. Cl Lemma 4.0.7 Suppose f 6 Diff" (M), (r > 0). Let 0( f ) be finite and satisfy the no cycle condition. Then, there exists an integer N > 0 such that Q], 76 (l) for k S N and Q], = 0 for k > N. Proof: By the definition of Oh, we know that $21 75 0. Now we prove (to 75 (0. Let S = {p 6 Q(f) such that W“(p)nW‘(fll) 3f (0} = {p1,p2, . . . ,p,}. Suppose, by the way of contradiction, that 92 = (0, then, for each i = 1, 2, . . . , 3, there exists a periodic points q, E 9( f) — 91 such that W"(pt) flW‘(q.-) 75 (0 Consider pl 6 S. W"(p1) nW’(q1) 79 Q), q1 6 Q(f) — {21. If ql E S, by relabeling the points in S, we can let ql be p2. Now we suppose that ql ¢ S. Since ql ¢ 01, W“(q1) # (b. By Lemma 4.0.5, there is a periodic point (12 E 9( f) such that W"(q1) flW’(q2) 3:4 (0. Since q] ¢ S, go 5! $21. If go 6 S, by relabeling the points in S, we can let p2 = go. If (12 e’ 43 S, we continue above process. Note that $2( f ) is finite, we must have a series of periodic points p1,q1,q2, . . .,qn E Q(f) such that qn = p2 and W“(p1)flW’(q1) at (0, W"(q,-) nW‘(q,-+1) 72 (b for i = 1, 2, . . .,n — 1, it means that p1» p2. We substitute p1 by p2 and repeat above argument, then, we have either p2 > p1 or p2 > p3. If p2 > p1 occurs, a cycle appears, this contradicts the fact that f satisfies the no cycle condition. So, we must have p1 >— p2 > p3. Continue this process, because S is finite, we must encounter a cycle, which is a contradiction. This proves 02 7t (ll. By repeating above argument, we finally can find a N > 0 such that SIN consists of all the sources of f on M. This proves the lemma. Cl Corollary 4.0.8 Suppose the f 6 Diff' (M), r > 0, 0( f) be finite and f sat- isfy no cycle condition. Let Q, = {0(p1),0(p2),...,0(pk,)}, where 0(pj) is the periodic orbit of periodic point p,- and 0(pj) fl 0(p1) = 0 for j 7’: I. Then, W:(0(p.-)> nwswm» = 0 forj e z. :1 Corollary 4.0.9 Suppose the f E Diffr (M), r > 0, 0( f) be finite and f satisfy no cycle condition. Then N 9U) = U 9:. k=l where Q,- 76 Q,- fori # j. C] Corollary 4.0.10 Suppose the f 6 Diff’ (M), r > 0, 0( f ) be finite and f satisfy no cycle condition. Then W"(Q,-) nil, 31$ (0 if and only if i 2 j C] Lemma 4.0.11 (Smale) [7] Suppose F be a compact f —invariant set and Q be a compact neighborhood of F such that nm>0 f'"(Q) = F. Then there is a compact neighborhood V of F such that V C int(Q) and f (V) C int(V). Cl 44 Definition A series of compact subset Mk, Mk_1, . . . , M1, Mo of M is called a filtra- tion of M associated with f if M=Mk3Mk_13-~3M13Mo=0 and f(Mi) C int(Mt) We denote it by [Mb Mk-1, . . . , M1, Mo]. Lemma 4.0.12 Suppose f E DiffT (M), (r > 0), (2( f ) be finite and f satisfy no cycle condition. Then, there exists a filtration [MN,MN_1, . . . ,M1,Mo] of M associated with f such that Q, C lflt(Mi—Mi_1) 9.- = r] f’(Mt-Mt_1) -—oo1f2j = Q(f). Then, if x E nngo f"Q1, then, fmx E Q1 for m > 0, it follows that the a—limit set d(x) of x is in Q1, i.e. a(x) 6 Q1090) = 91, thus x E W“(Ql) = 91. So, we have that (21 = “1120 anI. By Lemma 4.0.11, there is a compact neighborhood M1 of 91 such that o, c M1 Cint(Q1),Q1 c nnzo f"M1 c nnzo f"Q1 = 52,, and f(Ml) Cint(M1). 45 Let Q2 be a compact neighborhood of W“(S22). Since W"(Q2) H U 91' = ‘5 j>2 we can let Q2 be such that Q2 0 U,” Q,- = 0. We claim that n f"(Q2UM1) = U W“(Qt) n20 152 In fact, let x E flnzof"(Q2UM1), then, fmx E Q2UM2 for all m > 0, hence, d(x) 6 (Q2UM2) 09(f) = {21 U522, it follows that x E U152 W“(Q,-). So, 0 f"(anMi) = U W“(Q,) C int(Ql UMI) n20 1'52 By Lemma 4.0.11, there is a compact neighborhood M2 of U152 W“(Q,—) such that f(Mo) C int(Mo) and U152 WWI.) C M2 C int(QIUMI). We can suppose that M1 C M2. As a matter of fact, if M1 ¢ M2, we can substitute M2 by M2 U(M1 - M2). Now we check M2 has the required properties. First, we note that since 92 C lIlt(M2) and M1 002 C Q1 0522 = (0, we have 02 C int(Mo — M1). Because 92 is invariant under f, 522 C (Lam-<00 fj(M2 — M1), on the other hand, ifx 6 fl,- fj(M2 — M1), then d(x) C (M2 — M1)flQ(f) = 92 and w(x) C {22, x E W“(Qg) nW’mo) = fig. S0, 92 = fl_°°1 0(pj)) = 0 and Q U M,_1 C M,-. We claim that nnZO f"(QUM-_1) = UjSi-l W"(Qj) UW“(O(p1)). In fact, if x E nngo f"(QUM-_1), then f‘mx E QUM,_1, a(x) C (QUM,_1) 00(f), it follows that z e the.-. We.) uwuiom». So flf"(QUMt—1)= U W"(91)UW"(0(101)) n20 jgt—l By Lemma 4.0.11, there is a compact neighborhood M,, C int(Q U M -_1) such that M,-_1 C M,, C M, and f(M,,) C int(M, ). By using the similar argument as in the proof of Lemma 4.0.12, we have that 0(p1) C int(M,-l — M,_1) and 0(p1) = n-oo 0, there exists a Morse-Smale diffeomorphism g on M such that d(f, g) < 6. Proof: Let (2(f) = {0(q1), 0(q2), . . . , O(q,,)}. By relabeling the periodic points in 9(f), we can suppose that q,- be a hyperbolic periodic point for i < ko and q,- be a degenerate or non-hyperbolic periodic point for i 2 ko. Let [Mn,}lI,,_1,...,Ml,Mo] be the filtration defined in Proposition 4.0.14, then 0(qko) C int(M,co — Mko_1). Let V be neighborhood of 0(qko) such that 0(qko) C V C int(M,co — Mk0—1)- By Corollary 3.4.7, we know that for any 6 > 0, there is a diffeomorphism h defined on V such that h has no non-hyperbolic periodic points in V and d( f Iv , h) < e. 48 We define h(x) x E V gt(x) = h(x) x 6 (Mt, — M,,_1) — V (41) f(x) (I! E M - (Mk0 — Mk0_1) where h(x) is a diffeomorphism which makes g(x) as smooth as required. It follows that d( f, g1) < e on M and {2(g1) contains less degenerate periodic points than 9( f). Continue this process, we can find a diffeomorphism g such that g has no non-hyperbolic periodic points on whole M and d( f, g) < 6. Moreover, 52(5)) remains finite and satisfy the no cycle condition. Now, we prove that y can be approximated by a Morse-Smale diffeomorphism. Let p, q E 0(g) such that W“(p) (‘1 W" (q) 75 (0 and the intersection is not transversal. Let x E W“ (p)r1W’ (q) and N u(p) be a fundamental neighborhood for W"(p) at x. In N (p), we make a small perturbation such that the intersection becomes transversal. Because 5; has no cycles, this operation does not cause Q-explosion. We denote by g] the new diffeomorphism, then, Sl(g"1) is finite and hyperbolic and there are less non-transversal connections between stable manifolds and unstable manifolds than in the case of 9. Continue above process, since C(91) is finite, we finally can find a diffeomorphism g with finite, hyperbolic 9(g) which is close to f and all the connections between its stable manifolds and unstable manifolds are transversal. This proves the theorem. CI CHAPTER 5 Conclusion and Remarks Combine Theorem 2.0.11, Theorem 3.4.2 and Theorem 4.0.15, we have Theorem 5.0.16 Let M be a two dimensional compact manifold, f be an analytic diffeomorphism on M, its no-wandering set 9( f ) be finite and f satisfy locally nor- malized condition. Then, f can be approximated in C" (r > 0) by a Morse-Smale diffeomorphism. C] There is a long way to go to study the structures of systems that have zero en- tropies and to prove or disprove Newhouse conjecture. Following the path presented in this paper, we first have to remove the real analytic and locally normalized condi- tions, and then consider the diffeomorphisms that have finite Birkhoff centers instead of non—wandering sets, and then do further study. Also, we may study this problem from other angles. For instance, we may try to connect the structures of entropy zero systems with the non-exponential correlation decay under an invariant measure. 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