W518 4 I uk- 2_ (£1...) 3 4.1,. 5 Li 73%“ 73%, W Michth , State University This is to certify that the dissertation entitled Horseshoe-Type Diffeomorphisms with a Homoclinic Tangency at the Boundary of Hyperbolicity presented by Ulrich Hoensch has been accepted towards fulfillment of the requirements for Ph . D . degree in Mathematics vim Major professor 2003 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DaleDuepGS-p. 15 HORSESHOE—TYPE DIFFEOMORPHISMS WITH A HOMOCLINIC TANGENCY AT THE BOUNDARY OF HYPERBOLICITY By Ulrich A. Hoensch A DISSERTATION Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2003 ABSTRACT HORSESHOE-TYPE DIFFEOMORPHISMS WITH A HOMOCLINIC TANGENCY AT THE BOUNDARY OF HYPERBOLICITY By Ulrich A. Hoensch In 1979 Devaney and Nitecki showed that for certain parameters in the real Henon family, the set of points with bounded orbits is hyperbolic, and the dynamics are topologically equivalent to those of the full shift on two symbols. It was long known that this set of parameters could be enlarged by considering the geometry given by the invariant manifolds of one or both of the (hyperbolic) fixed points. In this paper we use this approach to extend Devaney and Nitecki’s results, and also to illustrate some methods and assumptions that are used in the process. In Chapter 2, we give results concerning the geometry and position of these invariant manifolds, in particular we investigate the situation before and at the first homoclinic tangency, and establish some sufficient conditions for quadratic contact. In Chapter 3, we illustrate the symbolic dynamics associated with the existence of a topological “horseshoe”; this is the first part on symbolic dynamics. The second part is given in Chapter 5, where we use a hyperbolicity condition to establish topological equivalence of the dynamics to the full shift on two symbols. Chapter 4 introduces an abstract class of maps - a class of maps that satisfy cer- tain geometric and hyperbolicity conditions. Here we give the main definitions and technical conditions needed; the strongest result in this chapter is that of proving hyperbolicity of a return map. Finally, Chapter 6 is devoted to applying the results of the previous chapters to the Henon map. We state our main results in this chapter. ACKNOWLEDGEMENTS I am most indebted to my thesis adviser, Sheldon Newhouse, for suggesting the topic of this thesis and for his guidance on the long and arduous journey towards its completion. I would like to express my gratitude for his patience and persistance in explaining some of the underlying concepts to me, both in the many excellent classes I took with him, and also in the many personal conversations we had. Without his advise - and the concurrent motivational effect - this thesis would not have been possible. Dr. Newhouse was also very helpful in providing me with references to the relevant sources in the literature. I would also like to thank the members of my dissertation committee, Kening Lu, William Sledd, Clifford Weil and Zhengfang Zhou for their time and interest in my academic progress. Special thanks belongs to Clifford Weil for his interest in the seminar talks I gave on subjects relating to this thesis. The knowledge that others were following the development of my academic work helped me greatly in its pursuit. Lastly, I would like to acknowledge the financial support of the Department of Math- ematics at Michigan State University, and in particular the granting of a research assistantship in the spring of 2003 which gave me the much needed time to complete this thesis. iii Contents 1 Introduction 1 2 The Henon Map 5 2.1 Introduction ................................ 5 2.2 Fixed Points and Images of Curves ................... 5 2.3 Invariant Manifolds ............................ 11 2.4 Quadratic Contact at the First Tangency ................ 16 3 Symbolic Dynamics (Part I) 19 3.1 Orientation-Preserving “Horseshoe” Maps before the First Tangency . 19 3.2 A Coding for A = n F “ (Q) before the First Tangency ......... 24 1162 3.3 Orientation-Preserving “Horseshoe” Maps at the First Tangency . . . 25 3.4 A Coding for A = n F “ (Q) at the First Tangency ........... 27 1162 4 The Abstract Model 30 4.1 Basic Definitions and Assumptions ................... 30 4.2 Hyperbolicity Results for [F2y|, |F1y| < [lel .............. 37 5 Symbolic Dynamics (Part II) 46 5.1 Assumptions and Definitions ....................... 46 5.2 Stable and Unstable Curves ....................... 47 5.3 Topological Equivalence ......................... 48 6 Application to the Henon Map 49 6.1 Geometric Conditions ........................... 49 6.2 The Region of (R, a)-Hyperbolicity ................... 50 6.3 Concavity Conditions ........................... 52 6.4 Main Results for Henon Maps ...................... 55 iv 1 Introduction In their 1979 paper, R. Devaney and Z. Nitecki proved that the set A of points with bounded orbits of the Henon map H (3:, y) = (rcc(1 — 1:) — by, 1:), b ¢ 0, is a hyperbolic set, and that H | A is conjugate to the full shift (a, E) on two symbols (cf. sections 3 and 5 for an explanation of these terms), provided the parameters are chosen so that r > (2 + J5)(1 + b) (cf. [DN]). Devaney and Nitecki’s method of proof uses real geometry and extends to include C2-perturbations of the Henon maps considered. These results involve only geometric estimates on the relative position of Q := [0, 1]”, H (Q) and H “(62) - it turns out that A is actually equal to the largest H -invariant set containing Q; i.e., A = fln E Z H"(Q). On the other hand, for a diffeomorphism F, Smale’s homoclinic point theorem gives a result about the existence of a (possibly small) hyperbolic set associated with the occurrence of a transverse homoclinic point q of a hyperbolic saddle point p. The hyperbolic set is the maximal F N -invariant set (for some possibly large N) of a tubular neighbourhood R about part of the unstable manifold of p, and R contains both p and q. Smale’s homoclinic point theorem relies on the geometry of the stable and unstable manifolds of the saddle point p. It is important to note that if the angle of intersection of the stable and unstable manifolds at q is small (q makes the transition from a transverse homoclinic point to a homoclinic tangency), R must be chosen to be very narrow. Relating [DN]’s result to the geometry of the invariant manifolds, we note that the main requirement would be that the homoclinic contact be quadratic, and - more restrictvely - that the distance d between unstable and stable manifolds between the two associated homoclinic intersections has to be rather large. We introduce an abstract class of maps that bridges the gap between having a large hyperbolic invariant set, but requiring that d be large, and allowing d to be small, with the trade-off that the hyperbolic invariant set then shrinks to consist simply of the hyperbolic saddle point and the orbit of the homoclinic point q. The abstract class contains the Henon family H (as, y) = (rx(1 — :r) — by,:r), for O < b << 1, and 02-perturbations. For a map F in this class, we denote by A the set of points with bounded orbits. We obtain hyperbolicity of a “return map” on A, which is a (possibly high) iterate of H, depending on in which region of A the initial point lies. This allows us to establish symbolic dynamics of F | A before and at the first tangency. The main technical assumption is on the relative concavity of the stable and unstable manifolds, related to their distance d. We devote the rest of this section to introduce some of the concepts just mentioned. We limit outselves to diffeomorphisms of R2 - the definitions and results can be naturally extended to general euclidian spaces, and finite dimensional manifolds. Hyperbolic saddle points, invariant manifolds, and homoclinic intersec- tions Let F be a Cr-diffeomorphism (r 2 1) of an open set U C IR2 onto V = F (U) C R). A fixed point is a point p 6 U such that F(p) = p. We say that the fixed point p is hyperbolic if none of the eigenvalues A1, A2 of the differential DFp has modulus 1; if 0 < |/\1| < 1 < |/\2|, then the fixed point is called a hyperbolic saddle point. Given a hyperbolic saddle point p, we consider the sets W“(p) 2: (q: Ip—F-"(q)| —>0 as n --)00} and W’(p) = {qr lp—F"(q)| —>Oas 71-wel- Then W"(p) and W3(p) are injectively immersed Cr-curves containing p (cf. e.g. [HK]). W"(p) is called the unstable manifold of the hyperbolic saddle point p, and W’(p) is called the stable manifold of the hyperbolic saddle point p. A homoclinic point is a point q ¢ p in the intersection of W“(p) and Ws(p). If the angle of intersection is not zero, then q is a transverse homoclinic point; otherwise q is called a homoclinic tangency. Hyperbolic sets and the cone criterion Let A be a compact F -invariant set; i.e., F (A) = A. Then A is called (uniformly) hyperbolic, if there exist A > 1, C > 0, such that for each 19 E A, there is a splitting TPIR2 = E; 69 E; such that: the splitting is DF—invariant: DFP(E,',‘) 2 E2“) and DFp(Es) = E2“), P the splitting depends continuously on p E A, ifv E E3, then IDF:(U)| 2 C - A” - |v| for all n > 0, if v E E5, then IDFP‘”(U)| 2 C - A" - |v| for all n > 0. If p is a hyperbolic fixed point, then A = {p} is a hyperbolic set. We also have invariant manifolds for hyperbolic sets. Assume for instance that A is a hyperbolic set of saddle type; i.e., dim(E;,‘) = dim(E;) = 1 for all p E A. Now we consider the sets W“(p) = {qr IF’”(p) - F"'(q)| -> 0 as n -> 00} and W’(P) = {61: IF"(p) - F"(€1)l-> 0 as n -> 00}- Then W“(p) and W‘(p) are again injectively immersed C'-curves ([HK]). W“(p) is called the unstable manifold of the point p E A, and W’(p) is called the stable manifold of the point p E A. In order to show that a given compact F -invariant set is a hyperbolic set, one can use the following cone criterion. A cone in R2 (or in TPR2) is a set of the form C=C(u,v) =2 {au+,8v:afiz 0}, where u,'v E R2 (or u, v E TPRQ). Cone Criterion Suppose A is a compact, F -invariant set, and suppose there exists a A > 1, and for each p E A there exists an unstable cone C; in Tle2 and a stable cone C; in T pR2 satisfying the conditions: 0 C; n C; = {0}, o the unstable cones are DF-invariant: DFP(C;,‘) C C'FRP)’ _1 . - , —1 the stable cones are DF -1nvar1ant. DFF(p)(Cfp(p)) C C5, the cones depend continuously on p E A, if v 6 C3, then IDFp(v)| 2 A- lvl, 0 if v E CISNP)’ then |DFE(;)(U)| Z A - lvl. Then A is a hyperbolic set. 2 The Henon Map 2.1 Introduction The Henon map we consider is given as Hb,r($7y) = (T$(1_ 1‘) — by?$)? For b 75 0, this is a diffeomorphism of the plane R2, and for b = 0, we have the logistic map Ho,r(x, y) = (7‘17(1 - 3:),x). In [DN], R. Devaney and Z. Nitecki use the following form for the Henon map: hA,B($a y) = (1 + y — A$27 Bx), whereas in [NY], H. Nusse and J. Yorke use the form ”p.423, 31) = (p - $2 + 031.22)- All these maps are conjugate via affine coordinate changes; for A, B 75 0, let 7' (2:1: — 1,2By — B)), and SA,B(:1:, y) 2 (Am, 3y). Then for r ¢ 2 + 2b, b 75 O, and A, B ¢ 0, we have Tr(r——2—2b) _bOHb,r = hr(r—2—-2b) _bOTr(r—2—2b) —b’ and SA,B°hA,B = ”Argos/1,3- 4 4 , 4 ’ 3 We want to investigate the dynamics of the map H (I, y) under iterates. 2.2 Fixed Points and Images of Curves We note that for b ¢ 0, the inverse of the Henon map is Hgflimy) = (y,%y(1- 31%;)- Where convenient, we write H(:r, y) = (H1(:r, y), H2(a:, y)) = (ra:(1 - 2:) — by, x) and thus omit the dependence on the parameters b and r. Also, r(1— 21‘) —b DHW, = : and -1 _ DHHCMI) _ Note that the Jacobian determinant det DHb, = b. Throughout this paper, we assume 0 < b < 1; i.e., that the Henon map is orientation-preserving and dissipative. The following result is easily verified. Proposition 2.1 (I) The Henon map Hb, has exactly two fixed points; namely, p0 = (0,0) and ( b+1 b+1) P1 = 1 — 7,1 " - T (2) DHWIP0 has the two eigenvectors (A1,1) and (A2,1), where 7‘ :l: \/r2 — 4b 2 /\1,2 z are the respective eigenvalues. (3} DH(,,,|,,,1 has the two eigenvectors 011,1) and (112,1), where H: f2-4b 2 ”1,2 = are the respective eigenvalues, and 7‘ 2: 2(b + 1) — r. (4) [fr > 1 + b, then p0 = (0,0) is a hyperbolic saddle point with 0 < A2 < 1 < A1. (5) [fr > 3(1+ b), then p1 = (1— “71,1— 2:1) is a hyperbolic saddle point withu2< —1 (rt(1 —t) +D, t). Let I = [0, 1], and Q = 12. Then the image of Q is a ”horseshoe”, with the left and right boundaries of Q being mapped to the bottom and top horizontal bounding lines of Hb,,.(Q) (with length b), and the bottom and top boundaries of Q being mapped to the left and right bounding parabolas of Hb,,(Q) (whose horizontal distance is b). A picture of Hb,,(Q) with r = 4.5 and b = 0.2 is given below. Picture2.1-0-2 0.2 0:41 0:6 0:8 i The next two results show that there are certain invariant classes of curves. Proposition 2.2 Suppose 7(t) = (rt(1 — t) + g(t),t) is a curve in R2 such that 2rlt— %l 2 1+ b for all t, and such that |g’(t)| S b and lg"(t)| g 12—32). Then Hb,,('y(t)) can be written in the form (rs(1 — s) + h(s),s), where |h’(s)| S b and We)! < 3‘31- - 1 — b. Proof: We have that for s 2: s(t) :2 rt(1 — t) + g(t), Hb,,(7(t)) = Hb,,(s(t),t) = (rs(1 — s) — bt, s), and ds d, = lr(1— 2t) + g’(t)l 2 2r I This means that s(t) has an inverse t(s). Letting h(s) = —b- t(s), we get Hb,,.('y(t)) : dh dt ._1-— h d-—=b~——-_ __ __ >__ =_ d, , (ru 2t) g (o) _ b (2 t) ,(b) _ b 1 b. , 1 d3 1 27‘ I 1 1+ b 1 ___—_—_. _ ’ >_ __ __ >__ =_ dt b (r(2t 1)+g(t))_b(t 2)+b( b)_ b 1 b’ , 1 1f2r(t—-—) _>_1+b. 2 In any case, filil Z %, and this means that s(t) has an inverse t(s). Letting h(s) = _1 dh dt , t(s), we get Hm, (y(t)) = (h(s),s), and d—s = 3 gives 3 S b and the statements about the sign of h’ (s). 3 The condition 0 < b g —1-— guarantees that E S 0; the formula fl = —id:§' (fl) fl dt2 ds2 clt2 ds gives d2h d2: dZs dt 3 1 2b2r 2b2r __=__=__._<_._ ” .3 4- (1 + b); i.e., A is a “topological horseshoe ”. (2) Let Sb =S'flQ. Then H(Q) r1822 2 0 ifand only ifr > (2+ «5) - (1 +b). Proof: The left boundary of H (Q) is given by the parabola I‘:=H({(x,1):OSer1})={(r:r(1—:r)—b,:r:):0§$g1}. F intersects the right boundary of Q precisely when r 2 4(1+ b), and I‘ intersects 86 precisely when r 5 (2 + \/_5-)(1 + b). C] We define: 1 bottom = {($,y)I2T(-2-—y) 2.1—Eb}, , , 1 top: (:r,y).2r y—§ 21+b. r I we also let QbottomJeft = Q n 8 bottom 0 Elefta and QtopJefta Qbottom,righti and Qtop,right along the same lines. Then we have the following lemma. Lemma 2.2 Suppose 0 < b S 1, and r 2 2(1 + 2b), then we have the following: (a) H“1 (£16,; 0 Q) 0 Q consists of two connected components Cleft and (fright. (b) Cleft l8 full-height m gleft (7 Q, and Cleft 2 1'1"1 (QbottomJeft) D Q (C) Cright i3 f’UIl-hCZght Z71 Eright 0 Q; and Cright : H—l (Qtop,left) 0 Q Proof: H “1 maps the left boundary of 6131'th to the parabola y +—> (y, %y(1 — y)), the bottom boundary of £18 ”HQ to a vertical line {0} x [0, -—D], and the top boundary of £18,, 0 Q to a vertical line {1} x [0, —D], for some D > 0. It remains to be checked whether the pre—image H 4(1) of the right boundary l of 518;; 0 Q avoids the region {(x,y) :0 S y S 1, 2r < 1 +b}. Let 22* be such that 2r (% —a:*) = 1+b. 1-x 2 Then I = (rat). 0 s t s 1, and H—‘(ll = (ti t(l—t)—x—;). _<_ 1 + b. Then we need to show that %t(1 — 1 2 r r2 — (1 +12)2 — — — — — > (t 2) , we get bt(1 t) _ 41” , 1 Suppose that t is such that 2r 5 —t * t) - 3;)— 2 1. Using that t(l — t) = and consequently Ali—t 10 1t 2_ 2 __ 2_ __ 2 %t(1—t)—%—ZT (1+b) _r (1+b):r 2r b+1. 4br 2br 4br We need r2 — 2r — b2 +12 4br. Since r 2 2(1+ 2b), we have r — (1+ 2b) 2 1+ 2b, and then [r — (1 + 2b)]2 2 (1+ 2b)2. This means r2—2r(1+2b)+(1+2b)22 (1+2b)2=1+4b+4b225b2+4b, because b S 1. This gives r2 — 2r — b2 + 1 2 4br, as required. [I] 2.3 Invariant Manifolds The two results that follow indicate the position of the stable and unstable manifolds, given certain condtions on b and r. Let W3(p,-) denote the stable manifold of the fixed point p,, and let W“(p,-) denote the unstable manifold of the fixed point p,, i = 0,1. Recall that Q = I 2 = [0,1]2, and let [in and If,2 be the first and second connected component (resp.) of Ws(p0) O Q; let 13,1 and 1.3.2 be the first and second connected component (resp.) of W’(p1) O Q. Proposition 2.4 Suppose 0 < b g 1, and r 2 3(1 + b). Then we can write (I) 1:1 I [011] -_) Q: y H (flq,l(y)iy)) 'LUIIB’I'C.’ (1a) 1111(0) = 0, o _<_ fan), and 2r (é — mm) 2 1+ b, (1b) 0 s (ff,1)’(y) s b. and 1 3 ,, 2b2r Q? then 0 5(f1,1)(y)31_b2- (2) 112 : [0,1] -> Q, 11 *-> (fish/Ly), where: (1c) if0 0, consider the curve 7(t) 2 (g(t), t) = (A2 - t, t), where 7mm 0 S t < 6 and A2 = 2 is the contracting eigenvector of DH”. If r > 1+b, 2b r+\/r2—4b r 2 3(1 + b), then r 2 2(1 + 2b). Thus, Proposition 2.3 and Lemma 2.2 give that the then we have that 0 < A2 = < b _<_ 1. We note that if 0 < b S 1 and first two connected components in Q of H ‘1(7(t)) and of all subsequent pre—images have the properties listed. It is well known in the theory of invariant manifolds (cf. for example [81]) that for some small 6 > 0, H‘”(I‘) —> W3(p0) as n —> 00, where I‘ = {7(t) : —6 < t < 6}. It is easy to check that if q = 7(t) for t < 0, H ’"(q) will not return to Q. D We also have results on parts of the unstable manifold of po = (0,0). First, we establish a set ’P of (b, r)-parameter values for which we have control over the unstable manifold. 12 Lemma 2.3 Let t v—> (rt(1 —t) +g(t), t) be a curve such that g(0) = 0 and |g'(t)| S b. Let t* be such that 2r (% — t") 2 1+ b, and let I" = rt* (1 — t*) + g(t‘). Let 'P = {(b, r) : r (rt*(1 — t*) + bt‘) (1 + bt" — rt*(1—t*)) S (1+ b)t*}. Then for every pair of parameters (b, r) E ’P, we have that (x*,t*)e {(x,y):2r 0, consider the curve 7(t) = (t, A_) , where 0 S t < 6 1 r+\/r2—4b. 2 IS the expanding eigenvector of DHPO. The first image of 7(t) and A1: t is H(7(t)) = (rt(1 — t) — b- -)\—,t). Let g(t) = —b- 3‘2. If r 21+ b, then we have 1 1 b 2b that |g'(t)|: /\—1: r+¢P——4_ Sb< 1. It follows from Proposition 2.2 that for n 2 1, H "(7(t)) has properties (1a)-(1c), at least as long as the y-range is within [0, yg]. If H "(7(t)) has y-range within [0,y3), the y-range will strictly increase under iterates (% > 1 in the proof of Proposition 2.2). Using Lemma 2.3, we may assume that for some n 2 1, H"('y(t)) has 2:- range [0,y’f], and hence H"+1(7(t)) has y-range [0,y‘f]. This proves part (1), since H"(7(t)) -+ W"(po) as n —> 00- Also, Lemma 2.3 and Lemma 2.2 give that H"+2(7(t)) has y-range contained in [y§,y‘f], which proves part (2). Finally, it is again easy to check that if q = 7(t) for t < 0, H"(q) will not return to Q. E] 14 92 Picture 2.4 It follows from Propositions 2.4 and 2.5 that for each there exists a curve b +—> r(b) with (b, r(b)) E ’P such that if (b, r) E ’P and r > r(b), the curves l3 and lig have two transverse intersections, and for r = r(b), l3 and ti 2 are tangent. The following pictures illustrate the previous results for r > r(b) and r = r(b). y /li,1 [l 1 u 1. [‘24 ......... vi 113 [121 ............ Picture 2.5 15 Picture 2.6 2.4 Quadratic Contact at the First Tangency If we restrict the extent of the region P, we can verify that the contact between I}; and [$92 is quadratic. m—l Definition 2.1 Let P be as in Lemma 2.3, except that additionally 0 < b < 6 II Proposition 2.6 Let 7“ : t 1—-> (g"(t),t) be a curve whose concavity K“ :2 (g“(t)) 2br satisfies ]K“ + 2r] S 1 , and let ’73 : t 1—> (gs(t),t) be a curve whose concavity 2b2 1 -—1 1_;2. Let0K” for all t in the common domain of 7“ and 78. b K” :2 (g3(t))” satisfies [K3] S The proof of the above Proposition is elementary. In particular, it gives us the following. Corollary 2.1 If (b, r(b)) E P (P as defined in Definition 2.1), then the tangency at (b,r(b)) is quadratic. We now investigate consequences of having a homoclinic tangency. 16 Suppose 7“(t) = t(l — t) + g"(t),t) and 73(t) = (gs(t),t) are two curves with r |(g“)'| S b and —b S (gs)’ S 0 such that '7“, 73 have a tangency at to; i.e., (*l rtrt(1-to)+ 9"(to) = 9300) and (**) r(l - 2150) + (9")'(to) = (9’)'(to)- Let A(t) = 98(1) — g"(t), then mm 3 2b, and (**) implies 2r [to — i] = |A’(t0)| 3 2b, in particular, since we assume 0 < b S 1, we have 2r 1 to — 5] S 1+ b. This gives the following result. Proposition 2.7 If (b, r(b)) E P, then the tangency between I; and [in occurs in the 1 1 We“, 2.9-9.114]. Now, we want to give estimates for the parameter r(b). Since rt0(1 — to) = region chnter,right = {(33, y) E Q I 21‘ Al‘? 1 2 1 r (to - 5) , (*) and the equation 2r to — § 2 |A'(t0)| give r2 — 4r/_\(t0) = [A’(t0)]2 where r = r(b) is understood to depend on to, the y-coordinate of the tangency. Note that we know that A(1 / 2) = 1. Hence we must solve the initial value problem [r(t)]2 — 4r(t)A(t) = [A’(t)]2 A(1/2) = 1, (Now r depends on t; note that using this notation, r(1 / 2) = 4.) Using the estimates t [A’(t)]2 2 0, A(t) 2 1+] A’ z 1 — 2b 1/2 1 t—§,and t1 (b, 2 + 2\/1 — 2b?) vs. the lower boundary of P. 17 Picture 2.7 This justifies the following proposition. Proposition 2.8 If0 < b < 0.07, then the tangency between 1‘; and 112 is quadratic. 18 3 Symbolic Dynamics (Part I) We have established that for (b,r) E P, and r > r(b), the Henon map exhibits a topological horseshoe, and for r = r(b), there is a first tangency between the stable and unstable manifolds of the fixed point (0, 0). We will consider such maps in their own right. 3.1 Orientation-Preserving “Horseshoe” Maps before the First Tangency Let F be a diffeomorphism of R2, and let p E R2 be a fixed hyperbolic saddle point of F. Suppose the stable and unstable manifold W3(p) and W“(p) of F at p have transverse homoclinic intersections only. Then F exhibits a “topological horseshoe” which can be illustrated as follows (for orientation-preserving F). W“(p) W509) Picture 3.1 19 Note that the dynamics of the points q,~, 3,, r,- and t,- in the picture above are given by F(Qi) = (1m, F(Si) = 31+1, etc. We want to define certain regions bounded by parts of the stable and unstable mani- folds. We let Q be the region enclosed by the part of W" (p) connecting p and qo, the part of W’(p) connecting qo and 31, the part of W"(p) connecting sl and q1, and the part of W"(p) connecting ql and p. We use the notation u s u s QZP—‘MIO—isl —*(11 —>Pc We also define the regions a s u s 1=p-—>q-1—>t_1 —+q1—+p and u s u s 2=SO—>q0——>sl -—>r_1 —>so. Note that _1_ U 2 = Q 0 F ’1 (Q). We define the following regions (also called blocks): For i1,i2,...,ik€ {1,2} and n1,n2,...,nk 6%, let 6221:1122" := {z e Q : F’W) E a1 S J' S k} = fl F442) ISjSk Then we get the following schematical pictures for certain blocks: 20 Picture 3.2 The blocks Q? = l and Q3 2 2. QI t-l T-i 31 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO .................................................... 0000000000000000000000000000000000000000000000000000 .................................................... 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO C II [_1 :9: ll IN) 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO .................................................... OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO Picture 3.3 The blocks o2; = F—1( 1') n 1'. iii = F_1(2_)01 Qgii =F"1(l)02 t_ _ (1132322225 sess]zs:"’l T“ ssss]ss: SI OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 00000000000000000000000000000000 00000000000000000000000000000000 ................................ 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ................................ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Q93 F“(1)ol 3:; = F“(2) o .2. 21 Picture 3.4 The blocks c2331)? = 17-209 m F’1(l')nt'. 31 oooooooooooooooooo OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 21. 2.2. 2.2. 2,1. v. ’ ” ” ’ ’ 02 02 02 02, Q Q. Q Q. = F(Zl- —l 2 = F(l) and Q Picture 3.5 The blocks QII OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO IIIIIIIIIIIIIIIIIIIIIIIIII OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 IIIIIIIIIIIIIIIIIIIIIIIIII OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO .......................... 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OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 00000000000000000000000000 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 00000000000000000000000000 .......................... 00000000000000000000000000 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO IIIIIIIIIIIIIIIIIIIIIIIIII OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000 22 (11 31 sass:as:sssssssssssssssssssisissssssssssssssssssssssis:ssssssasssssssssssssissss 1’2 _ 2231:2231:122133213123312332122212::1211322222222121322333331222111122223323313 2,2 — 82$ ............................................................................. 3.7.0 Q2Q22222213112222:2212:21:22:13:232232122323332}:12:2:22:22:tittitiiitiiiiiitifito 33¢ ............................................................................... Q —1 _ [5255255555255sasssssasssssssssssssssss52525asa525522222255ssssssssssssssissia‘ 1’1 ‘ P 40 ' _39_29_l _ 2 ' 3 ' Picture 3.7 The blocks Qty-J: — F(E) flF (1)0F (g). ql ............................................................................ #31 —3’_2’_1 oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 1,1,2 3122222222222331212231:2:2122:2221:1::322121122232322:33212222222123213313 —3’—2’_1 2,1,2 II:1:223:2222222312232223231221221222222213:23:2:2:I22231322121122132232: _3’—2’_1 2,2,2 '2:1:11:222:22:2223223221221:2:I:1:222:12:2222221122121232122122111222:3331? —3,-2,—I 32¢ ?T0Q1,2,2 —3 —2 —1 q2fi2123221232:123:131231321312I:32223232233:31:3:II:311312:23::11:2:::::2.Lt0Q12,1 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ‘3,-2’-1 83‘ ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 2,2,1 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO —3’-2’_1 oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 2,1,1 221222223:221::12:22:22322123232223:122222323321113222222222213312333? _3’—2’-1 r 4 1,1,1 P 40 23 F(2.) 0F2(l) F(2) o F2(_2.) We note the following. Proposition 3.1 Let i_k, . ..,i_1,i0,i1, . ..,ik 6 {1,2}. Then: (1) Each block R: Q0’-k1 """ . is a full— height sub- rectangle of QOM “““ _ In partic- to Ii-ui k 20 11. .ikl 1 ular R is a full-height sub-rectangle of Q. (2) Each block R = Q-‘kt’TkJ'lm”T1 is a full—width sub- rectangle on “1’1 . In L—kaz-Ic-li-l'uiz-l l- k+lr li‘l particular R is a full-width sub-rectangle of Q. (3) Consequently, each block Qk z: Q:k """ k- is non-empty. (4) Since every block Qk is compact and non—empty, we have that given any sequence (an) E {1,2}z, the set H F'" (a_,,) = n :2“, ...... " an is non-empty. nEZ nEZ 3.2 A Coding for A = n F" (Q) before the First Tangency. nEZ We now consider the set A = O F " (Q). This is a non-empty F -invariant set. For nEZ each :1: E A, we define a sequence a = (an)n€Z, on = 1 or on = 2, via uni: I if F”(:1:) 2 if F”(:r) Let 2 denote the space of bi—infinite sequences of 1’s or 2’s; i.e., E = {1, 2}Z. Then we have defined a map it : A —+ Z, 1,0(22) 2 (an). We define a metric d on 2 as N follows. If a = (on) ,b = (bu) e 23, then if a 7a b, we let d(a,b) = 5 , where N is such that a1, = bn for In] < N, and an ¢ bn for n = N or n = —N. For a = b, we let d(a,b) = 0. It is easy to check that this is a metric on 2. We .N,,N C" ._ have that two sequences are close if they agree on a cylinder set CN := ,_ N,” M .— {(a,,) E E : an = in for — N S n S N} for large N. The map 1/2 is called the coding map or simply the coding of A. The next two results show that this map is continuous and onto. Lemma 3.1 The coding map w : A —> E is continuous. 24 Proof: Suppose N Z 0 and :r E A are given. Let (an) 2 112(2). For each n = —N, . . . N, there exists a 6,, > 0 so that the (Sn-ball 36., (F”(:1:)) around F "(2:) satisfies 91m 3,, (F"(2:)) c 93. Let s(t) :2 n F‘" (3,, (F”(:r))). This is a non-empty InISN open set containing :12. Now, if y E B(:r), (w(2:))n = (1/2(y))n for In] S N. [:1 Lemma 3.2 The coding map w : A —-> Z is onto. Proof: This follows immediately from PrOposition 3.1, part (4). C] We now define the left shift 0' : Z —-+ E, (o(a,,))ic = ak+1. It is easy to see that o is a homeomorphism of Z, and that o o w = 1,1) 0 F. We have therefore established the following. Proposition 3.2 The map it : A —+ Z is a semi-conjugacy between the map F : A —> A and 0‘ : Z —> Z. This means that the diagram A L A W 1N >3 ii 2: commutes. 3.3 Orientation-Preserving “Horseshoe” Maps at the First Tangency We consider the same situation as in 3.1, except now the stable and unstable man- ifold W‘(p) and W“(p) of F at p have a homoclinic tangency. Then F also ex- hibits a “degenerate topological horseshoe” which can be understood as follows (for orientation-preserving F). 25 < T A z 1‘“ RS? ito W”! (12; ( // 33 i4,“ (p) [(1101 ‘ I'V“( ) , l. 0, l p p 3-1 q-l l" l . 1 W509) Picture 3.8 Again, the dynamics of the points q,, s,- and t, in the picture above are given by F(qi‘) = (1:41, F(Si) = Si+1i etc. Similar to what is done in 3.1, we define the regions '11 8 u 8 Q=p—->qo—>si—+qi—>p, u s u s 1=p—+q—i—>t-i—>qi—>p and u s u s Z=so—>qo—>sl—>t_1——+so. We still have that _1_U 2 = Q 0 F‘1(Q). What is different from the situation in 3.1 is that _1_ n 2 = {L1}. We again define blocks exactly as before. For i1,i2,...,i,c G {1,2} and n1,n2,...,n,c E Z, let 26 6233132" := {2: 6 Q : F’W) e 25.1 s j s k} = fl F‘"J'(z'.) _ 195k — These blocks look like the ones in 3.1, only with the points F"(r0) and F "(to) collapsed to F"(to), for each n E Z. Proposition 3.1 holds verbatim in the present situation. Pl'OpOSitiOn 3.3 LEI i_k, . . . ,I_1,’to, I1, . . . , ik E {1, 2}. Then: (1) Each block R: QO’-k1 """ - is a full- height sub- rectangle onO’ ,0 11...”. In partic— 201111: 1 ular R is a full-height sub-rectangle of Q. {2) Each block R = Q--’°’_~’k+1”"--l is a full-width sub- rectangle of Q “1 """ _1 . In 1_k,t_k+1...,1_1 t_ k+1,. .,i_1 particular R is a full-width sub-rectangle of Q. {3) Consequently, each block Qk:— — Q._k‘,‘"’k. is non-empty. (4) Since every block Q), is compact and non-empty, we have that given any sequence (on) E {1, 2}z, the set H F-" (£11) = n Q:,’,',T.’.’,lan is non-empty. nEZ nEZ 3.4 A Coding for A = H F” (Q) at the First Tangency. nEZ Let Q be as in section 3.3. We let A = n F" (Q), a non-empty F -invariant set. Our 7162 first objective is to define a coding for A. Let Z = {1, 2}z. We define the equivalence relation ~ on )3. We let t = (tn) 6 2 be the sequence such that t_3,t_4, . .. = 1, t_2 ‘2 2, t..1 =1,t0 = 2, t1,t2, . . . = 1; 1.8., t: (...,1,1,2,1,2,1,1,...) (the dot 0 denotes the 0th position). We also let r= (...,1,1,2,2,§,1,1,...). 27 Now, we define that o"(t) ~ o"(r) and o”(r) ~ o”(t) for all n E Z, and a ~ a for all a E 2. This is an equivalence relation. We denote by E the set of equivalence classes of ~, and we let 7r : E —+ E, a 1—) a be the canonical projection onto 23. Next, we let 0(t0) = {F”(t0) : n E Z}, and we define the map if} : A -—> I} as follows. 0 If :1: E A \ 0(t0), then define the sequence a = (on) E 2 by a. and then let 1,7203) 0 If :1: = F"(t0) for some n E Z, then let 15(2) 2 o"(t). Proposition 3.3 shows that tb : A —> E is onto. To show the continuity of the map 1]), we make the assumption that there is a continuous transition from the situation before the first tangency to the situation at the first tangency; more precisely, we assume (C) there exists a continuous, open and onto map r : A ——> A such that the diagram commutes. Using the quotient topology on I) (this means that a set G is open in E iff 7r‘1(G) is Open in D), we see that then if) : A —-> D is continuous; namely, if G is open in D, then 7r“1(G) is open in )3, and hence 1,0”107r’1(G) = r‘1 o iii-1(0) is open in A. Applying r to the left side of this equality gives that 1/3‘1(G) is open in A. ~ We define the left shift 6 on the quotient space I: simply by 6(a) = 0(a). It is easy to check that o is well-defined, a homeomorphism of D, and that 6 o if; = ti o F. We have the following version of Proposition 3.2 28 Proposition 3.4 Under the assumption (C), the map if) : A ——> E is a semi-conjugacy between the map F : A ——) A and 6 : E —> E. This means that the diagram 11 I. [i «it it 2L2 commutes. 29 4 The Abstract Model 4.1 Basic Definitions and Assumptions Suppose that F(LL‘, y) = (F1(;r,y),F2(:r,y)) is a C2 diffeomorphism of R2 onto its image. For (2:, y) E R2, the differential map DFUny) : 711,,”le —> Tp(,,,y)R2 is DF _ F1x(xvy) Fly/($33» (1:, l ’ ' y F2$($,y) PEI/($13!) Then the inverse of DFirv) is given by (an. rape-1 =__1. . W...) am) ,y F(Iiy) JF(IE, y) —F2I(CU, y) Fla-(1:, y) 7 where Jp(2:, y) :2 det DEL,” = F1I(27, y) . F2y(:r, y) — F242;, y) - F1y(:1:, y). We use the maximum norm [(vl,v2)| = max{|v1|, [v2]} for (v1,i)2) E R2 or (v1,v2) E TPRZ. Then we have [DP—[1’3”] : max{|F11(:r, 31)] + [FIE/($131“) IF2I($1y)] + [Ell/(xiyllI ° Note that for v E T(I,y)lR2, [DF(,,,y)(v)| S [DF(I,y)|-|v|, and also that for w E TRIMRZ, IDFIF(11;,y)(w)l Z ~|w|. 655] Definition 4.1 Suppose a Z 0 and p E R2. We define (a) the unstable a-cone at p to be K“(a,p) = {(vl,v2) E T,,lR2 : |v2| S alvll}; (b) the stable a-cone at p to be K‘(oz,p) = {(v1,v2) E Tle2 : [v1] S a|v2|}; (c) a K“(a)-curve is a curve 7(t) in such that ‘y(t) E K"(a,'y(t)) for all t; (d) a K’(a)-curve is a curve 7(t) in such that fit) E K’(a,'y(t)) for all t; (e) a K“(a)-line is a K“(a)-curve 7(t) such that curv(7)(t) = 0 for all t; (f) a K‘(a)-line is a Ks(a)-curve 7(t) such that curv('y)(t) = 0 for all t. 30 Definition 4.2 Let I = [0,1] C R, I = (0,1] C R, I = [0, 1) C R, or I = (0,1) C R and let 12 = I x I C R2 (i.e. there are 4 x 4 = 16 choices for IQ). A C2-rectangle Q is the image of I2 under a C2-difleomorphism \I'. We define bottom, top, left and right boundaries of Q by abottomQ = ‘1’“ X{0})1 atOPQ Z ‘1'“ X {1}): aleftQ = 2({0} X 1), (11ng = ‘1’({1} X I)- IfQ is a Cz-rectangle, then we say that R is a C2-subrectangle on ifR is itself a Cz-rectangle, and if R C Q. Moreover, we say that R is a full-height subrectangle of Q if dbottmnR C 01,0“an and ample C BtopQ; R is a full-width subrectangle of Q if aleftR C aleftQ and 6,,ghtR C Brigth. A curve 7 is a full-height curve in Q if 7 C Q and '7 connects 61,0;th and BtOPQ; a curve 7 is a full-width curve in Q if ”y C Q and '7' connects 6,8,,Q and a,,,,,,o. Let Q be a C2-rectangle in R2, and suppose that Q can be written as the union E1 U Q0 U E2, where E1, Q0, E2 are closed, full-height C2-subrectangles of Q with disjoint interiors, and such that a.,-,,,,E, = 61,. ”Q0, Brigtho = 81,, ftEg. In all that follows, 0 < a < 1, R > 1 and K > c > 0 are fixed constants. We assume the following geometric conditions for the map F. (G1) Both F (El) and F (E2) are full-width Cz-subrectangles of E1 U Q0 such that (a) F (abattomEl) : abottom (El U Q0), (b) F (abattomE2) = atop(El U Q0), (C) F(aleftEl) C aleftEla (d) F(arightE2) C azeftEl. (G2) F maps Q0 parabolically across E2. This means that the set F (Q0) 0E2 consists of two connected components that are full-width subrectangles of E2 (this is the situation “before the first tangency”), or F (Q0) 0 E2 consists of two full-width 31 subrectangles of E2 that intersect in one single point, which we denote by pt (this is the situation “at the first tangency”). Furthermore, there exists a full- height curve ’7 in Q0 such that F maps '7 outside of E2 (i.e., F (7) 0 E2 = 0), or - in the situation at the first tangency - we have F(y) 0 E2 = {pt}. We call such a curve 7 a critical curve. We also assume that (a) F (abottomQO) D abottornlS‘2 U atopE‘2a (b) F(aleftQO) C aleftE2a (C) F (arigthO) C aleftE'2 : arightCQO- Definition 4.3 Suppose 0 < 01 < 1 and R > 1. We say a difieomorphism F is (R, a)-hyperbolic on a set E, if for every p E E, we have: (I) if v E K“(a,p), then DFp(v) E K”(a,F(p)) and |DF,,(U)| Z Rlvl; (2) ifv e K3(a,F(p)), then _DF;(IP,(v) e K3(a,p) and [DF;(;,(U)| 2 Rlv]. The following lemma gives necessary conditions for (R, a)-hyperbolicity. Lemma 4.1 If F(Lr, y) = (F1(III, y), F2(:1:, y)) is (R, a)-hyperbolic on E, then we have the following estimates on E: [Fly] _<_ 0, [F227] SQ: [Fix] [le] . 1 u 1 F130?) Proof: Let p E E. Since E K (a, p) and hence BF? 2 E 0 0 F22: (P) I K“(a,F(p)), we get |F2x(p)| S a[F1m(p)|. Also, [F1x(p)| = DFp ( 0 ) 2 R - 1. . 0 s _1 0 1 —F1y(p) Since ( 1 ) E K (a,F(p)) and hence DFF(p) ( 1 ) _ Jp(p) . ( Fun?) ) E K’(a,p),weget|F1y(P)|SalFie(p)|-A180,|F13(p)l= Din-(1, 0 212-1. a lJF(P)l p l |F,,| 2 R, and |F1$| 2 R- |Jpl. 32 We also have sufficient conditions. The proof of the first lemma is elementary; the second lemma comes from [J N]. Lemma 4.2 Suppose 0 < a < 1 and R > 1. Suppose also that for all p E E, the diffeomorphism F satisfies the conditions: (1) ifv E K3(a,p), then |DF,,(v)| 2 Rlvl; and (2) ifv e Kant, F(p)), then [DF;(;,(e)] 2 R [v]. Then F is (R, a)-hyperbolic on E. Lemma 4.3 Suppose 0 < a < 1 and R > 1. Suppose also that for all p E E, the difieomorphism F satisfies the conditions: (I) |F2e(p)| + a|F2y(p)| + a2|F1y(p)l S alFie(p)l, (2} |F1e(p)| - a|F1y(p)| 2 R, (3) |F1y(p)l + a|F2y(p)l + a2lF2e(p)| S alFie(p)|, (4) |F1e(p)| - Cr|1"“2e(.v)| Z Jp(p)R- Then F is (R, a)-hyperbolic on E. We suppose that the following hyperbolicity condition holds. (H1) F is (R, a)-hyperbolic on E1 U E2. We define the sets E20 = E2 E2,1= E2 0 F_1(E1) Egg: 2 E2 0 F_1(E1) fl . . . fl F_k(E1). 33 Then each E” is a full-height C2—subrectangle of E2. Also, each E12,,C :2 E2,k\E2,k+1 is full-height in E2, and we have E2 = U Eu u anyway. k=0 Picture 4.1 We have that for each k 2 0, each of the two connected components of F‘1(E2,k) flQO is full-height in Q0. We denote these components by Eki. The following two pictures illustrate the geometry of these components. 34 Picture 4.2 The region Q0 before the first tangency E2+ E2- I 1 E0_ E1_ l l l l JHLI— E1+ Eo+ 35 L 1+ 130+ We make the following assumption on how certain curves intersect, and their concav- ity. (K1) If p E Eki, let q :2 F2(p) E F(EM). Then for every K"(1/a)-line l through p there exists a K‘(a)-line r; through q such that l’ = F(l) and n’ = F‘1(n) intersect in exactly two points (one of them being F (p) = F ‘1(q)). Furthermore, between these two points of intersection, (a) l’ can be parametrized as a curve (s(t), t), and —2K —6 S :ii(t) g —2K+€ for all t; (b) K.’ can be parametrized as a curve (g(t), t), and —e 3 gm 3 e for all t. The maximal distance of l’ and K.’ between these points of intersection is denoted by (1pm, l). Let dp(l) = maxdp(K.,l) and dp = min dp(l). K We also let 1 ~ szinf{DF:+l( )‘-\/3;:PEE2,I:} 0 fl = inf{|DFp(v)| :1) ¢ 0,2) E K"(1/a,p),p E Q0} I’UI We now assume ( 2) 1,6}?sz |F11.(p)| S > 1 and [1:3ng 06 [3 >1, 1U 36 4.2 Hyperbolicity Results for |F2yl, |F1y| << IleI We will be concerned with the situation when |F2yl, |F1y| are small when compared with IF 11.]. Then we have the following four results. Proposition 4.1 Suppose (H1), a,fl > 0 and 015 < 1. Given 6 > 0, there exists a F F 6 > 0 such that if lF2yl’ :Fly: 1:1: 1:: u E K’(fi,F(p)), then DFE(11))(’U) E K3(e,p). < 5 on E1 U E2, then we have that ifp E E1 U E2 and Proof: Consider v = (111,212) E sz, F(p)), and let v’ = (u’hug) = DF;(1’))(’U). Then 1 3: ' (—F21-'U1 + FIIUQ) . I _ I _ ’Ul — E -(F2yv1 — Fly’UQ) and U2 — Hence It’ll < lF‘Zyllvll + lFlyllv‘Zl < lFZylfl + lFlyl I'U’zl — lFlzll'Uzl - |F2x||v1| _ lFixl - |F2x|l3 lFZIl The (R, a)-hyperbolicity on E1 U E2 implies g a. This means I III wu 0. If lF2yl’ :Flyl| 1:: 1:: then for each p E E1 U E2 and v E K‘(e,p), we have that IDFp(v)| S (6 + 6) - Slip {lFlzla IF2xl} ' lvl EIUEz <60nE1UE2, Proof: For 1) = (121,122) E K‘(e,p) (i.e., lull g clvgl), let u’ = (111,16) = DFp(v). Then I I ’01 = le’Ul + Flyvg ’02 = FQI'UI + ngvg. 37 Hence MI E lleKe + 6)l'v2| lvél S (|F2xl6 + |F1x|5)) |v2|- Finally, note that M = |v2|. C] Proposition 4.3 Suppose (H1), 0,13 > 0 and afi < 1. Given M > 0, there exists a F F 6 > 0 such that if :F2yl’ :Fly: 11' 1.7: v E K3(fi,F(p)), then IDFE(;)(-v)' 2 AJ- |v| < (5 on E1 U E2, then we have that ifp E E1 U E2 and Proof: For v = (v1,v2) E Ks(l3,F(p)) (i.e., |v1| _<_ fllvgl), let v’ = (vi,v§) = DF;(;)(v). We have that 1 WI 2 I’Uél Z ——--(|F1x||v2| - lezllvill, lJFl where F”, ng and Jp are evaluated at p E E1 U E2. So we can estimate llellv'zl — lexllvll Z llellv’zl — lexl ‘ 5' lvzl 2 llel ' (1 ‘ 05) ' l’Uzl and 1 1 llellWl — lF2xllUIl 2 llel ' E ' lvll — lF2xllvll 2 llel ° 3 ‘ (1 — 05) ' l’Ull- In both estimates we used |F23| 3 allel (cf. Lemma 4.1). Now, |Jp| g 6 - IFMCI2 + alelxlz. Hence, > min(1,1/fi) . (1 — as) I I” l - 6-(1 +a) - |le| '7’" Using that IleI is bounded on E1 U E2, we get m11;(1(,11:flc)!)- (lF—izclw) _>_ M if 6 is small enough. El Proposition 4.4 Suppose (H1 ) T here exists a do > 0 such that if 0 < 6 S 60 and F 523+ :Fly: < 6 on E1 U E2, then we have that ifp E E1 U E2 and v E K“(1,p), then 1:: 11: Dew) e mu. m» and IBM)! 2 (1— 6)- |F13(p)|° lvl. 38 Proof: If v = (v1,v2) is such that |v1| 2 I'U2| and v’ = (14,225) 2: DFp(v), then WI 2 lvil Z (lFixl - lFlyl) ' It’ll Z (1 - 6)-|F11|°|vl| = (1- 5) ° |F1_,| ‘ lvl, and '1)ng(lF2Il+lF2yl)'lvllS(a + 6) ’ llel ' lull: Id! 1-6 . . 1—6 Hence -—, _>_ —. Since 0 < oz < 1, we can find a 60 > O Wlth 2 1 for all |v2| a+6 a+6 0<6360. 1:! We want to study the return map on Q0. We make the following definition: Definition 4.4 Suppose p E Eki. Then the return time ofp to Q0 is N(p) z: k + 2, 00 and (I) := FH?‘ is the return map on Eki. This defines the return map (I) : U Eki ———> k=0 Qa For the next result we assume (G1), (G2), (K1) and (H1), (H2). Theorem 4.1 There exists an ii with 1 > 6: > a, an R > 1, and there exists a 6 > 0 IFle lFlyl , < 6 on E1 U E2, then the map is (R, (1)-hyperbolic. llel lFlrrl such that if Proof: For Er, we may choose any number between 1 and a. We want to verify the conditions (1) and (2) in Lemma 4.2 for (I). (1) Let p E Eki, and suppose v E K3(&,p). We want to show that |D,,(v)| 2 Rlvl for some R > 1. Since v E K’(d,p), we have that v E K“(1/&,p) C K"(1/a,p). Let p’ = F(p) E Em and let v’ = DFp(v) = (vi, vé). By the definition of ,8, we have that lv’l 2 fllvl. If S > 1, then we are done; so we assume 0 < fl 3 1. We consider the two cases: 39 ° lvil 2 MI- Using (H2), we can choose A1 > A2 > 1 such that inf |F1x(p’)|-,8 2 A1. P’EEiUEz Since v1 2: v’ E K “(1, p’), Proposition 4.4 allows us to assume that vj :2 DFflv) E K“(1,Fj(p)) for 2 S j S k + 2. Furthermore, for 6 > 0 sufficiently small, |vj+1| Z (1 —— 6)-|F1I(Fj(p))|-|v1|for1$ j g k +1. /\ Ifdgl—i,wehave A1 I I A I W Z (1 - <5) - |F1x(p)| - Iv | 2 3f:- - |F12(p)| 13' lvl 2 A2 - lvl, and also 1+1 A2 j 3- /\2 ,- j /\2 ,- l’U l2 — - |F11(F (P))| ' I?) | Z 'lle(F (P))|'fl' Iv I Z — - lvl A1 A15 5 for2gjgk+r A ’“ ~ This means that |Dp(v)| = |vk+2| 2 (732—) '/\2 - |v| Z R- Ivl. ° I'U'1|< lvél- Let Q > 0, and let q = F2(p) = F(p’). Let u E K3(a, q) be such that for the curves l : t I—> p + tv and K : t H q + tu, we have dp(Is:,l) 2 dp. Also, let a = DFq‘1(u). If 6 > O is chosen to be small, (H1) and Proposition 1 ~ 4.1 give that v E Ks(el,p’). We can write v’ = wl ( + was Note 0 v ~ ' t* that |w2| = |v§| = |v’|. Using (K1), we can write Tg—I = y( ) and 1 , x(t*) , , x(t“) v = w2- , where x(t), y(t) are as 1n (K1), and p = = 1 t* y(t*) . , . , . Thus we have w2 -x(t ) = w1+ Ulz - g(t ). t* 40 The functions x(t) and g(t) satisfy the hypothesis of Lemma 4.4 below, so we get that Ix(t*) — g(t*)I 2 C6 - \/dpr. Since |x(t*) - g(t*)| = I—Z—l—I, we 2 have lel 2 C6 - I/dp/ - Ing 2 Ce - I/dpr - Iv’I. We have 1 - IDF’$+1(v)I_>_|w1|-DF",+1 —|ng-DF",+1 3’— . 0 P I'IT’I Using (H2), we choose A1 > A2 > 1 such that In?) Rk - Ce - fl 2 A1. k+l ~ ”I“ (I: I) (if 61,6 are chosen small). Hence we choose 61,6 such that DP":+1 3 I ” (IUI)< Proposition 4.2 asserts that may be chosen arbitrarily small <)\1 — A2 2L3 Now, 1 __ IDF:+1(,UI I > Cc. \/d—p’ IIUII DFIC+1(0) _ IU’I . A123A2 Al — A2 I > . — o 2 [mas—"1f? -lvl Z IA1+A2 2 ].lv|2R.lvl° (2) Let p E Eki, and suppose v E K“(61,F"+2(p)), i..e v E K’(1/61,F"+2(p)). We want to show that |Dg(1p)(v)I Z RIvI. Let w— - DF;,E’:;’(3(IJ) E TFIP)IR2. Note that we have that lwl, 1 41 where 117! := sup {IleI + IFlyI, IF2II + IngI}. Let [W > NI. Proposition 4.3 al- Q0 lows us to assume that Dng1+,(p)(v) Z M - IvI. Proposition 4.1 (with fl = i.) gives that for 6 > 0 small, v1 z: Dngl+2(p)(v) E Ks(a,F"+l(p)) and conse- quently vi 2:: DF;3+2(p)(iI) E K3(a, Fk‘j+2(p)) for 2 _<_ j _<_ k +1. (R, a)-hyperbolicity on E1 U E2 gives IijI 2 R. |va for 1 g j g k. F Combining these results, we get % 1 1 1 M ~ Dd)"1 v >-—..-- w =—~- 21"“ >-—..—-R’°° v1 >—..—-R’“° v >R- v. C] I.(,,,(>I_MIIMII_M II.M H- H The following lemma gives an estimate for the angle between curves with certain glr curvatures. This lemma is used in the proof of Theorem 4.1. Lemma 4.4 Let 2K > e > 0, and let x(t),y(t) be a C2 functions on some interval Ia, b] such that —2K — c S x(t) 3 -—2K + e and —e g y(t) S e for all t. Let to be a t—value with x(to) = y(to), d 2: x(to) — y(to) Z 0, and x(t*) = y(t*) for some t* E Ia, b]. Then 2K—26 \/K+6 Proof: If to = t*, then (1 = 0; so we may assume to 75 t*. We have If(t”) - WWI 2 515(7') — y(T) . (tar _ to)? 0=flrqu2=$Mfl-Mmht , x r — " for some r between to and t*, or equivalently, d = ——(—)—?i(T—) - (t* — t0)2. This \/c_l \/ K + 6. On the other hand, x(t*) — y(t*) = (x(r) —- 37(7)) - (t* — to) for some other T between 2 K — 2 to and a, i.e. my) — g(t*)| (2K — 26) - Is — tOI 2 317?: M3. [3 Next, we want to give sufficient conditions for (K1), conditions (a) and (b) to hold. means It* — toI 2 Concerning (K1) (a) we have the following result: 42 Proposition 4.5 Suppose 0 < a < 1. Let D be a bounded open subset of R2 and let F(x, y) be a Cz-difleomorphism of R2. Suppose that (a) IleI > 0 0n F‘1(D), and (b) IF2xI S alleI 0” F’1(D). Then for any 6 > 0 there is a (5 > 0 such that if (c) IngI < 5IF1xI and IFlyI < dIleI on F’1(D), and (d) IFlny < 6 and lF2ny < 6 on F’1(D), then the pre-image F ”(53) of every K s(a)-line K C D can be parametrized as a curve (31(3),8) with —e s 33(3) 3 5_ Proof: Let q = (x, y) and let Ii C D be a K 5(a)-line through q; we may parametrize K. as x(t) = (x + tu1,y+ t), where Iu1| g a. Let y(t) = (F‘1)2(n(t)) = (F‘1)2 (x + tu1,y + t). Then ' 1 ’ — a It’ l9(tll _>_ m ' [|F1e(n(t))l lF2x( (tllll- Conditions (a) and (b) imply that m := inf{IF13(z)| - aIF23(z)I : z E F_1(D)} > 0. So the function 3 = g(t) is invertible, and we can write F“1(I~:(t)) as (y(s), s), where 31(8) = (1W)l (Mg—1(3)» = (F"1)1($ + ui9‘1(8), y + 94(3)). , d _1 _ 1 Smce a; (g ) (s) — m, we have that . —1 —1 ‘1 “1 1 y(s)=(F mm >>--—+ 0 such that Concerning (K1) (b) we have the following result: Proposition 4.6 Suppose 0 < ,6. Let D be a bounded open subset of R2 and let F (x, y) be a C2-difl’eomorphism of R2. Suppose that (a) IF2xI — flIngI > 0 on D. 44 Then the image F(l) of every K“(/3)-line l C D can be parametrized as a curve (x(s), s), and furthermore, = lex + 2P‘Ixyv2 + Flyyvg _ (le + Flyv‘Z) ' (F2xx + 2F‘2xy’U2 + F2yyvgl (F21 + Fsz2)2 (F2;- + Fsz2)3 x(s) ) where v2 is the slope of the line 1. Proof: Let p = (x, y) and let I C D be a K “(m-line through p; we may parametrize l as l(t) = (x + t,y + tvg), where Iv2I g 5. Let y(t) = F2(l(t)) = F2(x+t, y+tv2). Then y(t) = F2$(l(t)) +F2y(l(t)) -v2, and also lg(t)I Z IF2x(l(t))l _ I6 ' IF2x(l(t))l Condition (3) implies that the function 3 2 g(t) is invertible, and we can write F (l (t)) as (x(s), s), where x(s) = F1(l(g"1(s))) 2 F1 (x + g_1(s),y + wag-1(8)). , d _1 _ 1 Since a; (g ) (s) — m, we have that __1_ y(t) where the partial derivatives of F are evaluated at l (g’1(s)) E D. ”()2 _ Flz‘i‘Fly'Ug + F1y(l(g—1(8))) ' "gm — F23 + F23, - v2, i3(8) = le(l(g-1(8)))' Now, : lex + 2F13yU2 'l’ Flyyvg _ (F11: + Flyv2) ' (F2xx + 215‘2xyv2 'l' 1723,3103) (F22.- + Fzyvz)2 (sz + Fzyvz)3 . Cl 115(8) 45 5 Symbolic Dynamics (Part II) 5.1 Assumptions and Definitions We look at the diffeomorphisms F : R2 ——> R2 as considered in section 3, only now we will assume certain hyperbolicity conditions that assure that the coding maps 1b : (Inez F" (Q) =: A -—> Z (before the first tangency) and 2/3 : flnez F" (Q) =: A —-> f) (at the first tangency) are actually homeomorphisms. We assume the the geometric conditions (G1), (G2) and the hyperbolicity condition (H1), as formulated in section 4. Recall that there we had two “hyperbolic regions” E1 and E2, and a “parabolic region” Q0. We defined the sets E2), = E2 0 F‘1(El) n . . . m F'k(El) and E2). := E2), \EM+1 (k 2 0), which are full-height rectangles in Q, and for each k 2 0, we let Ek_ denote the left component of F —1(E2’k) 0 Q0, and EH denote the right component of F ‘1(E2,k) (1 Q0. Furthermore, we let E00- denote the left component of F"1((9,,ng2) 0 Q0, and E004, denote the right component of F ‘1(6,,~ghtE2) 0 Q0. Then we have that Q0 (1 F‘1(E2) is “stratified” by the full-height (in Q) rectangles Eki; i.e., Q0 0 F_1(E2) = U Ekzt U Eooj: 1:20 We have the return map = F k” : Eki —> Q0. We now assume that this return map is uniformly hyperbolic; i.e., (I) is (R, (1)-hyperbolic, with the same R > 1 and 0 < o < 1 on each “stratum” Eki: (H3) For all k 2 O, the map F“2 : Eki —-> Q0 is (R, a)-hyperbolic. 46 Note that we can write Q1: EIUUEI.-UE..- and Q2 =EquEk.UE..+, kZO kZO and that in combination with (H1), (an appropriate power of) F can be thought as being uniformly hyperbolic on each “stratum” of Q1 U Q2. More precisely, by additionally letting (I) = F on E1 U E2 we have that (I) is (R, a)-hyperbolic on (Q1 U Q2) \ E'00:}:- We refer to the collection 8 := {E1,E2, EH: : k 2 0} as “strata”. For x E Q1 U Q2, let Sz denote the S E S with x E S. 5.2 Stable and Unstable Curves Forx E A: nF" (Q), let nEZ W3(x) = {y E Q1 U Q2 : SFn(x) = SFn(y) for all n 2 0} and W“(x) = {y E Q1 U Q2 : Sly—71cc) = SF—n(y) for all n 2 0}. L913 QU = E1 U (Q0 0 F(E1)) U E2 and QC 2 E1 U (Q0 0 F(El U E2». Then the hyperbolicity assumptions (H1) and (H3) give us that for each x E A, 0 We have F (W3(x)) C W5(F(x)) and W‘(x) is a continuous, full-height curve in Q, containing x, and it is a K 3(a)-curve in QU. If y E W"(x), then I"(x) —- @"(y)I —> 0 as n —-> oo. 0 We have F’1(W"(F(x))) C W“(x) and W“(x) is a continuous, full-width curve in Q, containing x, and it is a K“(a)-curve in QC. If y E W“(x), then I‘”(x) — '"(y)I —+ 0 as n —> 00. We are therefore justified in calling W‘(x) the stable curve of x E A, and W“(x) the unstable curve of x E A. 47 Furthermore, on QUflQC, W’(x)flW“(x) = {x}. By applying F, we get this pr0perty on all of Q. Now if y E A has the same coding as x; i.e., w(x) = rt(y), this last property gives that x = y. In other words, we now have that the coding map i/I : A -> E is one-to-one. This lets us improve upon the results in section 3. 5.3 Topological Equivalence We consider a family of Cl-diffeomorphisms Fa : R2 —> R2, a 2 ago with the following pr0perties: o the family depends continuously on the parameter a, 0 each F0, 0 > do, satisfies the geometric conditions (G1), (G2) “before the first tangency”, 0 Fan satisfies the geometric conditions (G1), (G2) “at the first tangency”, a each Fa, a 2 do, satisfies the hyperbolicity conditions (H1), (H3). Then we have the following resulting concerning the topological dynamics of F: Theorem 5.1 Let A0 = H(Fa)"(Q). (Q is defined in (G1), (G2)). nEZ (1) Ifa > ao, then there exists a homeomorphism we : Aa -—> 2 such that We. 0 Fa) (x) = (o 0 2%) (x) for all x E A0,. (2, a) is the left-shift on two symbols, as described in section 3. we, is the coding map. Also, the set A0 is hyperbolic. (2) There exists a homeomorphism 1b : Aao —> :3 such that (if 0 Foo) (17) = (5 0 1b) (x) for all x E Aao. (fljr) is the factor of the left-shift on two symbols, obtained by identifying the two possible codings for homoclinic tangencies. Refer to section 3 for full details. 48 6 Application to the Henon Map Recall that the Henon map is given by Him-(37,31): H(xiy) : (H1(.’L‘,y),H2($,y)) = (T1170. _ SB) - by,x). We want to show that H (x, y) satisfies the assumptions used for proving Theorem 4.1. 6.1 Geometric Conditions We observe that regarding the conditions (G1) and (G2), we need the following geometry for the invariant manifolds W“(po), W‘(po) and W’(p1): Picture 6.1 y /lf,1 I \ In I) 1 1 8 [1,2 3 s 12,2 12,1 Recall that l'l‘, lil, If,2 are parts of W“(po), W‘(po), and 13,1, L12 are parts of W’(p1). The geometry we need is present when these invariant manifolds are defined; i.e., when both fixed points are hyperbolic saddles. To make things precise, we let 49 E.={(x,y)= f,.(y)5xsf§,e(y),x3ry(1— y)+f1“( 21,.)y€[01]} Q0 = {(3341): i,2(l/) S 33 S fish/Lil? S 7'3/(1‘31)+f1()y E [01]} E2 = {(1830 3 f28,1(3/) S 33 S f1s,2(y)a$ S T.7/(1 " yl'l‘ f1(y )y E [0 ll} (, -3 ,(y) parametrizes lf j,ry(1 — y) + f,“(y ) parametrizes 1“; cf. section 2 .) Then Proposition 6.1 Ifr > 3(1 + b), and r > r(b), then the map F = Hg”. satisfies the conditions (G1) and (G2). In the next sections, we proceed to verify the conditons (H1), (H2) and (K1). 6.2 The Region of (R, (1)-Hyperbolicity We use the sufficient conditions given in Lemma 4.3 to determine a region where H (x y): Hb,(x, y) will be (R, a)- hyperbolic, for some R > 1, and some 0 < a < 1. Note that H1$(x,y) = r(1— 2x) H1y(x,y) = —b H23(x,y) =1 H2y(x,y) = 0. Then the conditions (1)-(4) in Lemma 4.3 become: (1)1+a2~b§a-2r 2 ._-I, (2) 2r 1 x—EI—a-bZR, (3) b+a2ga-2r JI—EI, 50 (4) 2r x—é—I-aZb-R. With the objective of choosing O z 1 and R z 1, we recall the definition of the closed region 1 ——>1b. x2_+} 8 = 8b,, = {(x,y) : 2r The interior of 8 is the complement of the closed vertical strip 1 ——<1b. x2_+} S = 8b,, = {(x,y) : 2r If p = (x, y) E 5, we see that we can choose R and a close to 1 so that the conditions (1)-(4) above hold. This gives the following result: Proposition 6.2 If R is any (possibly disconnected) closed region such that ’R n 85,,- = 0, then there exist R > 1 and O < a < 1 such that Hm, is (R, a)-hyperbolic on R. In conjunction with Lemma 2.1, part (2), this proposition gives an easy proof of [DN]’s results for the orientation-preserving case (b > 0). It is actually not difiicult to obtain the result for |bI instead of b, using the same simple geometric arguments. We have: Corollary 6.1 If r > (2+ E5) (1 +b), then there exists R > 1 and 0 < a < 1 so that H = H5, is (R, a)-hyperbolic on Q n H ‘1(Q). In particular, the set A = n H "(Q) ( which is also the set of points with bounded orbits) is a hyperbolic set and nEZ H I A is tapologically equivalent to the two-shift (Z, 0). Remark 6.1 This result uses the fact that for r > (2 -+- V5) (1 + b), the image H (l) of the line {(x,1) : 0 S x S 1} is to the right of the region {(x, y) : 2r|y — %I S 1 + b, O S x S 1}. It can be improved upon by considering the upper component l’ of H (l) D Q, and then estimating when H (l’ ) is to the left of this region. We omit the calculations and state only that by proceeding in this way, a better lower bound on r, valid for all b > 0, than the one in the previous corollary can be obtained. 51 We also have the following corollary: Corollary 6.2 There exists a be > 0 an R > 1, and a 0 < a < 1 such that if 0 < b S b0, and r > 3(1 + b), then Hg“. is (R, a)-hyperbolic on E1 U E2. 6.3 Concavity Conditions We verify condition (K1). Note that for H(x, y) = (rx(1 — x) — by,x), we have that on E ={(x,y):2rIx—%I 21+b}, IHsz 1 lH2xl _ IHlyl _b_ |Hlxl " 1+ b' 1 IH1$I=2TIIE— §I21+b>0, Furthermore, H21: :1: H2y = 0) Hlyy : H2yy : 0: Hlxy : Hlyy : H2xx : H2xy = 0 on all of R2. Now, we apply Proposition 4.5 with D a small open neighbourhood of F (8 ) fl [0, IV, and Proposition 4.6 with D a small open neighbourhood of 8 fl [0, 1]2 to get statement (b) and (a) (with K = r), respectively, of (K1), provided the lines I’ and It’ intersect as in (K1) for b > 0 small. To see this intersection property, we make the following argument: as b —-) O, the map H(x, y) = Hb,r(x, y) = (rx(1 — x) — by, x) limits to the logistic map H(x, y) = R,(x, y) = (rx(1—x), x). Also, as b —> O, we see from Proposition 1 that the pre—image of any K ’(a)-line K. will become a vertical line, whereas the image of any K u(fi)-line I will be a parabola s I—> (rs(1 - s), 3). So the intersection property holds for R, and since we think of H as a 02-perturbation of R, we have that this property also holds for b > 0 small. Hence, we have so far established that for b > 0 small, and r > 3(1 + b), the Henon map satisfies conditions (G1), (G2), (H1) and (K1) of the Abstract Model. We 52 define the sets E“, E2), and Eki, k = 0,1,2,..., as in the abstract case. Also, 00 (I) : U EMc —> Q0 will be the first-return map as in Definiton 4. 1:20 Concerning (H2), note first that for the Henon map H (x,y), IDHp(v1,v2)I 2 Ivll, hence ifv E K“(1/a,p), 0 < a < 1, we have that IDHp(v1,v2)I Z a - IvI. This means that DF a = “(l—Iii“ : v 75 0,.) e K“<1/e.p).p e on} .>_ a, and, since IHlxI 2 r—2 on E = {(x,y) : Ix—%I 2 1. —%},we have ianE IFiz(P)l'fl> 1 2 P631 To prove the second part of (H2), we need the following lemma to estmate the return time to Q0: Lemma 6.1 Let lf 2 l1, be the left and l; = [in be the right branch of the stable manifold of the fixed point (0, 0). Then for p E E“; we have 1 N+1 dist(p,l§) 2 dist(HN+l(P)Ili)' (r + b) Proof: Let u E T “MIR? Then IDH(a:,y)l = max{1,r|1 — 2x| + b} and we have the estimate IDH(3,y)(v)I S IDH(z,y)I - IvI, and hence IDH(x.y)(v)I S (r + b) - IvI. The last inequality gives the following result: dist(H’+1(P), l?) S dist(H’+1(p), HUD) S (7‘ + b) 'di8t(H’(P), 1?) fori=1,...,N, and dist(H(p), l‘i’) S (r + b) - dist(p, H‘1(l‘i’)) S (r + b) - dist(p, 1;) Hence dist(HN+1(p), l‘f) S (r + b)N+1 -dist(p,l§) C] Now, let us complete the proof of (H2); we assume b > 0 small, r > r(b) z 4, and we can choose 0 < o: < 1 as close to 1 as necessary. 53 suppose p : ($13,) 6 E2,kii'e'1p E E2IH(p) E El: ° ' 1Hk(p) E Elin+l(p) 6 Q0 for some I: Z 0. Let p0 = p = (x0,y0) and p,- = H‘(p) = (x,,y,-) for i = 1,...,k +1. We may make the following estimates: 1 1 $0+dp20.99, 27” TO—gI—b'027.8'(§"dp)a and fori=1,...,k, as,- g (41)" - (1,, 2r 1 1 ,. 1 1 k+1 1 Ic+2 . - = > _ . _ > — Usmg Lemma 4 W1th N k, we get that (1,, _ 4 (4.1) _ (4.1) , or " log(4.1) _ log(dp) Let m(d,,) — Floor I log(4.1) 2], and let m(dp) . p(dp) = log(7.8(0.5 — dpl) + Z [log(7.8(0.5 — (4.1)‘ - dp))I + log(\/d_,,). i=1 Letting C(x) = p(x) +log(2-1.9), we need only show that inf{((x) : O S x S 0.3} > O, to show (H2). The graph of C (x) (for 0 S x S 0.3) is shown below: 5 . 4 . \ lI/I/T’ ofos 0:1 0.15 0:2 of25 0:3 Summarizing the results of this section gives: 54 Proposition 6.3 There exists an O < a < 1, an R > 1, and there exists a be > 0 such that ifO < b < b0 and r 2 r(b), then the return map to Q0 of the Henon map Hg, is (R, (1)-hyperbolic. 6.4 Main Results for Henon Maps For each b > 0, there exists a unique value of the parameter r, denoted by r(b), such that for r > r(b), the invariant curves of p = (0,0) intersect transversely, whereas for r = r(b), they have their first homoclinic tangency. Let A = fl F”(Q) denote the set of (x, y) e R2 with bounded orbits. 1162 Now we state our results: Theorem 6.1 Let H(x, y) = (rx(1 — x) — by,x) be the Henon map. Then there exists a b0 > 0 such that for all 0 < b S b0, we have the following: (1) If r > r(b), then there exists a homeomorphism 1b : A —~> 2 such that the diagram A if» A III 10¢ 23 i) 2 commutes. Futhermore, the set A is hyperbolic. (2) If r = r(b), then there exists a homeomorphism if : A -—> :3 such that the diagram H A ——+ A ti t it t f: a i commutes. Where: 55 o (a, 2): the full shift on two symbols; 0 w : A —> Z: the coding map of points x E A; o ([7, :3): the quotient of (o, 2), obtained by identifying the two ambiguous codings for homoclinic tangencies; o y; : A —> L‘: the coding map of points x E A - sending each x to its equivalence class in E. Theorem 6.2 The results in the previous theorem also hold for CZ-perturbations of the Henon maps considered. 56 References [BS] Bedford, E., Smillie, J .: Real Polynomial Diffeomorphisms with Maximal En- tropy: Tangencies, http://front.math.ucdavis.edu/math.DS/0103038, 2001 [D1] Devaney, R.: An Introduction to Chaotic Dynamics, Addison-Wesley, 1989 [DN] Devaney, R., Nitecki, Z.: Shift Automorphisms in the Henon Mapping, Com- mun. Math. 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