On Hausdorff dimension of oscillatory motions of the Sitnikov and the restricted planar circular three body problem


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1 On Hausdorff dimension of oscillatory motions of te Sitnikov and te restricted planar circular tree body problem Part I: Conservative Newouse Penomena PRELIMINARY VERSION Anton Gorodetski 0.. Introduction. S. Newouse [N] sowed tat near every dissipative surface diffeomorpism wit a omoclinic tangency tere are open sets (nowadays called Newouse domains ) of maps wit persistence omoclinic tangencies. Moreover, in tese open sets tere are residual subsets of maps wit infinitely many attracting periodic orbits. Later C.Robinson [R] noticed tat tis result can be formulated in terms of generic one parameter unfolding of a omoclinic tangency. P. Duarte [Du], [Du2] sowed tat in area preserving case omoclinic tangencies also lead to similar penomena, te role of sinks is played by elliptic points. Here we prove a stronger one parameter version of te result: we can estimate te Hausdorff dimension of te yperbolic sets and omoclinic classes tat appear in te construction. We would like to mention te following related results. S.Newouse [N4] proved tat in Diff (M 2, Leb) tere is a residual subset of maps suc tat every omoclinic class for eac of tose maps as Hausdorff dimension 2. Moreover, Arnaud, Bonatti and Crovisier [BC], [ABC] improved tat result and sowed tat in te space of C symplectic maps te residual subset consists of te transitive maps tat ave only one omoclinic class (te wole manifold). Notice tat tis result can not be extended to iger smootness. T.Downarowicz and S.Newouse [DN] proved also tat tere is a residual subset R of te space of C r diffeomorpisms of a compact two dimensional manifold M suc tat if f R and f as a omoclinic tangency, ten f as compact invariant topologically transitive sets of Hausdorff dimension two. Initially our interest in te conservative Newouse penomena was motivated by te fact tat it appears in te tree body problem. Namely, let us try to understand te structure of te set of oscillatory motions (a planet approaces infinity always returning to a bounded domain) in te Sitnikov problem. It is a special case of te restricted tree body problem were te two primaries wit equal masses are moving in an elliptic From D.Turaev we learned recently tat later P.Duarte also proved a one parameter version of te result (unpublised)
2 2 orbits of te two body problem, and te infinitesimal mass is moving on te straigt line ortogonal to te plane of motion of te primaries wic passes troug te center of mass. Te eccentricity of te orbits of primaries is a parameter. After some cange of coordinates (McGeee transformation) te infinity can be considered as a degenerate saddle wit smoot invariant manifolds tat correspond to parabolic motions (te orbit tends to infinity wit zero limit velocity). Stable and unstable manifolds coincide in te case of circular Sitnikov problem (parameter is equal to zero). Dankowicz and Holmes [DH] sowed tat for nonzero eccentricity invariant manifolds ave a point of transverse intersection. Tis leads to te existence of omoclinic tangencies and appearance of all te penomenon tat can be encountered in te conservative omoclinic bifurcations, as described in te previous section. In particular, existence of yperbolic sets of large Hausdorff dimension implies tat tere is a sequence of open intervals in te space of parameters tat accumulates to zero, and residual subsets of tese intervals suc tat for corresponding parameters te set of oscillatory orbits in te Sitnikov problem as full Hausdorff dimension. Similar statement olds for te planar circular restricted tree body problem. Te existence of transversal omoclinic points in te latter case was establised in [LS], [X]. I would be extremely interesting to find out if Teorem 2 below can be strengtened by replacing te statement about te full Hausdorff dimension by a similar statement about positive Lebesgue measure of a omoclinic class. Te question is related to Kolmogorov s Conjecture wic claims tat te set of oscillatory motions as zero measure. Tis text is intended to become te part of joint paper wit V.Kalosin Main results. Teorem. Consider te family of area preserving Henon maps ( ( ) x y () H a : y) x + a y 2. Tere is a (piecewise continuous) family of sets Λ a suc tat te following properties old.. Te set Λ a is a locally maximal yperbolic set of te map H a ; 2. Te set Λ a contains a saddle fixed point of te map H a ; 3. Te set Λ a as an open and closed (in Λ a ) subset Λ a suc tat te first return map for Λ a is a twocomponent Smale orsesoe, and τ LR ( Λ a ) as a ; 4. dim H Λa 2 as a.
3 3 Remark. Te proof will essentially use: te construction by Duarte [Du], [Du2] wo proved persistency of omoclinic tangencies for conservative maps near te identity, and results regarding te splitting of separatrices for Henon family, [G], [G2], [G3], [GS] (see also [C] and [C2], were some numerical results are described). Remark 2. A similar statement olds also for any generic one parameter unfolding of an extremal periodic point (see [Du] for a formal definition) as soon as te form of te splitting of separatrices can be establised (see [G] for te relevant results in tat direction). Definition. Let P be a yperbolic saddle of a diffeomorpism f. A omoclinic class H(P, f) is a closure of te union of all te transversal omoclinic points of P. It is known tat H(P, f) is a transitive invariant set of f, see [N]. Moreover, consider all basic sets (locally maximal transitive yperbolic sets) tat contain saddle P. A omoclinic class H(P, f) is a smallest closed invariant set tat contains all of tem. Teorem 2. Let f 0 Diff (M 2, Leb) ave an orbit O of quadratic omoclinic tangencies associated to some yperbolic fixed point P 0, and {f µ } be a generic unfolding of f 0 in Diff (M 2, Leb). Ten tere is an open set U R, 0 U, suc tat te following olds: () for every µ U te map f µ as a basic set tat contains te unique fixed point P µ near P 0 and exibits persistent omoclinic tangencies; (2) tere is a dense subset D U suc tat for every µ D tere is some omoclinic tangency of te fixed point P µ ; (3) tere is a residual subset R U suc tat for every µ R (3.) te omoclinic class H(P µ, f µ ) is accumulated by f µ s generic elliptic points, (3.2) dim H H(P µ, f µ ) = 2, (3.3) dim H {x M P µ ω(x) α(x)} = Leftrigt tickness. Here we reproduce te definition of te left and rigt tickness from [Du2] and [Mo]. We will use tese onesided ticknesses instead of te standard definition. See [PT] for te usual definition of te tickness of a Cantor set.
4 4 Name dynamically defined Cantor set any pair (K, ψ) suc tat K R is a Cantor set and ψ : K K is a locally Lipscitz expanding map, topologically conjugated to some subsift of a finite type of a Bernoulli sift σ : {0,,..., p} N {0,,..., p} N. For te sake of simplicity, and because tis is enoug for our purpose, we will restrict ourselves to te case were ψ is conjugated to te full Bernoulli sift σ : {0, } N {0, } N. Also we will assume tat a Markov partition P = {K 0, K } of (K, ψ) is given. In our case tis means tat te following properties are satisfied: () P is a partition of K into disjoint union of two Cantor subsets, K = K 0 K, K 0 K = ; (2) te restriction of ψ to eac K i, ψ Ki : K i K, is a strictly monotonous Lipscitz expanding omeomorpism. For a general definition of Markov partition see [Mo], [PT]. Given a symbolic sequence (a 0,..., a n ) {0, } n, denote K(a 0,..., a n ) = n i=0 ψ i (K ai ), ten te map ψ n : K(a 0,..., a n ) K is a Lipscitz expanding omeomorpism. A bounded component of te complement R\K is called a gap of K. For a dynamically defined Cantor set (K, ψ) te gaps are ordered in te following way. Denote by Â te convex all of a subset A R. Ten te interval K\( K0 K ) is called a gap of order zero. A connected component of K\ (a0,...,a n ) {0,} n K(a0,..., a n ) tat is not a gap of order less tan or equal to n is called a gap of order n. It is straigtforward to ceck tat every gap of K is a gap of some finite order, and also tat, given a gap U = (x, y) of order n, for every 0 k n te open interval bounded by ψ k (x) and ψ k (y) is a gap of order n k. Definition 2. Given a gap U of K wit order n, we denote by L U, respectively R U, te unique interval of te form K(a0,..., a n ), wit (a 0,..., a n ) {0, } n, tat is left, respectively rigt, adjacent to U. Te greatest lower bounds { } LU τ L (K) = inf : U is a gap of K U { } RU τ R (K) = inf : U is a gap of K U are respectively called te left and rigt tickness of K. Similarly, te ratios τ L (P) = L U 0 U 0 and τ R (P) = R U 0 U 0,
5 were U 0 is te unique gap of order zero, are called te left and te rigt tickness of te Markov partition P. Definition 3. Given a Lipscitz expanding map g : J R, defined on some subset J R, we define distortion of g on J in te following way: { } g(y) g(x) z x Dist(g, J) = sup log [0, + ], x,y,z J g(z) g(x) y x were te sup is taken over all x, y, z J suc tat z x and y x; due to injectivity of g tis implies tat g(z) g(x) and g(y) g(x). Reversing te roles of y and z we see tat te distortion is always greater tan or equal to log = 0. If Dist(g, J) = c, ten for all x, y, z J wit z x and y x we ave c y x g(y) g(x) y x e ec z x g(z) g(x) z x. Definition 4. Te distortion of a dynamically defined Cantor set (K, ψ) is defined as Dist ψ (K) = sup Dist(ψ n, K(a 0,..., a n )) taken over all sequences (a 0,..., a n ) {0, } n. Lemma. (see [PT], [Du2]) Let (K, ψ) be a dynamically defined Cantor set wit a Markov partition P and distortion Dist ψ (K) = c. Ten e c τ L (P) τ L (K) e c τ L (P), e c τ R (P) τ R (K) e c τ R (P). Lemma 2. (Leftrigt gap lemma, see [Mo], [Du2]) Let (K s, ψ s ), (K u, ψ u ) be dynamically defined Cantor sets suc tat te intervals supporting K s and K u do intersect, K s (resp. K u ) is not contained inside a gap of K u (resp. K s ). If τ L (K s )τ R (K u ) > and τ R (K s )τ L (K u ) >, ten bot Cantor sets intersect, K s K u. Definition 5. Define F to be te set of all maps f : S 0 S R 2 suc tat: () S 0 and S are compact sets, diffeomorpic to rectangles, wit nonempty interior; (2) f is a map of class C 2, in a neigborood of S 0 S, mapping tis compact set diffeomorpically onto its image f(s 0 ) f(s ); (3) te maximal invariant set Λ(f) = n Z f n (S 0 S ) is a yperbolic basic set conjugated to te Bernoulli sift σ : {0, } Z {0, } Z ; (4) P = {S 0, S } is a Markov partition for f : Λ(f) Λ(f), in particular, f as two fixed points, P 0 S 0 and P S, wose stable and unstable manifolds contain te boundaries of S 0 and S ; (5) bot fixed points P 0 and P ave positive eigenvalues. 5
6 6 Te action of f and f respectively on te stable, and unstable, foliation of Λ, F s = {connected comp. of W s (Λ) (S 0 S )}, F u = {connected comp. of W u (Λ) (f(s 0 ) f(s ))}, can be described in te following way. Define I s = W s loc(p 0 ) S 0 and I u = W u loc(p 0 ) S 0. I s and I u are stable and unstable leaves of Λ respectively transversal to te foliation F u and F s. Ten te Cantor sets K s = Λ I u and K u = Λ I s, can be identified wit te set of stable leaves of F s, respectively unstable leaves of F u. Define te projections π s : Λ K s and π u : Λ K u in te obvious way: π s (P ) is te unique point in Wloc s (P ) Iu, and similarly π u (P ) is te unique point in Wloc u (P ) Is. Te maps ψ s : K s K s and ψ u : K u K u, ψ s = π s f and ψ u = π s f, describe te action of f, respectively f, on stable, respectively unstable leaves of Λ. Te pairs (K s, ψ s ) and (K u, ψ u ) are dynamically defined Cantor sets, topologically conjugated to te Bernoulli sift σ : {0, } N {0, } N, wit Markov partitions P u = {I u S 0, I u S } and P s = {I s f(s 0 ), I s f(s )}. Given a map f F we define te leftrigt tickness of Λ(f) to be τ LR (Λ) = min{τ L (K s )τ R (K u ), τ L (K u )τ R (K s )}. In order to estimate tis tickness we define te leftrigt tickness of P as τ LR (P) = min{τ L (P s )τ R (P u ), τ L (P u )τ R (P s )}. If te distortion on bot dynamically defined Cantor sets (K s, ψ s ), (K u, ψ u ) is small, say less tan or equal to c, ten it follows from Lemma tat 0.4. Duarte Distortion Teorem. e 2c τ LR (P) τ LR (Λ) e 2c τ LR (P). Definition 6. Given positive constants C along wit small small ε and γ, define F(C, ε, γ) to be te class of all maps f : S 0 S R 2, f F, suc tat: () diam (S 0 S ), diam (f(s 0 ) f(s )) ; (2) ( ) a b te derivative of f, Df (x,y) =, were a, b, c and d are C c d functions, satisfies all over S 0 S (a) det Df = ad bc =,
7 7 (b) d < < a C /ε, (c) b, c ε( a ); (3) te C functions on ( f(s 0 ) f(s ) ), ã = a f, b = b f, c = c f and d d = d f, i.e. Df b (x,y) =, satisfy c ã (a) b = d, b, c, ã = c γ( ã ), (b) (c) a ã = b, b, c, c = d γ( ã ),, d γ ã ( ã ), (d) a, d γ ã ( ã ); (4) te variation of log a(x, y) in eac rectangle S i is less or equal to γ( α i ), were α i = max (x,y) Si a(x, y) ; (5) finally, te gap sizes satisfy: dist (S 0, S ) ε γ and dist (f(s 0 ), f(s )) ε γ. Remark 3. For C = 2 tis definition coincides wit Definition 4 in [Du2]. Te nice feature of te maps from F(C, ε, γ) is tat te stable and unstable foliations ave small uniformly bounded distortion, as te following teorem sows. Teorem 3. For a given C > 0 and all small enoug ε > 0 and γ > 0, given f F(C, ε, γ), te basic set Λ(f) gives dynamically defined Cantor sets (K u, ψ u ) and (K s, ψ s ) wit small distortion, bounded by D(C, ε, γ) = 4(C +3)γ +2ε. In particular, e D(C,ε,γ) τ L (P s ) τ L (K s (f)) e D(C,ε,γ) τ L (P s ), e D(C,ε,γ) τ R (P s ) τ R (K s (f)) e D(C,ε,γ) τ R (P s ), e D(C,ε,γ) τ L (P u ) τ L (K u (f)) e D(C,ε,γ) τ L (P u ), e D(C,ε,γ) τ R (P u ) τ R (K u (f)) e D(C,ε,γ) τ R (P u ). Remark 4. Again, for C = 2 tis Teorem coincides wit Teorem 2 in [Du2]. Notice tat conditions (2b) and (2c) of definition 6 imply tat b, c C, and for C = 2 tis gives an unreasonable restriction on te class of maps tat could be considered. We will need to apply Teorem 3 for a map wic belongs to te class F(C, ε, γ) wit larger value of C, see Proposition 20.
8 8 Proof of Teorem 3. Te straigtforward repetition of te proof of Teorem 2 in [Du2] wit te necessary adjustments needed to take te constant C into account proves Teorem 3. Te only place in te proof of Teorem 2 from [Du2] were te condition a 2 is used is te inequality (3) from Lemma 4.2. If we use te inequality ε a C instead, 6γ sould be replaced by 3 ε 2 (C + 2) tere. Due to tis cange, in Lemma 4. one sould take 2(C + 2)γ instead of 8γ as an upper bound of Lipscitz seminorm Lip(σ s ) and Lip(σ u ) of functions σ s and σ u tat describe stable and unstable foliations. Tis leads to similar canges in te statement of Lemma 4.4 and in te estimate of te distortion. Finally we get te above statement. Teorem 3 is proved Te definition of te Hausdorff dimension. Let us recall te definition of te Hausdorff dimension. Let K R be a Cantor set and U = {U i } i I a finite covering of K by open intervals in R. We define te diameter diam(u) of U as te maximum of U i, i I, were U i denotes te lengt of U i. Define H α (U) = i I U i α. Ten te Hausdorff αmeasure of K is ( ) m α (K) = lim ε 0 inf H α(u) U covers K, diam(u)<ε It is not ard to see tat tere is a unique number, te Hausdorff dimension of K, denoted by dim H (K), suc tat for α < dim H (K), m α (K) = and for α > dim H (K), m α (K) = Large tickness implies large Hausdorff dimension. Proposition 3. Consider a Cantor set K, denote τ L = τ L (K) and τ R = τ R (K), and let d be te solution of te equation Ten dim H (K) d. τ d L + τ d R = ( + τ L + τ R ) d. Remark 5. One can consider Proposition 3 as a generalization of te Proposition 5 from Capter 4.2 in [PT], were te relation between te usual tickness and te Hausdorff dimension of a Cantor set was establised. Proof of te Proposition 3. We will need te following elementary Lemma. Lemma 4. For any d (0, ) and τ L, τ R > 0 min { x d + y d x 0, y 0, x + y, x τ L ( x y), y τ R ( x y) } = ( ) d ( ) d τ L τ R = +. + τ L + τ R + τ L + τ R
9 9 Te proof of Lemma 4 is left to te reader. We sow tat H d (U) (diam K) d for every finite open covering U of K, wic clearly implies te proposition. We can assume tat U is a covering wit disjoint intervals. Tis is no restriction because wenever two elements of U ave nonempty intersection we can replace tem by teir union, getting in tis way a new covering V suc tat H d (V) H d (U). Note tat, since U is an open covering of K, it covers all but finite number of gaps of K. Let U, a gap of K, ave minimal order among te gaps of K wic are not covered by U. Let C L and C R be tat bridges of K at te boundary points of U. By construction tere are A L, A R U suc tat C L A L and C R A R. Take te convex all A of A L A R. Ten and Or, equivalently, A L C L τ L U τ L ( A A L A R ) A R C R τ R U τ R ( A A L A R ). A L A and A R A Lemma 4 now implies tat ( ) A L d ( A R + A A ( ) τ L AL AR A A ( ) τ R AL AR. A A ) d ( τ L + τ L + τ R ) d + ( τ R + τ L + τ R ) d =, and A L d + A R d A d. Tis means tat te covering U of K obtained by replacing A L and A R by A in U is suc tat H d (U ) H d (U). Repeating te argument we eventually construct U k, a covering of te convex all of K wit H d (U k ) H d (U). Since we must ave H d (U k ) (diam K) d, tis ends te proof. Proposition 3 can be used to find an explicit estimate of te Hausdorff dimension via onesided ticknesses. In particular, wen one of te onesided ticknesses is very large and anoter one is small, te following Proposition gives an estimate tat is good enoug for our purposes. Proposition 5. Denote by τ L and τ R te left and rigt ticknesses of te Cantor set K R. Ten ( ) ( ) dim H K > max log + τ R +τ L log + τ L +τ R ( ), ( ). log + +τ R τ L log + +τ L τ R
10 0 Proof of Proposition 5. We will use te following Lemma. Lemma 6. Assume tat for some x, y > 0, x + y <, and some d, d 2 (0, ) te following relations old: Ten d 2 > d. y = ( x) d, x d 2 + y d 2 =. Indeed, ( x) d = y = ( x d 2 ) d 2 < ( x) d 2, so due to our coice of x we ave d 2 > d. Let us apply Lemma 6 to x = τ L +τ L +τ R and y = τ R +τ L +τ R. If y = ( x) d, x d 2 +y d 2 = for some d, d 2 (0, ) ten by Proposition 3 we ave ( ) ( ) log( x) log τ L +τ L +τ R log + τ L +τ R dim H K d 2 > d = = ( ) = ( ). log y τ log R +τ L +τ R log + +τ L τ R In a similar way one can sow tat dim H K > log + τ R +τ L. Proposition 5 is proved. log + +τ R τ L Remark 6. Assume tat τ R, τ λ L (λ ) ν. Ten ( ) ( log + τ R +τ L log + lim ( ) = lim λ +0 log + +τ R λ +0 τ L log ) λ +(λ ) ν ) = ( + + λ (λ ) ν + ν. So if ν is small enoug and λ is close to one, ten dim H K is close to. Moreover, if τ R, τ λ L log(λ ) ten ( log + τ R lim λ +0 and terefore dim H K as λ. log +τ L ) ( + +τ R τ L ) =, 0.7. Uniqueness of te area preserving quadratic family. In [H], [F] it was establised tat up to te cange of parameter and coordinates tere exists only one oneparameter area preserving quadratic family wit some conditions on te fixed points (Henon family). In particular, we can consider te family (2) F ε : (x, y) (x + y x 2 + ε, y x 2 + ε) instead of (). In tis form it is a partial case of a so called generalized standard family, and it was considered in [G].
11 0.8. Rescaling and te family of maps close to identity. Let us consider te following family of te affine coordinate canges: ( ) ( ) ( ) ( ) u δ 2 δ 2 0 u Υ δ = + v 0 0 δ 3, v were δ = ε 4. Ten ( ) ( ) u Υ u + δv δ F δ 4 Υ δ = v v + δ(2u u 2 ) ( ) 2u u + δ 2 2, 0 Now we ave a family of maps close to identity. For eac of tese maps te origin is a saddle wit eigenvalues λ = + δ 2 + δ 4 + 2δ 2 = + 2δ + O(δ 2 ) >, λ 2 = λ = + δ 2 δ 4 + 2δ 2 = 2δ + O(δ 2 ) <. Set = log λ. By definition = 2δ + O(δ 2 ), and δ can be given by implicit function of. Define te following (rescaled and reparametrized) family (3) F : (u, v) (u, v) + δ(v, 2u u 2 ) + δ 2 (2u u 2, 0) Birkoff normal form. First of all let us recall tat te real analytic area preserving diffeomorpism of a two dimensional domain in a neigborood of a saddle wit eigenvalues (λ, λ ) by an analytic cange of coordinate can be reduced to te Birkoff normal form ([S], see also [SM]): (4) N(x, y) = ( (xy)x, (xy)y), were (xy) = λ + a xy + a 2 (xy) is analytic. We need a generalization of tis Birkoff normal form for oneparameter families. Teorem 4. ([FS], Proposition 3.; [Du2], Proposition 6.) Tere exists a neigborood U of te origin suc tat for all (0, 0 ) tere exists a coordinate cange C in U wit te following properties:. If N = C F C ten N (u, v) = ( (uv)u, (uv)v), were (uv) = λ() + a ()uv + a 2 ()(uv) is analytic. 2. C 3 norms of te coordinate canges C are uniformly bounded wit respect to te parameter. 3. (s) is a smoot function of s and.
12 2 Remark 7. Te second property is not formulated explicitly in [FS] but it easily follows from Caucy estimates. Indeed, it follows from te proof tere tat te map C is analytic and radius of convergence of te corresponding series is uniformly bounded from below. Also we will need te following property of te parametric Birkoff normal form for te family F. Lemma 7. For some constant C > 0 and small enoug 0 > 0 and s 0 > 0 te following olds. For all [0, 0 ) and s [0, s 0 ). log (s) C, 2. (s) C, 3. (s) C. Remark 8. Tis Lemma is similar to Lemma 6.3 from [Du2], but in our case we ave one, not two parameter family, and terefore tose two statements are essentially different. Proof of Lemma 7. Consider g(s, ) = log (s). We ave g(s, 0) = 0, g(0, ) =, and g is a smoot function of (s, ). Tis implies tat for small enoug s 0 > 0, 0 > 0 and large C > 0 we ave g(s, ) C for all s [0, s 0 ] and [0, 0 ]. From te explicit form of te family F (3) we see tat F Id as 0 in C r norm for every r N. Since C 3 norms of C and C are uniformly bounded, tis implies tat N Id in C 2 norm as 0. In particular, ( (xy) + DN (x, y) = (xy)xy x2 (xy) ) ( ) 0 (xy) (xy)xy 0 2 (xy) (xy)y2 2 (xy) as 0 uniformly in (x, y) U. Tis implies tat (s) 0 as 0 uniformly in s [s, s 0 ] for every s (0, s 0 ). Also 0 (s) = for every [0, 0 ), so 0(s) = 0. Since (s) is a continuous function, tis implies tat (s) 0 as 0 uniformly in s [0, s 0 ]. Since (s) is a smoot function of (s, ), tis implies tat (s) C if C > 0 is large enoug. Similarly one can sow tat (s) C. Lemma 7 is proved Gelfreic normal form. Te family F is closely related to te conservative vector field { ẋ = y, (5) ẏ = 2x x 2.
13 Namely, due to Teorems A and A from [FS] (see also Proposition 5. from [FS]) te separatrix pase curve of te vector field (5) (let us denote it by σ) gives a good approximation of some finite pieces of W s (0, 0) and W u (0, 0). Denote by σ a segment of separatrix σ tat contains some points P u Wloc u (0, 0) U and P s Wloc s (0, 0) U (see Fig.?) and by V a neigborood of σ. Denote by W s (0, 0) te finite piece of W s(0, 0) between te points were W s (0, 0) leaves U for te first time and te first point were W s(0, 0) returns to U again. Define W u (0, 0) in a similar way. Ten W s(0, 0) and W u(0, 0) are always (for all (0, 0)) in V. Te restriction of te map F on te local separatrix W u F (0) is conjugated wit a multiplication ξ λξ, ξ (R, 0). Let us call a parameter t on W u F (0) standard if it is obtained by a substitution of e t instead of ξ into te conjugating function. Suc a parametrization is defined up to a substitution t t + const. Denote Π r0,e 0 = {(t, E) R 2 t < r 0, E < E 0 }. Te following Lemma follows from Teorem 4 in [G3]. Lemma 8. Tere are neigborood V of te segment of σ between points P u and P s and constants r 0 and E 0 suc tat for some 0 > 0 and all (0, 0 ) tere exists a map Ψ : Π r0,e 0 R 2 wit te following properties:. Ψ (Π r0,e 0 ) V ; 2. Ψ is real analytic; 3. Ψ is area preserving; 4. Ψ conjugates te map F wit te sift (t, E) (t +, E); 5. Ψ ( W u ) = {E = 0}, and t gives a standard parametrization of te unstable manifold; 6. C 3 norms of Ψ and Ψ are uniformly bounded wit respect to (0, 0 ). Remark 9. In [G3] uniform estimates of first derivatives only are included to te statement, but te uniform boundedness of te second and tird derivatives easily follows from te Caucy type estimates since te radius of convergence of te power series tat represent Ψ is uniformly bounded from below. We are grateful to V.Gelfreic for tis remark. 0.. Splitting of separatrices. In [G] for te initial family F ε (2) te form of W s is described in some normalized coordinates (see also Teorem 7. and Proposition 7.2 from [G2]). 3
14 4 Denote G R = {(t, E) R 2 t 0, E 9 }. Te formal statement is te following: Teorem 5. ([G], [G2]) Tere exists 0 suc tat for all (0, 0 ) tere exists a map Φ : G R R 2 suc tat. Φ is real analytic; 2. Φ is area preserving; 3. Φ conjugates te map F ε() wit te sift (t, E) (t +, E); 4. Φ ( W u ) = {E = 0}, and t gives a standard parametrization of te unstable manifold; 5. te second projection of te inverse map E = pr 2 Φ as te derivatives of te first order bounded by const and te derivatives of te second order bounded by const 0 ; te first derivatives of te first projection are bounded by const 2 ; (6) (7) 6. Φ ( W s ) is a grap of some realanalytic periodic function Θ(t); 7. Te function Θ(t) as te following form: Θ(t) = 2 Θ e 2π2 / sin 2πt + O(e 2π2 / ); Θ(t) = 4π Θ 2 e 2π2 / cos 2πt + O(e 2π2 / ). Here Θ is a constant tat does not depend on. Remark 0. Θ = 0, see [GS] for te proof. Remark. Also te following asymptotic formula olds for te second derivative of te function Θ(t) ([G]): (8) Θ(t) = 8π 2 Θ 3 e 2π2 / sin 2πt + O( e 2π2 / ). Remark 2. In [G2] complex values of t and E were considered (wic is an essential part of te proof). Since we do not use complexified invariant manifolds, we state only te real part of te claim ere. We would like to obtain formulas similar to (6) in te previous Teorem for te stable manifold for te family F. Te following statement olds:
15 Proposition 9. Stable manifold Ψ ( W s ) can be represented as a grap of a realanalytic periodic function Θ (t) suc tat (9) (0) () Θ (t) = 8 2 Θ 6 e 2π2 / sin 2πt + O( 5 e 2π2 / ); Θ (t) = 6 2π Θ 7 e 2π2 / cos 2πt + O( 6 e 2π2 / ); Θ (t) = 32 2π 2 Θ 8 e 2π2 / sin 2πt + O( 7 e 2π2 / ). Remark 3. To simplify te notation define te function (2) µ() = 6 2π Θ 7 exp ( 2π 2 /). Notice tat te angle between W u and W s at te omoclinic point in te normalized coordinates is equal to µ()( + O()). Formulas (9) can be now rewritten in te following way: Θ (t) = 2πt µ() sin 2π + O(2 µ()), Θ (t) = µ() cos 2πt + O(µ()), Θ (t) = 2π µ() sin 2πt + O(µ()). Te idea of proof of Proposition 9. Te only difference wit respect to te formula (6) is due to te fact tat transformation to te closetoidentity form is not areapreserving, so te E coordinates gains a factor equal to te Jacobian of te transformation if we want a symplectic system of coordinates. So te first step is to scale E te original E in suc a way tat new (T,E) are symplectic coordinates for te closetoidentity map. Ten te proof sould follow te following lines. Assume you ave (9) in some coordinates (T,E) and want to sow tat te same formula olds in any oter coordinates, say (t,e) suc tat ) (t,0) corresponds to te unstable manifold, wic implies (T, 0) (t, e) = (T, 0) 2) (T, E) (t, e) is areapreserving 3) te cange of coordinates commutes wit te translation (T, E) (t +, E) 4) te derivatives of (x, y) (t, e) do not grow too fast as 0 (e.g. uniformly bounded) )3) imply (t, e) = (T + O(E), E + O(E 2 )), 5
16 6 were bot O are periodic in T. Of course, te constants in te O estimates depend from. Now take into account 4) te coordinate cange (T, E) (t, e) comes as a composition of two canges ((T, e) te original (x, y) and back (t, e)). Terefore te constants do not grow faster tan say 0. Now take te image of te grap given by (9). In its neigborood te map (T, E) (t, e) is very close to identity as E = θ exponentially small. Consequently, te form of te grap is te same in te new coordinates (t, e) Construction of te domain for te first return map. Let q u be te closest to P u σ point of intersection of W u and W s. Consider a finite sequence of images of q u under te map F tat belong to te neigborood V, {q u, F(qu ), F2 (q u),...}. Let qs be te point of tis sequence closest to te point P s σ. Define k() N by F k() (q u) = qs. Take te vector v = (, 0) T q uu and consider te vector w = (w, w 2 ) = D(C Ψ H k() C Ψ )v T q s U. Witout loss of generality we can assume tat w > 0 (oterwise just take te omoclinic point between q u and F(qu ) instead of qu ). Scaling, if necessary, we can assume tat in te Birkoff normalizing coordinates we ave q u = (, 0), qs = (0, ). Fix small ν > 0. Recall tat λ = (0) = e. Set [ ] n = log(µ()+ν ) (3). 2 Due to tis coice λ 2n µ() +ν. More precisely, λ 2n [µ() +ν, λ 2 µ() +ν ). Remark 4. Notice tat tis coice of n for ν = is analogous to te formula (7) in 2 [Du2]. Define te following lines: τ + (,0) = {x = λ 0 }, τ (,0) = {x = λ 0 }, τ + (0,) = {y = λ 0 }, τ (0,) = {y = λ 0 }. Denote by S te square formed by Wloc s (0), W loc u (0), N n(τ + n (0,)) and N bottom and left edges of S as te size (τ + (,0) ). Te (4) l = λ n+ 0. ( ) λ 0 Notice tat since DN is close to te linear map 0 λ, te curve N n(τ + (0,) ) (resp., N n (τ + (,0) )) is C close to a orizontal (resp., vertical) line. Denote by R u and R s te rectangles formed by W u (0), τ + (,0), τ 2n (,0) and N (τ + (0,) ), and by W s (0),
17 τ + (0,), τ 2n (0,) and N (τ + (,0) ), respectively. Notice tat Ru = N n (S) {x λ 0 } and R s = N n (S) {y λ 0 }. Denote by R te intersection (see Fig.?): Now consider te rectangles R = H k() C Ψ (Ru ) C Ψ (Rs ). 7 (5) S 0 = S N (S) and S = N n Ψ C H k() (R ) and define te first return map { N (x, y), if (x, y) S T (x, y) = 0 ; N n C Ψ H k() C Ψ N n(x, y), if (x, y) S Renormalization. We are going to prove tat te map T as a yperbolic invariant set in S and to investigate its properties (Hausdorff dimension, lateral ticknesses) wit respect to te parameter. In order to get a family of maps in te square of a fixed size we rescale te map T. Namely, we consider a map ρ : S [0, 2] [0, 2], ρ(x, y) = (l x, l y). Denote ρ(s 0 ) = S 0, ρ(s ) = S, and define (6) T : S 0 S [0, 2] [0, 2] by T = ρ T ρ. One can obtain te following two Lemmas by a straigtforward calculation using Lemma 7 and te mean value teorem. Lemma 0. For (x, y) S 0 we ave (7) D T e S0 (x, y) = were λ = e. ( (l 2 xy) + (l2 xy)l 2 xy (l2 xy)l 2 x 2 (l2 xy)l 2 y 2 (l 2 xy) = ( ) λ 0 0 λ + ) = (l2 xy) (l2 xy)l 2 xy (l 2 xy) ( O(µ() 2+ν ) O(µ() 2+ν ) O(µ() 2+ν ) O(µ() 2+ν ) ), Notice tat if (x, y) R u, (x, y) S, or (x, y) R s ten xy 4λ 2n. If (x, y) S, ten also x 2 4λ 2n and y 2 4λ 2n.
18 8 Lemma. For (x, y) R u we ave ( ) n (8) DN n (xy) + n n (xy) (xy)xy n n (xy) (x, y) = (xy)x2 n (xy)y2 n n+ (xy) (xy) n (xy)xy = n+ (xy) ( ) ( ) λ n 0 O( µ() +ν log(µ() = 0 λ n + +ν ) ) O((µ() +ν ) 3 2 log(µ() +ν ) ) O(. µ() +ν log(µ() +ν ) ) O((µ() +ν ) 3 2 log(µ() +ν ) ) For (x, y) S we ave (9) DN n (x, y) = ( ) ( λ n 0 O( µ() +ν log(µ() = 0 λ n + +ν ) ) O( ) µ() +ν log(µ() +ν ) ). O((µ() +ν ) 3 2 log(µ() +ν ) ) O((µ() +ν ) 3 2 log(µ() +ν ) ) 0.4. Cone condition. Te coordinate canges C Ψ and Ψ C ave uniformly bounded C 3 norms. Assume tat teir C 3 norms are bounded by some constant C 0. Let us introduce te following cone fields in S 0 S : (20) (2) K u (x, y) = { v = (v, v 2 ) T (x,y) Si v > 0.00C 6 0 ν v 2 }, and K s (x, y) = { v = (v, v 2 ) T (x,y) Si v 2 > 0.00C 6 0 ν v }. Lemma 2. (Cone condition for S 0 ) Assume tat is small enoug. For every vector v K u (x, y), (x, y) S 0, we ave D T (x,y) ( v) K u ( T (x, y)), and if D T (x,y) ( v) = w (w, w 2 ) ten w λ 0.9 v. For every vector v K s (x, y), (x, y) T ( S 0 ), we ave D and if D T (x,y) ( v) = w (w, w 2 ) ten w 2 λ 0.9 v 2. Proof of Lemma 2. Tis follows directly from Lemma 0. Lemma 3. (Cone condition for S ) Assume tat is small enoug. T (x,y) ( v) Ks ( T (x, y)), For every vector v K u (x, y), (x, y) S, we ave D T (x,y) ( v) K u ( T (x, y)), and if D T (x,y) ( v) = w (w, w 2 ) ten w 0.0C 4 0 ν v and w 25C 4 0 ν v. For every vector v K s (x, y), (x, y) T ( S ), we ave D T (x,y) ( v) Ks ( T (x, y)), and if D T (x,y) ( v) = w (w, w 2 ) ten w 2 0.0C0 4 ν v 2 and w 25C0 4 ν v.
19 Before to begin te proof of Lemma 3 we will formulate and proof two extra lemmas tat give estimates of te angle between images of vectors under linear maps. Lemma 4. For any two vectors ū, ū 2 and any linear map A : R 2 R 2 te following inequality olds: sin (Aū, Aū 2 ) A A sin (ū, ū 2 ) Proof of Lemma 4. Take two vectors s and s 2 suc tat s 2 ( s s 2 ) and s ū, s 2 ū 2. In tis case sin (ū, ū 2 ) = s s 2 s. Now we ave sin (Aū, Aū 2 ) A s A s 2 A s Lemma 4 is proved. A s s 2 A s = A A sin (ū, ū 2 ) Lemma 5. For any vector ū R 2, ū 0, and any linear maps A, B : R 2 R 2 te following inequality olds: sin (Aū, Bū) A A B. 9 Proof of Lemma 5. Lemma 5 is proved. sin (Aū, Bū) Aū Bū Aū A B A = A A B. Proof of Lemma 3. We will prove te first part of te statement. Te proof of te second part is completely te same. Take a vector v (v, v 2 ) K u (x, y), (x, y) S. Consider te following points: P = ρ (x, y) S, P 2 = L n (P ) R u, P 3 = Ψ C (P 2) Π r0,e 0, P 4 = H k() (P 3 ) Π r0,e 0, P 5 = C Ψ (P 4 ) R s, P 6 = L n (P 5 ) T (S ) S, and denote by (x i, y i ) te coordinates of te point P i, i =,..., 6. We will follow te image of te vector along tis sequence of points and estimate te angle between tat image and coordinate axes and te size of te image. Step. Te vector v () = Dρ ( v) T P U as coordinates (lv, lv 2 ) and v () K u (P ). Notice tat v () = l v. Step 2. By Lemma te vector v (2) = DN n ( v() ) as coordinates (λ n lv +O( µ() +ν log(µ() +ν ) )l v, λ n lv 2 +O((µ() +ν ) 3 2 log(µ() +ν ) )l v ).
20 20 Since v > 0.0C 6 0 v 2, we ave v v v + v 2 ( + 00C 6 0 +ν ) v, and ence 4 lλn v < lλ n v 2( + 00C 6 0 +ν ) < 2 λn l v v (2) 2λ n v () = 2λ n l v. Te angle between v (2) and te vector ē (, 0) (we will assume tat tis angle is acute, oterwise consider te vector (, 0) instead) is not greater tan v(2) 2 v (2) 4λ 2n v 2 < v 400C6 0 +ν λ 2 µ() +ν = (400C0λ 6 2 )µ() 2+2ν. Step 3. Set v (3) = D P2 (Ψ C ) v(2). Ten C0 v (2) v (3) C 0 v (2). Let us estimate te angle between v (3) and te vector ē = (, 0). Let P be a projection of te point P 2 to te line {y = 0}. Ten dist(p 2, P ) lλ n = λ 2n+ 0. Since te image of te line {y = 0} under te map Ψ vector ē = (, 0) under te differential D(Ψ we ave ( D P2 (Ψ C C is a line {E = 0}, te image of te ) as te form (s, 0) = sē. Now ( v (3), ē ) = ( v (3), sē ) = ( D P2 (Ψ C ) v(2), D P (Ψ C )ē ) C ) v(2), D P2 (Ψ C )ē ) ( + DP2 (Ψ )ē, D P (Ψ C C )ē Now let us estimate eac of te summands. Since all te angles tat we consider are small, we can always assume tat α < 2 sin α < 2α for all angles α tat we consider. Due to Lemma 4 we ave (22) ( D P2 (Ψ C ) v(2), D P2 (Ψ C )ē ) 2 sin ( D P2 (Ψ C ) v(2), D P2 (Ψ C )ē ) 2C 2 0 sin ( v (2), ē ) 2C 2 0 (400C 6 0λ 2 )µ() 2+2ν = 800C 8 0λ 2 µ() 2+2ν Due to Lemma 5 we ave (23) ( D P2 (Ψ C )ē, D P (Ψ C )ē ) 2 sin ( D P2 (Ψ C )ē, D P (Ψ C )ē ) 2C 0 C 0 dist(p 2, P ) 4C 2 0λ 2n+ 0 4C 2 0 λ 2. µ() +ν Finally (if is small enoug and λ = e is close to ) we ave (24) ( v (3), ē ) 800C 8 0λ 2 µ() 2+2ν + 4C 2 0λ 2. µ() +ν 5C 2 0µ() +ν Step 4. Since H(t, E) = (t+, E), te estimates for v (3) work for v (4) = DH k() ( v (3) ) also. Step 5. Consider v (5) = D P4 (C Ψ ) v (4) T P5 U. Notice tat ) (25) v (5) C 0 v (4) C 2 0 v (2) > 4 C 2 0 lλ n v and
21 2 (26) v (5) C 0 v (4) C 2 0 v (2) 2C 2 0lλ n v. Now let us estimate te angle between v (5) and te ax Oy. Let P # be a projection of te point P 5 on te line {x = 0}. Take te vector ē 2 = (0, ) T P #U and consider te image D P #(Ψ C )ē 2 T Ψ C (P # ) Π r 0,E 0. Te vector D P #(Ψ C )ē 2 is tangent to te grap of te function Θ(t), and due to (9) (27) From (24) we ave 2 µ() < (D P #(Ψ 5 µ() < (D P #(Ψ C )ē 2, ē ) < 2µ(). C )ē 2, v (4) ) < 5µ(). Notice tat dist(ψ C (P # ), P 4 ) 2C 0 λ 2n+ 0 4C 0 µ() +ν. Tis implies (in te way similar to Step 3) tat for small enoug (28) ( v (5), ē 2 ) < 5C 2 0µ() + C 0 C 0 4C 0 µ() +ν < 6C 2 0µ(), (29) ( v (5), ē 2 ) > 5 µ()c 2 0 4C 3 0µ() +ν > 6 C 2 0 µ(). Step 6. Set v (6) = DN n v(5) T P6 U. Let us estimate v (6) first. If v (6) = (v (6), v (6) 2 ) ten by Lemma v (6) = ( λ n + O ( µ() +ν log(µ() +ν ) )) ( ) v (5) +O (µ() +ν ) 3 2 log(µ() +ν ) v (5) 2 and ( ) ( ( )) v (6) 2 = O µ() +ν log(µ() +ν ) v (5) + λ n + O (µ() +ν ) 3 2 log(µ() +ν ) v (5) 2. From (29) we ave (30) Terefore v (5) 2 C 2 0 µ() v (5) > 48 C 4 0 lλ n v µ() and v (5) 2 v (5) v (6) 2 λn v (5) 96 C 4 0 lλ 2n µ() v, v (6) 2 2λ n v (5) 4C 2 0l v. Now let us sow tat v (6) K u (P 6 ). Indeed, if is small enoug (and λ = e is close to ) ten v(6) v (6) C 6 0 λ 2 ν > 0.00C0 6 ν. Terefore v (6) K u (P 6 ), and Dρ( v (6) ) = D T ( v) K u (ρ(p 6 )). Also if w = (w, w 2 ) = D T ( v) ten w = l v (6) > 96 λ 2 C 4 0 ν v > 0.0C 4 0 ν v
22 22 At te same time we ave (3) D T ( v) = l v (6) l ( v (6) + v (6) 2 ) l (2λ n v (5) + 4C 2 0l v ) l (2λ n v (5) sin ( v (5), ē 2 ) + 4C 2 0l v ) l (2λ n 2C 2 0lλ n v 6C 2 0µ() + 4C 2 0l v ) 24C 4 0λ 2n µ() v + 4C 2 0 v 25C 4 0 ν v. Lemma 3 is proved Markov partition and its tickness. Standard arguments of te yperbolic teory (see, for example, [IL]) sow tat te Cone condition (Lemmas 2 and 3) togeter wit te geometry of te map T imply te existence of te yperbolic fixed point Q of te map T in S T ( S ). Our coice of te omoclinic points q u and qs implies tat te eigenvalues of Q are posilive. Denote te eteroclinic point were Wloc s (Q) intersects W u (O) = {(0, x) x R} by (x s, 0), and te eteroclinic point were Wloc u (Q) intersects W s (O) = {(y, 0) y R} by (0, y u ). Denote te segments of stable and unstable manifolds tat connect te fixed points O and Q wit tese eteroclinic points by γ u (O) connects O and (x s, 0), γ s (O) connects O and (0, y u ); γ u (Q) connects Q and (0, y u ), γ s (Q) connects Q and (x s, 0). Notice tat γ s (Q) S and γ u (Q) T ( S ). Let S be te square formed by γ u (O), γ s (O), γ u (Q) and γ s (Q), S S. Now define S 0 = ρ(ρ (S) N ρ (S)) S 0 and S = ρ(s ρ (S)) S, see Fig?. Notice tat one of te vertical edges of S is γ s (Q) and anoter is an intersection of S and a vertical edge of S, and terefore it intersects W u (O) at te point ρ(n n (, 0)) = ρ(λ n ) = λ n 0 λ n = λ 0. Similarly, T (S ) as a vertical edge [λ 0, y u ] Oy. Define now T = T S. Te maximal invariant set of T in S, Λ = n Z T n (S), is a orsesoe type basic set wit Markov partition P = {S 0, S }. Te map T : S 0 S S belongs to class F (see definition 5). Consider now te Markov partitions P s = {[0, λ x s ], [λ 0, xs ]} and P u = {[0, λ y u ], [λ 0, yu ]}
23 of te Cantor sets K s Ox and K u Oy associated wit te yperbolic set Λ. We ave τ L (P s λ x s ) = 0 λ x, τ R (P s ) = xs λ 0, s 0 λ x s τ L (P u ) = λ λ 0 λ y u, τ λ R (P u ) = y u λ. 0 λ y u λ 0 λ y u Lemma 6. Te following estimates old for all (0, 0 ) if 0 is small enoug: 2 τ L (P s ) 0 9, 0.0C0 4 ν τ R (P s ) 900C0 4 ν, 2 τ L (P u ) 0 9, 0.0C0 4 ν τ R (P u ) 900C0 4 ν. Proof of Lemma 6. We will prove only estimates for te partition P s (for P u everyting is te same). Notice first tat λ 0 λ x s x s ( λ ). Terefore τ L (P s ) = λ x s λ 0 λ x s since e 2 for small. On te oter and, since x s τ L (P s ) = λ λ x s 0 λ x s λ x s x s λ x s = λ λ 0 λ = λ λ = λ = e 2, λ 9 0 = e Now let us estimate τ R (P s ). Denote by I te segment of Wloc u (O) between te points (λ 0, 0) and (x s, 0) (i.e. te bottom orizontal edge of S ), lengt(i) = x s λ 0. Due to Lemma 3 Since 2 Hence 0.0C 4 0 ν lengt(i) lengt(t(i)) 25C 4 0 ν lengt(i). lengt(t(i)) 2 (tis follows from te Cone condition again), we ave τ R (P s ) since e for 0, and Lemma 6 is proved. τ R (P s ) 200C4 0 +ν λ 50 C 4 0 +ν lengt(i) 200C 4 0 +ν 50 C 4 0 +ν x s ( λ ) 50 C 4 0 +ν λ 9 0 λ 270C4 0 +ν 0 0 ν e 0.0C 4 = 270C4 0 +ν e C4 0 ν. 23
24 24 Lemma 7. If 0 > 0 is small enoug ten for all (0, 0 ) we ave dist(s 0, S ) 0. and dist(t(s 0 ), T(S )) 0.. Proof of Lemma 7. We will prove te first inequality only, since te proof of te second one is completely similar. Notice tat te vertical boundaries of S 0 and S are tangent to te cone field {K u }. Consider te left vertical edge of S and te rigt vertical edge of S 0. Teir lowest points are (λ 0, 0) and (λ x s, 0), and te distance between tem is equal to λ 0 λ x s λ 0 λ = λ (λ 9 0 ) = e (e 9 0 ) e if (0, 0 ) and 0 is small enoug. From te cone condition we ave tat te difference between xcoordinates of any two points on tose edges is greater tan C6 0 +ν 9 ( ) C ν 9 40 > 0. if is small enoug. Lemma 7 is proved Estimates of derivatives: verification of te conditions of Distortion Teorem. We proved tat te map T : S 0 S S as an invariant locally maximal yperbolic set Λ wic is a twocomponent Smale orsesoe (i.e. T belongs to te class F) and obtained estimates of te lateral ticknesses of te corresponding Markov partitions. In order to get estimates of te lateral ticknesses of te related Cantor sets we need to estimate te distortion of te corresponding mappings. ( ) a b Denote te differential of te map T : S 0 S S by DT =, were a, b, c c d and d are smoot functions over S 0 S. Ten te ( differential ) of te inverse map d b T : T(S 0 ) T(S ) S as te form DT =, were ã = a T c ã, b = b T, c = c T and d = d T. Notice tat tis notation agrees wit te notation of Definition 6. Lemma 8. Consider te restriction of te map T to te rectangle S. Tere exists a constant C > (independent of ) suc tat () d λ 2n C, b, c C, (2) C ν a C ν,
25 (3) b, b, c, c, (4) d λ 2n C, d a C, λ 4n C, (5) a C 2 ν, (6) b, b, c, c, ã C, (7) (8) d ã λ 2n C, C 2 ν. d λ 4n C, 25 Proof of Lemma 8. We denoted ( ) a0 b 0 = D(C c 0 d Ψ H () Ψ 0 terefore DT S (x, y) = C ), ( ) ( a(x, y) b(x, y) λ = 2n a 0 (λ 0 x, λ 2n+ 0 y) b 0 (λ 0 x, λ 2n+ 0 y) c(x, y) d(x, y) c 0 (λ 0 x, λ 2n+ 0 y) λ 2n d 0 (λ 0 x, λ 2n+ Te C 3 norm of te map C Ψ H () Ψ C (as well as of its inverse) is uniformly bounded by some constant independent of. Now all te inequalities (), (3), (4) follow for C large enoug. Inequalities (2) follow from Lemma 3, we just need to assume tat C > 00C0. 4 Now let us prove te inequality (5). Since a(x, y) = λ 2n a 0 (λ 0 x, λ 2n+ 0 y), we ave a (x, y) = λ2n+ 0 a 0 (λ 0 x, λ 2n+ 0 y) = λ 2n+ 0 a 0 (λ 0 x, 0) + O(). In order to distinguis te points from R u and from R s let us denote te coordinates in R u by (x, y) and te coordinates in R s by (X, Y). Ten te map C Ψ H k() : R u U can be represented as a composition Ψ were C (x, y) (t(x, y), E(x, y)) (X(t, E), Y(t, E)), (t, E) Π r0,e 0, (t(x, y), E(x, y)) = H k() Ψ We ave 0 y) C (x, y) and (X(t, E), Y(t, E)) = C Ψ (t, E). a 0 (x, y) = X(t(x, y), E(x, y)) and a 0 a 0 = 2 X(t(x, y), E(x, y)). 2 ).
26 26 In particular, since E(x, 0) = 0, (32) a 0 (x, 0) = d2 d X(t(x, 0), 0) = dx2 dx = 2 X (t(x, 0), 0) t2 ( t = 2 X (t(x, 0), 0) t2 (x, 0) ( t (x, 0) ( ) X t (t(x, 0), 0) (x, 0) = t ) 2 + X t (t(x, 0), 0) 2 t (x, 0) = 2 ) 2 ( ) t + a 0 (x, 0) (x, 0) 2 t (x, 0) 2 From te Cone condition (more precisely, from Steps 35 of te proof of Lemma 3) we know tat a 0 (x, 0) 7C0µ(). 4 Also since C 3 norms of maps C Ψ and H k() Ψ C are bounded by C 0, we ave a 0 (x, 0) 2 X (t(x, 0), 0) t2 C C0µ() 4 C0 2 Now we need to estimate 2 X (t(x, 0), 0) t. Notice tat te image of te Oy ax under 2 te map C Ψ is a grap of te function E = Θ (t), and terefore X(t, Θ (t)) = 0. Tis implies tat and (33) d dt (X(t, Θ (t)) = 0 = X t (t, Θ (t)) + X E (t, Θ (t)) Θ (t) d 2 dt 2 (X(t, Θ (t)) = 0 = = 2 X t 2 (t, Θ (t))+2 2 X t E (t, Θ (t)) Θ (t)+ 2 X E 2 (t, Θ (t))( Θ (t)) 2 + X E (t, Θ (t)) Θ (t) Since for Θ (t), Θ (t) and Θ (t) we ave asymptotics (9) (see Proposition 9 and Remark 3), we get 2 X t (t, 2 Θ (t)) 2C 0 2µ() + C 0 (2µ()) 2 + C 0 4π µ() < 20C 0 µ() if is small enoug. At te same time by te mean value teorem we ave 2 X (t, 0) t2 2 X t (t, 2 Θ (t)) + C 0 Θ (t) < 20C 0 µ() + C 0 π µ() 40C 0 µ() Finally we ave a 0 (x, 0) 40C 0 µ() C C 4 0µ() C C 3 0 µ()
27 27 and (setting x = λ 0 x) (34) a(x, y) λ 2n+ 0 50C 3 0 µ() + O() for large C (independent of (0, 0 )). (µ()) ν λ 0 50C 3 0 µ() + O() C 2 ν Properties (6) (8) are symmetric to te properties (3) (5). Lemma 8 is proved. Lemma 9. Te variation of log a(x, y) in S is less tan 600C 2 C 6 0 ν. Proof of Lemma 9. Take two points (x, y ) and (x 2, y 2 ) from S. We want to estimate log a(x, y ) log a(x 2, y 2 ) by using te mean value teorem. Generally speaking, te set S is not convex, so we need some preparations to apply it. Let γ be te intersection γ = T(S) {x = 2 }. Ten ˆγ = T ( γ) is a smoot curve tangent to te cone field {K s }, ˆγ S. Denote ˆx = {y = y } ˆγ and ˆx 2 = {y = y 2 } ˆγ. Notice tat te wole interval wit te end points (x, y ) and (ˆx, y ) belongs to S, as well as te interval wit end points (x 2, y 2 ) and (ˆx 2, y 2 ). Now we ave (35) log a(x, y ) log a(x 2, y 2 ) log a(x, y ) log a(ˆx, y ) + log a(ˆx, y ) log a(ˆx 2, y 2 ) + log a(ˆx 2, y 2 ) log a(x 2, y 2 ) Due to te Cone condition te widt of S is not greater tan 200C 4 0 +ν. By te mean value teorem we ave (36) log a(x, y ) log a(ˆx, y ) a a(x, y ) (x, y ) x ˆx C +ν C 2 ν 200C0 4 +ν = 200CC ν Similarly log a(ˆx 2, y 2 ) log a(x 2, y 2 ) 200C 2 C 4 0 ν Now parameterize te curve ˆγ by te parameter y, ˆγ = ˆγ(x(y), y), y [y, y 2 ] (or y [y 2, y ] if y 2 < y ). Consider a function g(y) = log a(ˆγ(x(y), y). Since ˆγ is tangent to te cone field {K s }, for some y [y, y 2 ] we ave (37) g(y ) g(y 2 ) = g (y ) y y 2 = a(ˆγ(x(y ), y ) a ˆγ x + a ˆγ y y y 2 ( ) C +ν a ˆγ x + a ˆγ y C +ν (C 2 ν 00C0 6 +ν + C ) C ν (00C C C ) 200C 2 C 6 0 ν
28 28 Finally we ave log a(x, y ) log a(x 2, y 2 ) 400C 2 C 4 0 ν + 200C 2 C 6 0 ν < 600C 2 C 6 0 ν Lemma 9 is proved. Te following Proposition directly follows from Lemmas 2, 3, 7, 8 and 9. Proposition 20. Te map T : S S 2 S belongs to te class F(C, γ, ε) (see definition 6), were C = 20CC 4 0, 6 γ = 200CC ν and ε = 20CC ν Proof of te first of te Main Teorems. Properties. and 2. of Teorem clearly follow from te construction and te Cone condition. Let us combine now Proposition 20 wit Duarte Distortion Teorem (Teorem 3) and Lemma 6. Let us assume tat is small enoug so tat e D(C,ε,γ) < 2. Ten we ave Terefore 4 τ L (K s ) 20 9, 4 τ L (K u ) 20 9, 200 C 4 0 ν τ R (K s ) 800C 4 0 ν, 200 C 4 0 ν τ R (K u ) 800C 4 0 ν. τ L (K s )τ R (K s ) 800 C 4 0 +ν as 0 (i.e. a ). Similarly τ L (K u )τ R (K u ) as a, so τ LR (Λ) as a. To ceck te property 4., we apply Proposition 5 (notice tat we are exactly in te setting of Remark 6). We ave ( ) ( ) log + τ L(K s ) (38) dim H K s +τ R (K s 4 log + ) +800C0 ( ) 4ν ) = log + +τ L(K s ) τ R (K s ) log ( C 4 0 ν = log [ ( C 4 0 ν )] log [ ν ( +ν + 200C 4 0( ))] = O(log ) + ν O(log ) > + 2ν if is small enoug. Since ν could be cosen arbitrary small, dim H K s as 0 (i.e. a ). Similarly dim H K u as a. Since dim H Λ = dim H K s +dim H K u ([MM], see also [PV]), we ave Teorem is proved. dim H Λ 2 as a.
29 Sketc of te proof of te second of te Main Teorems. Step. Coose a sequence of parameters {µ n }, µ n 0, suc tat f µn as a quadratic omoclinic tangency and a transversal omoclinic points. Step 2. Using te renormalization tecnics by MoraRomero [MR] and Teorem one can sow tat near eac µ n tere is an open interval U n suc tat for µ U n te map f µ as an invariant locally maximal transitive yperbolic set Λ µ wit persistent omoclinic tangencies. Step 3. Te yperbolic saddle P µ and te set Λ µ are omoclinically related, see Lemma 2 from [Du]. Terefore for every µ U n tere exists a basic set Λ µ suc tat P µ Λ µ and Λ µ Λ µ. Since Λ µ as persistent omoclinic tangencies, so does Λ µ. Tis proves te part (). Step 4. Tere is a dense subset D n U n suc tat for every µ D n tere is some omoclinic tangency for te fixed point P µ. Tis is a consequence of a standard arguments, see [PT]. Te part (2) is proved. Step 5. Consider local invariant manifolds W u ε (P µ ) and W s ε (P µ ), and set Wµ u (k) = fµ(w k ε u (P µ )) and Wµ(k) s = fµ k (Wε s (P µ )). We ave W u (P µ ) = k 0 Wµ u (k) and W s (P µ ) = k 0 Wµ(k). s Since we consider a generic unfolding {f µ }, we can assume tat for an open and dense set O(k) U n we ave Wµ(k) s Wµ u (k). From [MR] it follows tat tere exists an open and dense subset V (k) O(k) U n suc tat for every µ V (k) and every x Wµ(k) s Wµ u (k) tere exists an elliptic periodic point p suc tat dist(p, x) <. k Terefore for every µ from a residual set R = k V (k) te omoclinic class H(P µ, f µ ) is accumulated by elliptic points, and tis proves (3.). Step 6. From Teorem and [MR] it follows tat for every m N tere exists an open and dense subset A(m) U n suc tat for every µ A(m) tere exists a yperbolic set Λ # µ suc tat dim H Λ # > 2. From Lemma 2 from [Du] it follows m tat P µ and Λ # µ are omoclinically related. Terefore tere exists a basic set µ suc tat P µ µ and Λ # µ µ. In particular, for µ R 2 = m A(m) we ave dim H H(P µ, f µ ) = 2. Set R = R R 2. Tis proves (3.2). Step 7. Te last property (3.3) follows from te following Lemma Lemma 2. Let Λ M 2 be a basic set of a surface diffeomorpism. Ten dim H {x Λ O + (x) is dense in Λ and O (x) is dense in Λ} = dim H Λ. Teorem 2 is proved.
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