DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 4, Number 4, October 1998 pp. 765–782

FIRST HOMOCLINIC TANGENCIES IN THE BOUNDARY OF ANOSOV

Maria Carvalho Centro de Matem´atica, Universidade do Porto (Communicated by Mark Pollicott )

Abstract. In dimension two, there are no paths from an Anosov diffeomorphism reaching the boundary of the stability components while attaining a first quadratic tangency associated to a periodic point. Therefore we analyse the possibility to con- struct an arc ending with a first cubic homoclinic tangency. For several reasons that will be explained in the sequel, we will restrict to area preserving diffeomorphisms. Introduction. Let f : M → M be an Anosov diffeomorphism on a compact connected Riemannian smooth manifold M of dimension two. After [M], it is known that the manifold must be the bidimensional torus and f conjugate to an automorphism of the quotient group R2/Z2. That is, f is the projection of a matrix acting in R2 with integer entries and absolute value of the determinant equal to one. Therefore we will start with a linear Anosov diffeomorphism f. The purpose of this note is the analysis of the generic ways f has to lose hyperbolicity and the construction of a path, from f to the boundary of its stability component, that ends with a diffeomorphism exhibiting a cubic homoclinic tangency. Generically to reach the boundary of the set of Anosov diffeomorphisms, we will denote by B, the diffeomorphism f either passes through a first elementary bifurcation (saddle - node, flip or Hopf) or one of its periodic points is involved in a quadratic homoclinic tangency. See [NPT] for a complete description. In both cases, the main role is played by a hyperbolic periodic , whose set of eigenvalues cross the unitary circle or whose invariant manifolds become non-transversal. The first kind of perturbation does not intend to alter the stable/unstable foliations, its effect is to be felt on the dynamics within these foliations. The diffeomorphisms so obtained share most of the dynamical properties of f (it is actually expansive and conjugate to f) and also its ergodic features. See [C] for details. In the sec- ond possibility, the foliations themselves are changed while the stable and unstable manifolds of a periodic orbit gain a non-transversal contact. Here the most impor- tant alterations are due to occur to the structure of the foliations and this is the main difference to be underlined. Among Anosov diffeomorphisms in dimension two, a generic (quadratic) tan- gency is dismissed as a first, isolated, bifurcation, meaning that we cannot reach B without previously leaving the Anosov world. This is due to the denseness of the stable/unstable foliations of all points of the manifold M, which is the nonwander- ing set of f and also the closure of the periodic points. The creation of premature tangencies is inevitable if a quadratic perturbation in one of these foliations is performed. A complete and elegant characterization of the possible tangencies in dimension two may be found in [PT].

1991 Mathematics Subject Classification. Primary 58F11, 28D05. The author acknowledges finantial support from JNICT through the project PRAXIS/2/2.1/Mat/19/94.

765 766 MARIA CARVALHO

We therefore started looking for cubic tangencies. This kind of contact is not generic but it is topologically transversal, hence persistent, and a generic unfolding may be pursued. It may be achieved, roughly speaking, if, given a periodic orbit P of f, we choose a transversal intersection, say R, of its Ws(P ) and the unstable one Wu(P ) - these obviously exist and are even dense in M - and we slightly bend one of the manifolds until the contact becomes not differentiably s u transversal. In this way the tangent spaces TRW (P ) and TRW (P ) intersect along a subspace of dimension one. Locally, in some well chosen coordinates in a neighbourhood of R, the stable/unstable manifolds of P look like the curves y = 0 and y = αx3, where α 6= 0 to avoid degeneracy. The idea seems simple but the technicalities are oddly numerous. Several problems have to be tackled, of which the following are a mighty sample. Is it possible to construct a path (ft)t∈[0,1] from f to B using this type of bifurcation without losing control of the foliations ? May the tangency be isolated so that it appears as the unique bifurcation to handle ? How is the diffeomorphism we reach in B ? 0 Concerning the first question, if the C - distance along an isotopy (ft)t is small enough, then f1 is semi conjugate to f ≡ f0 as asserted in [W]. But this is not enough. To control the foliations, we would like to manage a conjugacy that could take the stable/unstable manifolds by f0 onto stable/unstable sets for f1, preserving their denseness and their topological indices as foliations. But to guarantee that the above semiconjugacy is a homeomorphism, it is essential to establish previously that f1 is expansive, which is obviously related to the way the stable/unstable foliations behave. More precisely, the existence of such a conjugacy depends on where these foliations intersect. However, fortunately, expansiveness does not depend on how they contact. The initial effort must therefore be devoted to obtain stable/unstable foliations for the perturbations of f and afterwards try to conclude expansivity. The second question demands more care. The perturbation must ensure that changes only appear near R and hyperbolicity is lost only at the orbit of R and only when t = 1. For that we have to exhibit a system of invariant cones on the whole torus when t < 1 and on the complement of the orbit of R when t = 1. A first attempt on this subject is due to Lewowicz in [L]. Here he constructs an arc of diffeomorphisms (gt)t∈[0,1] starting at an Anosov one and touching B while having a flat tangency between the stable and unstable manifolds of a periodic point P , the contact being accomplished exactly at the point P . The perturbation is global and at the same time the spectrum of Dg1 becomes non - hyperbolic. The result is an expansive diffeomorphism in B, not Anosov but conjugate to the initial hyperbolic system. This kind of phenomenon is not what we are looking for, since we want to isolate the bifurcation and get a first tangency that is simultaneously the unique bifurcation occurring. Another contribution came recently on a work by H. Enrich, [E]. He proved that, given a fixed Anosov diffeomorphism f0, it is possible to modify it by an isotopy (ft)0≤t≤1 arriving at B, through a local perturbation, with a first cubic heteroclinic s u tangency (between W (P ) and W (Q), P and Q distinct periodic points of f0, the tangency being accomplished at a point R different from the above ones). The diffeomorphism f1 is expansive, transitive and conjugate to f0, exhibiting a Pesin region of total probability. As expected in dimension two, see [NP], the nonwan- dering set is the closure of the periodic points; but it is no more a hyperbolic set: for instance, R is nonwandering but is not assigned a hyperbolic splitting, although it has stable and unstable fibers which however do not form a direct sum. It is HOMOCLINIC TANGENCIES 767 precisely the argument in [E] we would like to strengthen in order to achieve a first homoclinic tangency. For this reason we will start with a sketch of it. Consider an Anosov diffeomorphism f with two distinct periodic points P and Q and a heteroclinic intersection point R of Ws(P ) and Wu(Q). Assume that R be- longs to the intersection of two neighbourhoods V of P and U of Q, with linearizing coordinates associated to the hyperbolicity of P and Q that turn the corresponding local stable and unstable manifolds orthogonal. Since f is an automorphism, these are in fact the natural coordinates on the torus. We then choose a flow (θt)0≤t≤1 that rotates points around the circles centered at R by an amount that decreases π with the distance from R and is zero except near R. Besides, θ1 rotates by 2 and thereby interchanges the stable and unstable directions. If S is a point near R and u s E (S) and E (θ1(S)) are the unstable subspace at S and the stable subspace at u s θ1(S) for f, then the angle between Dθ1(E (S)) and E (θ1(S)) varies smoothly and has a minimum equal to 0 at R. Then, if θ1 is suitably chosen, there is a constant C > 0 such that

 u s  2 2 tan ∠ Dθ1(E (S)),E (θ1(S)) = C[dist (S, R)] + O ([dist (S, R)] )

Since the stable and unstable directions are orthogonal, we also have

 s u  2 2 tan ∠ Dθ1(E (S)),E (θ1(S)) = C[dist (S, R)] + O ([dist (S, R)] )

As long as the orbit of S stays in the region V of linearization associated to P , the derivative of f will expand the unstable direction by a factor of µP and shrink the stable direction by a factor of λP . Thus

 n n+1 u n+1 s  tan ∠ Df Dθ1(E (S)), Df E (θ1(S))}

|µP |  n n u n s  = ) tan ∠ Df Dθ1(E (S)), Df E (θ1(S))} . |λP |

The number mS of iterations of f before the orbit of S leaves V depends on how large the unstable coordinate is; since this coordinate is initially bounded above by mS dist (S, R), mS verifies (µP ) dist (S, R) =diameter of V, so

− log(dist (S, R)) mS −1 (µP ) = O [(dist (S, R)) ] and mS = O ) log |µP | and

 n  mS u mS s tan ∠ Df Dθ1(E (S)), Df E (θ1(S))} m |µP | S  n u s  = tan ∠ Dθ1E (S),E (θ1(S))} |λP | whose major component is − log( dist (S,R)) ! log |µP |− log |λP | !  mS   log |µ | |µP | = O |µP | log |µP | = O 1 P |λP | |λP | dist (S,R)

log |µP |− log |λP | If P is a dissipative point, that is, if |µP | |λP | < 1, then > 2 hence log |µP | 768 MARIA CARVALHO

m |µP | S  u s  tan ∠ {Dθ1E (S),E (θ1(S))} |λP |

log |µP |− log |λP |  1  log |µ |  2  1 b > P C dist (S, R) > dist (S, R) dist (S, R) where b = log |µP | − log |λP | − 2 > 0. log |µP | If S is close to R, this implies that the tangent above is reasonably large. And ensures that, at the moment the orbit leaves V, the vectors, although distorted by the perturbation inside V, have norms and directions that suit hyperbolicity. One also has to perform a similar argument when the roles of Es and Eu are interchanged, and to manage that we repeat the proof for f −1 and the periodic point Q. Notice that the assumption that Q is an expansive periodic point (that −1 is |λQ| |µQ| > 1) ensures that Q is dissipative with respect to f and previous arguments apply entirely to it. These estimates are essential to the construction of invariant cones for points different from R, in order to prove that ft is an Anosov diffeomorphism for all 0 ≤ t < 1; that is, to ensure that a first bifurcation is happening at t = 1. The saddle points P and Q work as ushers to the orbits near the tangency when we iterate backwards or forwards. Accordingly, the orbits of points close to R, the ones affected by the perturbation, share the rates they find dominating in the neighbourhoods of P and Q connected with the hyperbolicity at P and Q. The bifurcation occurring at t = 1 may be enhanced through a simple map P : C → R, where C is the local component of Ws(P ) containing R (this is a curve embedded in M), given in a suitable system of coordinates by the tangent of the angle made by the directions of Ws and Wu along C. The modifications in P, expressed as the pair (p1, 0), must ensure that the contact between the graph of p1 and y ≡ 0 is generic (quadratic), which means that ∂ p ∂ p p (R) = 0, 1 (R) = 0 1 (R) = 0. 1 ∂ x ∂ y These three conditions define a submanifold inside B and justify the assignment to it of a codimension equal to three. The main ingredient for these conclusions is the distinct dynamical nature of the two periodic points P and Q and we were not able to free them from this demanding. But under additional hypothesis, we would like to make sure that the hypothesis 0 < |λP | < 1 is enough to obtain similar, although weaker, deductions. This way the dissipativeness demanded from P would become unnecessary and Q would be dispensable. The easiest way to overcome this difficulty is to consider a conservative periodic point P , that is, where |λP | |µP | = 1 and restrict our family to area preserving diffeomorphisms. In this case the existence of invariant families of cones that induce hyperbolic splittings for all points but R is easier to guarantee, according to the work of Wojtkowski, [Wj]. He showed there that the existence of a family of cones in the tangent bundle which is mapped into itself by the linearized version of the given is sufficient for the non-vanishing of the Lyapounov exponents. Therefore our concern is to find natural candidates for this family of invariant cones. The best hint to follow is the initial family of cones associated to the Anosov diffeomorphism; we will use them for orbits that do not cross the region where the perturbation is acting; and analyse the alterations they endure for orbits that come near R. This way we conclude that HOMOCLINIC TANGENCIES 769

Theorem. If f is a linear Anosov diffeomorphism on the two-torus and P is a fixed point of f, it is possible to construct an arc of diffeomorphisms (ft)0≤t≤1 such that (i)f0 = f and

(ii) ft is area preserving; (iii) f1 exhibits a first homoclinic tangency associated to P ; (iv) ft, 0 ≤ t < 1, is an Anosov diffeomorphism; (v) f1 has non-zero Lyapounov exponents Lebesgue almost everywhere. A final word on the dimension of M. Surely it seems too restrictive, but several main ideas fail to work (like the classification of quadratic tangencies, the stratifi- cation of the eigenvalues or the work of Wojtkowski) and do not have immediate analogues if we rise the dimension of M. So far, generic unfoldings of cubic tangen- cies have not received much attention, although the famous H´enon family of planar 2 dissipative diffeomorphisms Ha, b(x, y) = (1−ax + y,bx) shows how frequent and promising these tangencies are. Numerical results indicate that, for example, at the parameters (a1,b1) ' (1.095, 0.388) and (a2,b2) ' (1.203, 0.417) cubic tangencies, of different type, are detectable. After a convenient renormalization scheme and corresponding diminishing of dimension−till we reach the one-dimensional cubic 3 model maps given by Ga,b(y) = (ay ± y + b)−the numerical approximations of the unfolding of cubic homoclinic tangencies stay close to theoretical predictions and bring the classical diagram of creation of invariant structures, like infinitely many sinks, cascades of period doubling and strange .

2. An arc with a cubic homoclinic tangency. Let f : M → M be a linear Anosov diffeomorphism and P a periodic point. We may assume that P is the fixed point of f and that its eigenvalues are 0 < | λP | < 1 < | µP |. To construct an isotopy (ft)0 ≤ t ≤ 1 such that f0 = f, ft is Anosov for all t ∈ [0, 1[ and f1 exhibits a cubic homoclinic tangency associated to P , we need to locate the perturbation and estimate its iterations by f. If the perturbation of f is small enough, the hyperbolic point P persists along the arc (ft)t and we denote by P its continuation. Consider the hyperbolic splitting of TP M with respect to f, say Eu(P, f) ⊕ Es(P, f), where dimension of Eu =dimension Es = 1, and an open neighbourhood V of P with natural coordinates on the torus (x∗, y∗). With these coordinates, P = (0, 0) and these are the linearizing coordinates associated to the hyperbolicity at P . Denote by TS,x∗ and TS,y∗ the fibers of the bundles tangent to x∗ = constant and y∗ = constant. Take in V a point R = (r∗, 0), other than P but close to it, of intersection of the f-stable and f-unstable manifolds of P . Such an intersection R is persistent, so we may assume that the perturbation ensures the survival of R through the perturbation. 2 2 Suppose that the square ∆ = (x∗, y∗): x∗ ≤ 1 and y∗ ≤ 1 is a subset of V and contains both P and R. Let U be an open neighbourhood of R, contained in V and where we may use C∞ coordinates (x, y), independent of the other coordinates (x∗, y∗), such that xR = 0, yR = 0 and (i) the component of Ws(P ) in U that contains R is given by y ≡ 0; (ii) the component of Wu(P ) in U that contains R is given by x ≡ 0; 2 2 (iii) there is a constant K such that |v1(S)| ≤ K (x +y ) and 0 < K < |v1|+|v2|

for ∀ S ∈ U ∀ v(S) = (v1, v2): kvk = 1 and v ∈ TS,x∗ ; 2 2 (iv) there is a constant L such that |w2(S)| ≤ L (x +y ) and 0 < L < |w1|+|w2|

for ∀ S ∈ U ∀ w(S) = (w1, w2): kwk = 1 and w ∈ TS,y∗ Properties (iii) and (iv) are a consequence of the local product structure of the stable and unstable foliations of f in V. The existence of coordinates verifying 770 MARIA CARVALHO these estimates and continuous with small perturbations of f was established in [E] - Lemma 1.1. Let  be a positive number, chosen small enough in order to ensure that the set E = {(x, y): ρ ≤ } is contained in ∆, where ρ is an abbreviation for the distance of (x, y) to R measured in the coordinates inside U. Notice that, since R 6= P , there is a Ω > 0, depending only on  and r∗, such that (x∗, y∗) ∈ E implies that |x∗| ≥ Ω. We can iterate by f each S = (x∗, y∗) ∈ U without leaving V a number of times which is not less than

( log |x∗| b− c if x∗ 6= 0 i(S) = log |µP | ∞ if x∗ = 0 n n In fact, using the coordinates (x∗, y∗), we have f (S) ∈/ ∆ ⇔ |µP | |x∗| > 1 ⇔ log |x∗| nlog |µP |+ log |x∗| > 0 ⇔ n > − , where the first equivalence is due to the log |µP | way the orbit of S evolves in ∆, according to the linearized system. If x∗ = 0, then S belongs to the component inside V of the local stable manifold of P (which contains P ) and so, for all n ∈ N, we have f n(S) ∈ ∆ and i(S) = +∞. In a similar way, it is certain that we may iterate S by f −1 without leaving V a number of times which is not less than

( log |y∗| b c if y∗ 6= 0 n(S) = log |λP | ∞ if y∗ = 0 r For each 0 ≤ t ≤ 1, define ft = f ◦θt where θt : M → M is a C diffeomorphism which fixes R, θ0 ≡ identity, θt is equal to the identity out of E for all t and, in the coordinates (x, y), θt may be written as

 x cos(tγ(ρ)) + y sin(tγ(ρ))   cos(tγ(ρ)) sin(tγ(ρ)) x θ (x, y) = = t −x sin(tγ(ρ)) + y cos(tγ(ρ)) − sin(tγ(ρ)) cos(tγ(ρ)) y π ∞ π where γ : [0, +∞[ → [0, 2 ] is a C map satisfying γ(0) = 2 ; γ(ρ) = 0 ∀ ρ ≥ ; γ is strictly decreasing in [0, ]. θt in E is thus a rotation of angle tγ(ρ) whose value depends on t and the distance of (x, y) from R. Notice that the map ρ is invariant along θt orbits, that is, ρ(x, y) = ρ(θt(x, y)).   c11,t c12,t Besides, for each t ∈ [0, 1] and S = (x, y) ∈ U, Dθt(x, y) = , where c21,t c22,t x d γ c11,t = cos(tγ(ρ)) + ρ t d ρ [−x sin(tγ(ρ)) + y cos(tγ(ρ))] y d γ c12,t = sin(tγ(ρ)) + ρ t d ρ [−x sin(tγ(ρ)) + y cos(tγ(ρ))] x d γ c21,t = − sin(tγ(ρ)) + ρ t d ρ [−x cos(tγ(ρ)) − y sin(tγ(ρ))] y d γ c22,t = cos(tγ(ρ)) + ρ t d ρ [−x cos(tγ(ρ)) − y sin(tγ(ρ))] This yields (i) for each t ∈ [0, 1], Dθt has Jacobean equal to Jac Dθt = c11,t ∗ c22,t − c12,t ∗ c21,t = 1; therefore θt is area preserving. Since f0 preserves the Lebesgue measure, this probability is also invariant by ft for all t ∈ [0, 1].

(ii) Dθ1(R) (1, 0) = Dθ1(0, 0) (1, 0) = (0, −1) and Dθ1(R) (0, 1) = Dθ1(0, 0) (0, 1) = (1, 0). That is, θ1 sends the stable direction at R into the unstable one, and neigh- bour directions move accordingly but with a not so huge rotation; this produces the desired unique tangency. We still have to define explicitly the map γ, but let us first complete the program of this proof. To detect the formation of a cubic tangency for t = 1 at R and to HOMOCLINIC TANGENCIES 771 confirm that it happens only at R and that no other bifurcations are occurring we have to find out how Dθt behaves in a neighbourhood of R. It would be useful to conclude that, for each S in E, Tθt(S),y∗ is close to Dθt(TS,x∗ ) and Tθt(S),x∗ is close to Dθt(TS,y∗ ). The closeness between the fibers indicates mainly how small are the angles be- tween unitary vectors of those bundles since θt does not alter the distance ρ. Con- sider then a unitary vector w(S) = (w1, w2) of TS,y∗ , a fiber whose direction is close to the unstable direction of f at S; in the linearizing coordinates in V, TS,y∗ appears almost horizontal if S is near R; this yields w1 ' 1 and w2 ' 0. Therefore   Dθt(S)(w) = c11 w1 + c12 w2, c21w1 + c22w2) ' c11 w1, c21 w1). π As, near R and for t close to 1, θt is a rotation of an angle of approximately 2 , we expect to find Dθt(S)(w) almost in the direction of TS,x∗ , which means that Dθt(S)(w) appears close to the stable direction of f at θt(S). This should be revealed by a small absolute value of the first coordinate of Dθt(S)(w) contrary to what happens to the second coordinate. Similarly, a unitary vector v(S) = (v1, v2) of TS,x∗ , a fiber whose direction is close to the stable direction of f at S which, in the linearizing coordinates in V, appears almost vertical if S is near R, must have v1 ' 0 and v2 ' 1. Therefore   Dθt(S)(v) = c11 v1 + c12 v2, c21v1 + c22v2) ' c12 v2, c22 v2).

We thus expect to find Dθt(S)(v) almost in the direction of TS,y∗ , which means that Dθt(S)(v) appears close to the unstable direction of f at θt(S). So, by dual- ity, this should be noticed by a comparatively small absolute value of the second coordinate of Dθt(S)(v). The contents of next Lemma establishes bounds for the inevitable distortions in the expectations above; the estimates make appeal of the size of the neighbourhood E, parametrized here by ρ. Lemma 2.1. Given a ∈ R+, there are A > 0, B > 0, (a) > 0 and a C∞ map π π γ : [0, +∞[ → R such that (i) (a) → 0 as a → +∞, (ii) γ(0) = 2 ; 0 < γ(ρ) < 2 ∀ 0 < ρ < (a); γ(ρ) = 0 ∀ ρ ∈ [(a), +∞[; and (iii) ∀ t ∈ [0, 1] ∀ ρ ∈ [0, (a)] we have

d γ(aρ2) A(1 − t + taρ2) ≤ cos(tγ(aρ2)) + t [−x2 sin(tγ(aρ2)) + xy cos(tγ(aρ2))], ρ d ρ d γ(aρ2) sin(tγ(aρ2)) + t [y2 cos(tγ(aρ2)) − xy sin(tγ(aρ2))] ≤ B. ρ d ρ

Proof.√ A map√γ as demanded in (ii) is easy to describe once (a) is found. Denote u = a x, v = a y and change the variable ρ into c = aρ2. Then, if ρ 6= 0, we have

d γ(aρ2) d γ(c) 1 d γ(c) d c 1 d γ(c) d γ(c) = = = 2aρ = 2a ρd ρ ρ d ρ ρ d c d ρ ρ d c d c 2 2 d γ(aρ ) 2 2 2 and cos(tγ(aρ )) + t ρ d ρ [−x sin(tγ(aρ )) + xy cos(tγ(aρ ))] = cos(tγ(c)) + d γ(c) 4 2t [−u2 sin(tγ(c)) + uv cos(tγ(c))]. Let (a) = √π and γ given by d c 2a

(  2c − 1 π − 7 1 π8 [1 − b )] [ − c + b(2c)(c − 2 8 c 8 )] if c ∈ [0, ] γ(c) = π8 − 1 2 2 π8 0 if c ∈] 2 , +∞[ 772 MARIA CARVALHO

∞ π8 where b is any C real used to smooth the joints of γ at c = 0 and c = 2 ; for instance, b may be defined in order to verify •b(z) = 0 ∀ z≤ 0 •b(z) = 1 ∀ z≥ 1 0 •0 ≤ b (z) ≤ 4 for all z 3 ( exp ( − 1 ) if c ∈ [0, δ] •there is δ > 0 such that b(c) = c2 0 if c = 0 1 ∞ In spite of the exponent 8 of c in its expression, γ is a C map due to the flatness of b along the curve c = 0. π8 To prove (iii) of the Lemma, we have to analyze γ in [0, 2 ]. Consider first c in 7 1 1 π − 8 8 π [0, 2 ]. γ is given by γ(c) = 2 − c + b(2c)(c − 2 c ), which is in [0, 2 ]. Besides π 2 (I) ∀ 0 ≤ γ ≤ 2 1 − π t γ(c) ≤ cos(tγ(c)); 7 1 7 1 2 2 π − 8 8 2 − 8 8 (II) 1− π t γ(c) = 1− π t { 2 −c+b(2c)(c−2 c )} = 1−t+t {c π −b(2c)(c−2 c )} 2 2 ≥ 1 − t + t c π ≥ π (1 − t + tc). d γ(c) d b(2c) 7 1 7 7 − 8 8 − 8 − 8 1 (III) If c 6= 0, then d c = −1 + 2 d c (c − 2 c ) + b(2c) (1 − 2 c 8 ) d b(2c) 7 1 7 − 8 8 − 8 1 = −1 + 2 d c (c − 2 c ) + b(2c) [1 − (2c) 8 ], which is non-positive because d b(2c) 7 1 7 7 − 8 8 1 − 8 − 8 1 2 d c (c − 2 c ) ≤ 0, ∀ c ∈ [0, 2 ] (2c) ≥ 1 and b(2c) [1−(2 c) 8 ] ≤ [1 − 1 7 d γ(c) 2 8 ] = 8 . Therefore 0 ≤ 2t d c [− u sin(tγ(c))] since sin x is positive on arguments π γ(c) ∈ [0, 2 ]. √ √ x2+y2 ρ2 c (IV) |uv| = | a x a y| = a|xy| ≤ a ( 2 ) = a ( 2 ) = 2 . d γ(c) 2 d γ(c) (V) cos(tγ(c))+2t d c [−u sin(tγ(c))+uv cos(tγ(c))] ≥ cos(tγ(c)) [1+2t d c |uv|] d γ(c) c ≥ cos(tγ(c))) [1 + 2t d c 2 ]. Moreover

d γ(c) c d b(2c) − 7 1 − 7 − 7 1 [1 + 2t ] = 1 + tc [−1 + 2 (c − 2 8 c 8 + b(2c) (1 − 2 8 c 8 )] d c 2 d c 8 d b(2c) 2 − 7 9  − 7 c = 1 + t [−c + 2 (c − 2 8 c 8 ) + b(2c) c − (2 c) 8 )] d c 8 n  − 7 c n d b(2c) 2 − 7 9 o = 1 + t −c + b(2c) c − (2 c) 8 )Bigr} + t 2 (c − 2 8 c 8 ) 8 d c n − 31 1 o n8 2 − 7 9 o ≥ 1 + t min {−c, 2 8 c 8 } + t min {c − 2 8 c 8 } 1 1 c∈[0, 2 ] 3 c∈[0, 2 ]

1 n8 2 − 7 9 o = 1 + t (− ) + t min {c − 2 8 c 8 } 1 2 3 c∈[0, 2 ] 1 8 1 1 1 > 1 + t (− ) + t (− ) ≥ 1 − − > 0. 2 3 10 2 3 and so

d γ(c) c 2 d γ(c) c cos(tγ(c)) [1 + 2t ] ≥ (1 − t + tc) [1 + 2t ] d c 2 π d c 2 2 1 1 ≥ (1 − − )(1 − t + tc) = A (1 − t + tc). π 2 3 1 π8 Now, if c is in [ 2 , 2 ], some minor changes have to be made on the above esti- 2c − 1 mates. In this interval, π8 − 1 is greater than zero and 1 ≤ 2c, therefore b(2c) = 1  7 1 2c − 1 π − 8 8 π and γ is given by γ(c) = [1 − b π8 − 1 )] [ 2 − 2 c ] ∈ [0, 2 ]. HOMOCLINIC TANGENCIES 773

7 1 1 π − 8 8 2c − 1 1 8 Besides, since 0 ≤ [ 2 − 2 c ], −1 ≤ −[1 − b( π8 − 1 )], 2 ≤ c ⇒ 1 ≤ (2c) , and π8 2 1 2 ≤ c ⇒ c π9 ≤ π , 7 1 7 1 2 2 2c − 1 π − 8 8 2 π − 8 8 (VI) 1 − π t γ(c) = 1 − π [1 − b( π8 − 1 )] [ 2 − 2 c ] ≥1 − π t [ 2 − 2 c ] = 1− 1 1 8 1 8 2 8 2c 1 2 2 2 2 t+ π c t = 1−t+t π ≥ 1−t+t π ≥ 1−t+tc π9 ≥ π9 (1−t)+tc π9 = π9 (1−t+tc). (VII) If c 6= 0, using 7 7 2c − 1 − 8 − 8 1 d b 2c − 1 2 π 0 ≤ [1 − b( π8 − 1 )], [−2 c 8 ] ≤ 0,[− d c ( π8 − 1 )( π8 − 1 )] ≤ 0 and 0 ≤ [ 2 − − 7 1 2 8 c 8 ], we deduce

d γ(c) 2c − 1 − 7 − 7 1 = [1 − b( )] [−2 8 c 8 ] d c π8 − 1 8 d b 2c − 1 2 π − 7 1 + [− ( )( )] [ − 2 8 c 8 ] ≤ 0. d c π8 − 1 π8 − 1 2

d γ(c) 2 Therefore 0 ≤ 2t d c [−u sin(tγ(c))] since the sine is positive on arguments π γ(c) which lie in [0, 2 ]. d γ(c) 2 (VIII) Hence cos(tγ(c))+2t d c [−u sin(tγ(c))+uv cos(tγ(c))] ≥ cos(tγ(c))+ d γ(c) d γ(c) d γ(c) 2t d c uv cos(tγ(c)) = cos(tγ(c)) [1 + 2t d c uv] ≥ cos(tγ(c)) [1 + 2t d c |uv|] ≥ d γ(c) c cos(tγ(c)) [1 + 2t d c 2 ]. Moreover d γ(c) c [1 + 2t d c 2 ] 7 7 7 1 2c − 1 − 8 − 8 1 d b 2c − 1 2 π − 8 8 = 1 + tc {[1 − b( π8 − 1 )] [−2 c 8 ] + [− d c ( π8 − 1 )( π8 − 1 )] [ 2 − 2 c ]} 31 1 7 9 2c − 1 − 8 8 d b 2c − 1 2 π − 8 8 = 1 + t {[1 − b( π8 − 1 )] [−2 c ] + [− d c ( π8 − 1 )( π8 − 1 )] [ 2 c − 2 c ]} 31 1 7 9 − 8 8 8 2 π − 8 8 ≥ 1 + t {min 1 π8 {−2 c }} − t { 3 ( π8 − 1 ) max 1 π8 { 2 c − 2 c }} c∈[ 2 , 2 ] c∈[ 2 , 2 ] 1 1 1 1 = 1 + t (− 4 ) − t ( 4 ) ≥ 1 − 2 − 4 > 0. d γ(c) c 2 d γ(c) c 2 1 1 So cos(tγ(c)) [1+2t d c 2 ] ≥ π (1−t+tc)[1+2t d c 2 ] ≥ π (1− 2 − 4 ) (1−t+tc) = A (1 − t + tc). This proves the first inequality of (iii) of the Lemma. Concerning the second inequality, we have just to notice that 2 2 d γ(aρ ) 2 2 2 sin(tγ(aρ )) + t ρ d ρ [y cos(tγ(aρ )) − xy sin(tγ(aρ ))] d γ(c) 2 ≤ | sin(tγ(c)) + 2t d c [u cos(tγ(c)) − uv sin(tγ(c))]| d γ(c) 2 d γ(c) ≤ | sin(tγ(c))|+|2t d c [u cos(tγ(c))|+|2t d c uv sin(tγ(c))| d γ(c) 2 d γ(c) ≤ 1 + 2 | d c | |u |+2 | d c | |uv| 2 2 c c π8 π8 ≤ 1 + 2 Υ |u + v |+2 Υ 2 = 1 + 2 Υ c + 2 Υ 2 ≤ 1 + 2 Υ 2 + 2 Υ 4 = B.(Υ is d γ(c) π8 an upper bound for the values of | d c | in [0, 2 ])

3. f t is an Anosov diffeomorphism for all t ∈ [0, 1[ Our aim now is to find stable and unstable vector bundles at S ∈ M, invariant by ft and forming a local product structure. By last section, they must be looked after near TS,y∗ and

TS,x∗ respectively. Their existence will be assured if the angle between the fibers

Dθt(TS,y∗ ) and Tθt(S),x∗ (and also between their duals) only vanishes if t is zero π and S = R. Denote by ∠ (F1, F2) the angle in [0, 2 ] between the fibers F1 and F2. Next Lemma will show that the perturbation considered guarantees that the angle ∠ (Dθt(TS,y∗ ),Tθt(S),x∗ ) varies in a quadratic way with the distance to R and linearly with t. Notice that, as we are specially interested on what happens inside the neighbourhood of linearization V, we have 774 MARIA CARVALHO

∠ (Dθt(TS,y∗ ),Tθt(S),x∗ ) = ∠ (Dft(TS,y∗ ),Tft(S),x∗ ) Lemma 3.1. Given t0 ∈]0, 1], there is h > 0 such that for all positive κ, there exists a0(κ) verifying h (1 − t) + κ ρ2(S) ≤ tan( (Dθ (T ),T )) ∠ t S,y∗ θt(S),x∗

∀ a > a0,S ∈ Ea and t ∈ [t0, 1]. Proof. The essential ingredient here is the corresponding inequality for t = 1 which says that, in spite of vanishing at S = R, the tangent of the angle between the fibers is bounded below by ρ2(S) and so, in particular, only vanishes at R.A small perturbation of this inequality will not create additional zeros and moreover the dependence on t will be the simplest and the most useful in this context.

Fix S ∈ Ea. There are C1 > 0 and C2 > 0 such that for all w ∈ TS,y∗ and t −1 we have 0 < C1 ≤ kDθt(S)(w)k ≤ C2, since the matrices Dθt(S) and Dθt (S) vary continuously with S and t. Assume that there exist t0 in ]0, 1] denying the conclusion of the Lemma, that is, for which tan( (Dθ (T ),T )) < h (1 − t ) + κ ρ2(S ) ∠ ta Sa,y∗ θta (Sa),x∗ a a

∀ h > 0 ∃ κ = κ(h) > 0 : ∀ a0 ∃ a > a0 ∃ Sa ∈ Ea ∃ ta ∈ [t0, 1]. So for any large a, there are two unitary vectors v (θ (S )) = (v , v ) ∈ T ta a 1 2 θta (Sa),x∗ and w (Sa) = (w1, w2) ∈ TSa,y∗ , making an angle ∠ (Dθta (w (Sa)), v (θta (Sa))) whose tangent is less than h (1 − t ) + κ ρ2(S ). But a a

D θta (w(Sa)) v (θta (Sa))− kD θta (w(Sa)) k tan( (Dθta (w (Sa)), v (θta (Sa))) ≥ ∠ || v(θta (Sa)) || D θ (w (S )) ta a = v (θta (Sa)) − kD θta (w (Sa))k q [D θta (w (Sa))]1 2 [D θta (w (Sa))]2 2 = {|v1(θt (Sa)) − |} + {|v2(θt (Sa)) − |} a kD θt (w (Sa))k a kD θt (w(Sa))k ( a a ) [D θta (w (Sa))]1 2 [D θta (w (Sa))]2 ≥ max |v1(θta (Sa)) − |} , |v2(θta (Sa)) − | . kD θta (w (Sa))k kD θta (w(Sa))k If we fix h = A , this h determines a κ(h) such that we may choose an arbitrarily 2 C2 large a verifying 2 h (1 − ta) + κ ρ (Sa) > tan(∠ (Dθta (w(Sa)), v(θta (Sa))) n o [D θta (w (Sa))]1 [D θta (w (Sa))]2 ≥ max |v1(θta (Sa)) − |, |v2(θta (Sa)) − | . kD θta (w (Sa))k kD θta (w(Sa))k Thus

[D θta (w (Sa))]1 2 |v1(θta (Sa)) − | < h (1 − ta) + κ ρ (Sa)(∗) kD θta (w (Sa))k and [D θta (w (Sa))]2 2 |v2(θta (Sa)) − | < h (1 − ta) + κ ρ (Sa). kD θta (w(Sa))k 2 But, as mentioned in section § 2, |v1(θta (Sa))| ≤ K ρ (θa(Sa) and |w2(Sa)| ≤ 2 L ρ (θta (Sa)). Therefore, for instance, [D θta (w (Sa))]1 c11 w1+c12 w2 |v1(θta (Sa)) − | = |v1(θta (Sa)) − | = kD θta (w (Sa))k kD θta (w (Sa))k 2 2 d γ(aρ ) 2 2 2 {cos(tγ(aρ ))+t ρ d ρ [−x sin(tγ(aρ ))+xy cos(tγ(aρ ))]} w1 = |v1(θta (Sa)) − kD θta (w (Sa))k 2 {sin(tγ(aρ2))+t d γ(aρ ) [y2 cos(tγ(aρ2))−xy sin(tγ(aρ2))]} w − ρ d ρ 2 | kD θta (w (Sa))k 2 {cos(tγ(aρ2))+t d γ(aρ ) [−x2 sin(tγ(aρ2))+xy cos(tγ(aρ2))]} w ≥ | ρ d ρ 1 kD θta (w (Sa))k HOMOCLINIC TANGENCIES 775

2 2 d γ(aρ ) 2 2 2 {sin(tγ(aρ ))+t ρ d ρ [y cos(tγ(aρ ))−xy sin(tγ(aρ ))]} w2 − |−|v1(θta (Sa))| kD θta (w (Sa))k 2 {cos(tγ(aρ2))+t d γ(aρ ) [−x2 sin(tγ(aρ2))+xy cos(tγ(aρ2))]} w ≥ | ρ d ρ 1 kD θta (w (Sa))k 2 2 d γ(aρ ) 2 2 2 {sin(tγ(aρ ))+t ρ d ρ [y cos(tγ(aρ ))−xy sin(tγ(aρ ))]} w2 2 − |−Kρ (Sa) kD θta (w (Sa))k 2 1 2 d γ(aρ ) 2 2 2 ≥ |{cos(tγ(aρ )) + t [−x sin(tγ(aρ )) + xy cos(tγ(aρ ))]} w1 C2 ρ d ρ 2 2 d γ(aρ ) 2 2 2 2 −{sin(tγ(aρ )) + t ρ d ρ [y cos(tγ(aρ )) − xy sin(tγ(aρ ))]} w2| − K ρ (Sa) 2 1 2 d γ(aρ ) 2 2 2 ≥ |{cos(tγ(aρ )) + t [−x sin(tγ(aρ )) + xy cos(tγ(aρ ))]} w1| C2 ρ d ρ 2 2 d γ(aρ ) 2 2 2 2 −|{sin(tγ(aρ )) + t ρ d ρ [y cos(tγ(aρ )) − xy sin(tγ(aρ ))]} w2| − K ρ (Sa) 1 2 1 2 ≥ A (1 − ta + ta(aρ (Sa)))|w1|− B |w2|−K ρ (Sa) C2 C2 A 2 1 B 2 ≥ (1 − ta + ta(aρ (Sa))) − |w2|−K ρ (Sa) C2 2 C2 A 2 B 2 2 ≥ (1 − ta + ata ρ ) − L ρ − K ρ 2C2 C2 A 2 a ta A B = (1 − ta) + ρ [ − L − K]. 2 C2 2 C2 C2 If we fix a such that a t0 A > B L − K + κ(h), i.e., a > 2 B L − 2 C2 K + 2 κ C2 , and 2 C2 C2 t0A take into account that h ≤ A , then we get 2 C2

A 2 a ta A B 2 (1 − ta) + ρ [ − L − K] > h (1 − ta) + ρ κ(h) 2 C2 2 C2 C2 which yields

[D θta (w (Sa))]1 2 |v1(θta (Sa)) − | > h (1 − ta) + ρ κ(h) kD θta (w (Sa))k clearly contradicting (∗). This Lemma may be reformulated in terms of the aforementioned map P. The

fibers TS,y∗ and TS,x∗ , for S in E, vary differentiably with S because M has dimen- r sion two and the arc ft is made up by C diffeomorphisms, r ≥ 2. Thus P is of class C1. According to the definition, P may be expressed in the coordinates (x, y) as the pair (p1, 0) and this Lemma is assuring that for ∀ t0 ∈]0, 1] ∃ h > 0 : ∀ κ > 0 ∃ a0(κ): ∀ a > a0 ∀ S ∈ E ∀ t ∈ [t0, 1] 2 h(1 − t) + κ ρ (S) ≤ tan(∠ (Dθt(TS,y∗ ), ) ≤ kv − wk = |p1|, which ultimately means that the map p1 varies in a quadratic way. Since it is at least a C1 map, this yields ∂ p ∂ p p (x, y) = p (R) + 1 (R)x + 1 (R)y + O(2) + ··· = O(2) + ··· 1 1 ∂ x ∂ y where O(2) is equal to ∂2 p ∂2 p ∂2 p 1 (R)x2 + 1 (R)y2 + 1 (R)xy. ∂ x2 ∂ y2 ∂ x ∂ y if these terms exist, in which case they do not all vanish but at R. Notice that we have only concluded that, at t = 1, a tangency is occurring in the arc (ft)t and at a unique point R. To prove that it is the first bifurcation, that is, that ft is an Anosov diffeomorphism for all t in [0, 1[, we need also to check that other possible bifurcations did not accompany the appearance of the tangency. This will be done exhibiting, for each t in [0, 1] two families of invariant cones that will lead to the hyperbolic splitting for all S if t < 1. Lemma 3.1 controls the angle between the bundles that are close to the candidates to be the stable and unstable spaces; we are left to analyze their inner dynamics and establish that they 776 MARIA CARVALHO are effectively the skeleton of the hyperbolicity for t < 1. Since we started with an Anosov diffeomorphism f, there is t0 > 0 such that ft is an Anosov diffeomorphism, in the same class of conjugacy of f, for all t ∈ [0, t0[. So we are left to consider ft for t in [t0, 1[. Definition. Let S be a point in M such that the tangent fiber to M at S is a direct sum of two subbundles E1 and E2. Then, fixed  > 0, an  - cone made by E2 around E1 is the set

CS,(E1) = {(v1, v2) ∈ E1(S) ⊕ E2(S): kv2k ≤  kv1k} Notice that, given the point S as the beak of the cone and the space E, the width  completely determines the cone around E. Following [NP1], let us consider a first property concerning cones, apparently weaker than hyperbolicity. Definition. A diffeomorphism g : M → M has a quasi-hyperbolic splitting if for each S in M the tangent space to M at S is a direct sum of two subbundles E1 and E2 satisfying (i): S −→ (dim E1(S),dim E2(S)) is a constant map; and + (2i): there are bounded maps 1, 2 : M → R , σ > 1 and q ∈ N such that q q q q (2i.1) Dg (CS, 1(S)(E1(S))) ⊂ Cg (S), 1(g S)(E1[g (S)]); −q −q −q −q Dg (CS, 2(S)(E2(S)) ⊂ Cg (S), 2(g S)(E2[g (S)]); q (2i.2) σ kvk ≤ kDg (v)k ∀ v ∈ CS, 1(S)(E1(S)); −q σ kwk ≤ kDg (w)k ∀ w ∈ CS, 2(S)(E2(S)). The metric may be adapted in order to get the above conditions written with q = 1 and this is the contents of next Lemma. It asserts that quasi-hyperbolicity is an indirect reformulation of global hyperbolicity. It may be found in [NP1]. Lemma. 3.2. g is an Anosov diffeomorphism ⇔ g has a quasi-hyperbolic splitting.

Our next step is to prove that, for all t ∈ [t0, 1[, ft has a quasi-hyperbolic splitting. Obviously the Anosov diffeomorphism f0 has assigned to each point S such a pair of cones, around the stable and unstable spaces, with the advantage that their widths may be chosen independent of the point. Let us find the cones for the other maps.

Lemma 3.3. (a) For ∀ t ∈ [t0, 1[ ∀ S ∈ M ∃ pair of cones at S of a quasi- hyperbolic splitting for ft. (b)∀ S 6= R ∃ pair of cones at S for f1 verifying (2i.1). (c)f1 has non-zero Lyapounov exponents Lebesgue almost everywhere. Proof. The arguments to prove (i) and (2i.1) for (a) and (b) are close enough to be presented together. The demand (2i.2) deserves extra attention when t < 1. For t = 1 the best we may expect is the non-vanishing of all Lyapounov exponents at Lebesgue almost every point, following [Wj] in the context of area preserving surface diffeomorphisms with families of invariant cones that satisfy (2i.1). See also [BK]. (a), (b) Fix t in [t0, 1]. Given S in M pick a normalized vector v whose direction u is close to TS,x∗ , hence near the direction of Ef (S). Only the points S whose orbits cross E are affected by the perturbation. Out of E the hyperbolicity of f0 guarantees the requested growth of the norm of the vectors. After entering V, the orbit may pass by E and here the rate of expansion of v undergoes possible reductions. After some iterations, whose number depends on how close to R the positive orbit of S goes, it leaves V; we will prove that, at the moment it gets out of V and by the u effect of the expansion within TP W (P ), the norm of the corresponding iterate of v by Dft(S) is rather big, having recovered from the distortions within E. HOMOCLINIC TANGENCIES 777

The perturbation θt is essentially a rotation of some angle that depends on the point. But, for the argument we are structuring and by reasons already explained, we need to consider the components of the vectors along the directions of TS,x∗ and

TS,y∗ . For t > 0 the rotation θt do alter the angles between the vectors and these directions; this is after all the essence of the creation of a tangency. So the absolute value of the considered components may even contract while inside E if the tangent of these angles is reduced. Start denoting the cones associated to each point S and the Anosov diffeo- morphism f0, with range β1(t), β2(t) to be chosen later, by CS, β1(t)(E1(S)) and u s CS, β2(t)(E2(S)), where E1(S) = E (S) and E2(S) = E (S) of the hyperbolic split- ting. We are looking for similar cones for each ft satisfying properties (i) and (2i), which we will denote by

CS, 1(S,t)(E1(S, t)) and CS, 2(S,t)(E2(S, t)).

The changes in the family of cones of f0 have to be made only along orbits that cross E. Outside E, they are the best candidates to C and so we must also ensure that, when the orbit leaves V, the iterates of the cones C fall inside the original family C in order to suit the definitions of C globally. We will construct the cones around E1, an analogous reasoning being applied to the other family of cones.

For S ∈ E, define E1(S) = TS,x∗ (respectively E2(S) = TS,y∗ ) and consider an 1(S) - cone associated to this bundle, where

1 1 t < 1 ⇒  (S, t) = ; t = 1 & S 6= R ⇒  (S, 1) = . 1 h(1 − t) + κ ρ2(S) 1 κ ρ2(S)

In the sequel, we will use the fact that any vector of TSM making an angle with the direction TS,y∗ bigger than a number £ must be in a fixed cone at S around E1(S). As we will see, for t < 1 and all S, the cones so defined correspond to a quasi- hyperbolic splitting. For t = 1 and S = R, the sum of TR,x∗ with TR,y∗ is not a direct one, and this is exactly the reason why the quasi-hyperbolicity fails at R. As proved in the beginning of this section, the tangent of the angle between each unitary vector v in the cone CS, 1 (E1(S)) and the direction of TS,y∗ is at least 2 h(1 − t) + κ ρ (S) and this estimate is preserved by iteration of ft while the iterates do not leave E. Let m0 ∈ N be small enough (depending on ft) in order to ensure that, for all m ∈ N smaller than m0 and v in CS, 1 (E1(S)), the orbit of S by ft remains in E and we have

m |µP | m 2 2 m tan(∠ D(ft) (v),T(ft) S,y∗ ) ≥( ) [h(1 − t) + κ ρ (S)] = [h(1 − t) + κ ρ (S)]. |λP | We may reduce the neighbourhood E if necessary and get, for all S in E,

( log |x∗| b− c if x∗ 6= 0 log |µP | m0 < i(S) = +∞ if x∗ = 0 2 m m Hence if t 6= 1, we have [h(1−t)+κ ρ (S)] ≤ tan(∠ D(ft) (v),T(ft) S,y∗ ) and (a) if ρ is bigger than some number R, then [h(1 − t) + κ ρ2(S)] ≥ κ ρ2(S) ≥ κ R2 and, since κ may be taken arbitrarily large, this rate is as big as needed. The increasing of κ yields a diminishing of the size of E (see their connection in Lemma 2.1) and therefore the hypothesis that this obliges the iterates of the vector to proceed close u to the direction of TP W (P )−where the expansion is very strong in spite of θt−and 778 MARIA CARVALHO

|y∗| is near zero. This compensates the initial possible bad position of the vector, which could produce a small component in the direction of TS,x∗ ; (b) if 0 < ρ ≤ R, then [h(1 − t) + κ ρ2(S)] > h(1 − t).

Both estimates mean that, if S is close enough to y∗ = 0, then its fiber TS,x∗ is in the interior of CS, 1(S,t)(E1(S)) and j(S) j(S) j(S) D(ft) (CS, 1(S,t)(E1(S))) ⊂ C(ft) (S), β1(t)(E1(ft) (S)) j(S) 1 where (ft) (S) is the first positive iterate of S out of V and β1(t) = h(1−t) . To the positive iterates of S that are inside V, we define j(S) j(S) j(S) j(S) •C(ft) (S), 1[(ft) (S),t](E1(ft) (S))) = D(ft) (CS, 1(S,t)(E1(S))) j(S) if D(ft) (CS, 1(S,t)(E1(S))) is not contained in CS, 1(S,t)(E1(S)); j(S) j(S) j(S) •C(ft) (S), 1[(ft) (S),t](E1(ft) (S))) = CS, 1(S,t)(E1(S))) otherwise. If t = 1 but S 6= R, then the rate [h(1 − t) + κ ρ2(S)] is just κ ρ2(S) and we may 1 pursue an argument similar to the one in (I) but now using β1(1,S) = κ ρ2(S) as the range of the cones C for f0. This means that while t < 1 or S 6= R, after the selected positive iterates, the cone does not vanish and proves (2i.1) with q = 1 for each ft , 0 ≤ t < 1 and all S, or t = 1 and S 6= R. Concerning (2i.2) for ft, t0 ≤ t < 1, we may find σ1 > 1 such that for all S out u of E and all vector v in CS, 1(S,t)(E1(S)) = CS, β1 (E (S)) we have kDft(S)(v)k ≥ σ1 kvk. Therefore the property (2i.2) holds. It is the path inside E that brings trouble. Fortunately this piece of orbit is uniformly bounded, that is, there is a positive integer N0,t independent of S such that any orbit leaves E after a number of iterates by ft that do not exceed N0,t. For instance, if S is in E, then |x∗| 6= 0 (since we assumed R 6= P , we may conclude that S ∈ E ⇒ |x∗|= 6 0) and so, using the dynamics of f0 inside the neighbourhood of linearization V, we deduce that the successive iterates of S remain in E if and only if n n n (f0) (S) ∈ E ⇔ ρ ((f0) (S)) ≤  ⇔ |µP | |x∗| ≤  ⇔ n log |µP |+ log |x∗| ≤ log() log()−log |x∗| log()−log |Ω| ⇔ n log |µP | ≤ log()−log |x∗| ⇔ n ≤ ≤ = N0,0 where log |µP | log |µP | Ω was defined in section § 2. An analogous conclusion may be obtained for each ft, getting a similar starting moment N0,t out of E. Fix now t in [t0, 1[ and consider first a point S in M \V such that ft(S) ∈ V u but S/∈ Wloc(P ). While the orbit remains out of E, the norms of the iterates of a vector v ∈ CS, 1 (E1(S)) expand at a rate assigned by f0; after entering V, the orbit may pass by E and here the rate of expansion is not the same: the perturbation θt, although being a rotation, may shrink the norm of vectors by rates that depend on the point. But after some iterations, whose number depends on how close to P the positive orbit of S reaches, it leaves V. Assume that there is a natural integer j(S) j(S) J(S) such that (ft) (S) belongs to E and find an iterate (ft) (S) such that

n J(S)+1 (ft) (S) ∈ V ∀ 1 ≤ n ≤ J(S) and (ft) (S) ∈/ V. We will analyze the itinerary of the vector v from the first iterate inside E - where the perturbation θt acts - until the orbit leaves V and study afterwards the piece of the orbit of S before it enters E, where ft is still the Anosov f0; the union of both arguments will allow us to check that, immediately after the route inside the region where ft differs from f0, the iterate of v did not diminish and in fact duplicated. Next Sublemma tells us how the vectors are changed at the moment the orbit leaves V. HOMOCLINIC TANGENCIES 779

J(S) Sublemma 1. 2kvk ≤ kD(ft) (S)(v)k for S as above and all vector v in

CS, 1 (E1(S)) \{0}.

Proof. Let v be in CS, 1 (E1(S)) \{0}. The component of v in the direction of

TS,x∗ is the one with more chances to contribute to the wanted expansion, due to u the closeness to TP W (P ). This component is given by kvk sin(∠ (v, TS,x∗ )). The ratio between the angle ∠ (v, TS,x∗ ) and the one made by the directions defined by TS,x∗ and TS,y∗ is bounded below by a positive number Θ0 uniformly in M \V.

This means that, up to a constant, we may replace kvk sin(∠ (v, TS,x∗ )) by

kvk sin(∠ (TS,x∗ ,TS,y∗ )) which, by Lemma 3.1, verifies 2 kvk sin(arctan[h(1 − t) + κ ρ (S))]) ≤ kvk sin(∠ (TS,x∗ ,TS,y∗ )). Therefore, as the map z 7→ √ z is increasing, 1 + z2 κ ρ2(S) h(1 − t) + κ ρ2(S)) kvk ≤ kvk q p 2 2 1 + (κ ρ2(S))2 1 + [h(1 − t) + κ ρ (S)] 2 ≤ kvk sin(arctan[h(1 − t) + κ ρ (S))]) ≤ kvk sin(∠ (v, TS,x∗ )). n Let us first compute the effect on kvk of the action of D(ft) , n = j(S), ··· ,J(S). s Since |x∗|= 6 0−because S/∈ Wloc(P )−, the number of iterates necessary for S to leave V is at least i(S) = b− log |x∗| c. The above component of v is essentially log |µP | multiplied in each forward iteration by a factor |µP |, since we are in the linearized neighbourhood associated to P . Thus, at the moment ft(S) leaves V, the rate of expansion is bigger than 2 2 2 log |x∗| κ ρ (S) 1 κ ρ (S) κ ρ (S) − log |µ | i(S) |µP | P = ≤ |µP | p 2 2 |x | p 2 2 q 2 1 + (κ ρ (S)) ∗ 1 + (κ ρ (S)) 1 + (κ ρ2(S))

n On the other hand, the action of D(ft) for n= 0, ··· , j(S) must also be taken s into account. If S is in E and S does not belong to Wloc(P )−that is, if |y∗| 6= 0−and we iterate backwards until the orbit of S leaves V, we need a number of iterates that is bigger than n(S) = b log |y∗| c. Since t < 1, if a vector w belongs to log |λP | −m −m −m C(ft) (S), 2 (E2((ft) (S)))) \{0}, w makes an angle with T(ft) (S),y∗ which is less than a number Θt uniformly for S in E. The component of w that will grow more by positive iterates is given by

−m kwk cos(∠ (w, T(ft) (S),y∗ )) −m which is bigger than kwk cos(Θt). As before, iterating (ft) (S) forward till we reach S, the above component of w is multiplied by a factor |µP |, the inner expan- sion in V of the Anosov diffeomorphism f0. Thus, at the end of this piece of orbit, the rate of expansion is bigger than

log |y∗| log |µP | log |λ | log |λ | n(S) |µP | P cos(Θt) = |y∗| P cos(Θt) ≥ |µP | cos(Θt) Joining both estimates above (the ones corresponding to the path of the orbit while it enters V, passes by E and leaves V), we then get a lower bound for the rate of expansion of the norms of the vector v : after all these iterates, the norm of v is log |µP | 2 log |λ | 1 κ ρ (S) multiplied by a factor greater than cos(Θt)|y∗| P √ . |x∗| 1+(κ ρ2(S))2 780 MARIA CARVALHO

Moreover, since ρ (S) ≤ |x∗| and ρ(S) ≤ |y∗|, 2 2 log |µP | 1 κ ρ (S) log |µP | 1 κ ρ (S) log |λ | log |λ | |y∗| P p ≤ |y∗| P p ρ(S) 1 + (κ ρ2(S))2 |x∗| 1 + (κ ρ2(S))2

log |µP | log |µ | − − P 2 log |λP | 2 1 log |λP | 1 κ ρ (S) 1 1 κ ρ (S) p ≤ p ρ (S) ρ (S) 1 + (κ ρ2(S)))2 |y∗| ρ (S) 1 + (κ ρ2(S)))2

log |µ | 2 log |µ | P −1 κ [ρ (S)] ( P + 1) κ ρ (S) log |λP | [ρ (S)] = ρ (S) log |λP | . p1 + (κ ρ2(S)))2 p1 + (κ ρ2(S)))2

log |µP | Notice χ = = −1 since |µP | |λP | = 1. Therefore log |λP | κ ρ (S)(χ + 1) κ = p1 + (κ ρ2(S))2 p1 + (κ ρ2(S))2 κ ≥ p1 + 2κ ρ2(S) + (κ ρ2(S))2 κ = 1 + κ ρ2(S) because κ is non zero and ρ(S) 6= 0. As κ goes to +∞, ρ(S) goes to 0 (or in other terms, the neighbourhood E diminishes); since ρ(S) < 1, we finally conclude that, 1 lim ( )2 = +∞. S→R ρ (S) So by choosing κ big enough (or the neighbourhood E conveniently small), we manage a rate of expansion independent of t and bigger than 2, as promised. Notice the calculations above separated the orbit of the point S in three pieces: (1) the first one, while S enters in V till it reaches E, after which the selected log |µP | log |λ | 1 vector expanded at least τ1 = cos(Θt) |y∗| P = cos(Θt) ; |y∗| (2) the second one, corresponding to the itinerary in E until the departure from V, at the end of which we ensure that the vector expanded at least τ2 = 2 1 √ κ ρ (S) ; |x∗| 1+(κ ρ2(S))2 (3) the last one, associated to the itinerary out of V during which we are dealing with the rate of expansion of the Anosov diffeomorphism f0.

After Sublemma 1, what is missing in order to prove (2i.2) for ft, t0 ≤ t < 1, is to check that the estimates above confirm that the distortions on the geometry of the cone CS, 1 (E1(S)), during the passage within E due to action of the rotation θt, are compensated by the other components of the orbit of S.

Sublemma 2. For all t ∈ [t0, 1[ there are q(t) ∈ N and σ(t) > 1 such that q σ kvk ≤ kD(ft) (S)(v)k ∀ S ∈ M ∀ v ∈ CS, 1 (E1(S)) ∀ q ≥ q(t).

Proof. Denote by σ1 the rate of expansion of the Anosov diffeomorphism f0 along u 1 E and consider σ2(S) = [τ(S)] J(S) , where τ(S) comes from Sublemma 1 and is bigger than 2. σ2 measures the average expansion in each iterate while crossing V and Claim: σ2(S) is strictly bigger than 1. We first prove this claim. If there is a sequence of points Si such that σ2(Si) comes arbitrarily close to 1, with i going to +∞, then we must have J(Si) going to infinity with i. But then

1 1 N3(Si) J(Si) J(S ) J(S ) [ζ(t) σ1 − N3(Si)] i ≤ σ2(Si) = [τ(Si)] i HOMOCLINIC TANGENCIES 781 where ζ(t) is a uniform positive lower bound for the factor that multiplies the absolute value of the component of a vector of CSi, 1 (E1(Si)) along the direction of

TSi,x∗ while iterating inside E (since t < 1, ζ(t) is less than 1); and N3(Si) counts the iterates inside E. Therefore N3(Si) ζ (t) J(Si) σ1 ≤ σ2(Si) σ1 which converges to σ1 with i going to infinity because N3 is bounded for each t fixed. But σ1 > 1, hence σ2(Si) cannot approach 1. So the claim is valid. Denote byσ ˆ the number infS∈M {σ1, σ2(S)}, which is strictly bigger than 1 by earlier considerations, and consider a natural number q(t) verifying log( σˆ ) q(t) > 2( ζ(t) ) N logσ ˆ 0,t

q(t) ζ (t) 2 N0,t and σ(t) given by σ(t) = (ˆσ) ( σˆ ) . These choices were motivated by the region of influence of the saddle P , near which the orbits go before crossing E, and where in average the expansion rate is not less thanσ ˆ; and the long visits to the complement of V where the expansion rate is σ1.We will check that, for these values, Sublemma 2 holds. Notice first that, by definition of q(t) and the fact that ζ (t) 1 σˆ < σˆ < 1, we have

q(t) ( q(t) log(ˆσ) ) ( log(ˆσ) ) ) ζ(t) log( σˆ ) ζ(t) log( σˆ ) σ(t) > (ˆσ) q(t)( ) ζ(t) = (ˆσ) q(t)( ) ζ (t) σˆ σˆ ζ (t) log(ˆσ) q(t) = (ˆσ) q(t) exp{log( )( )} σˆ σˆ log( ζ (t) ) = (ˆσ) q(t) exp{− log(ˆσ) q(t)} = 1.

Fix S ∈ M and take a vector v in CS, 1 (E1(S)) and a natural number q bigger than q(t). The worst situation for the expansion of the norm of the iterates of v happens when the orbit of S crosses the problematic region E; and in this most adverse situation we already ensure that ζ(t) σ(t)|vk = ( )2 N0,t (ˆσ) qt kvk ≤ [ζ(t) 2 N0,t [(ˆσ) q(t)− 2 N0,t ]kvk ≤ kD(f ) q(S)(v)k. σˆ t While the orbit does not visit E, the rate of expansion is at least σ1 which is bigger than σ(t). This ends the proof of Sublemma 2. To complete the proof of Lemma 3.3, similar considerations have to be made concerning backward iterates of ft and the cones CS, 2(S,t)(E2(S)). The previous argument only used the saddle nature of P and the fact that |λP | |µP | = 1; therefore −1 we may apply it again to (ft) and P inside the neighbourhood V. Now the eigenvalues are 1 and 1 verifying 0 < 1 < 1 and 1 1 = 1. µP λP |µP | | λP | |µP | | λP | (c) After last section and the construction of the cones, the existence of non-zero Lyapounov exponents for f1 at Lebesgue almost every point is a consequence of [Wj]. Therefore the Pesin region has full measure and we may integrate the Os- eledec’s bundles associated to the non-zero exponents, obtaining stable and unstable manifolds Lebesgue almost everywhere. 782 MARIA CARVALHO

The manifolds mentioned in (c) are topologically transversal and locally only intersect at the defining point, that is, at Lebesgue almost all x we have s \ u Floc(x) Floc(x) = {x}. Besides, for t = 1, something else may be said. As already observed, along the orbits by f0 that do not enter E, the hyperbolicity is inherited by f1. So these points have stable and unstable manifolds. For the other orbits, except the one of R, we may use the (intersection of all iterates of) invariant cones to define a pair of subspaces that may be integrated producing a pair of foliations, F1 and F2. These are the stable and unstable leaves of each point. For the orbit of R, the foliations that are performing the tangency are the candidates to F1 and F2, although not defining a continuous extension of the previous family of leaves. These leaves are also topologically transversal and locally only intersect at the T considered point; that is, for each point x, we have F1, loc(x) F2, loc(x) = {x}. According to [W], since the perturbation of f0 is small and f0 is stable, the dif- feomorphism f1 is semiconjugated to f0. But the properties of the leaves F1, F2 yield more: f1 is in fact conjugate to f0. This conjugacy forwards to f1 topological properties of f0, like the denseness of the foliations F1 and F2 (which are sent by the conjugacy onto the stable/unstable manifolds of the Anosov diffeomorphism) and the transitivity of the dynamics. The existence of a conjugacy between f0 and f1 confirms what was already shown in [C] about the regularity of diffeomorphisms at the boundary of the stability components of an Anosov system: they may be non-hyperbolic, so the conjugacy may fail differentiability, but they keep a good memory of the initial point of the arc. Acknowledgments. Thanks are due to the referee for many helpful comments which did much to improve this paper.

REFERENCES . [C] Carvalho, SRB measures for n-dimensional derived from Anosov diffeomorphisms, Ergod. Th. & Dyn. Sys., 13 (1993), 21–44. [E] Enrich, “A Heteroclinic Bifurcation of Anosov Diffeomorphisms,” Ergod. Th. & Dyn. Sys., 18, (1998), 567–608. [BK] Burns and Katok, Infinitesimal Lyapunov functions, invariant cone families and sto- chastic properties of smooth dynamical systems, Ergod. Th. & Dyn. Sys., 14 (1994), 757–785. [L] Lewowics, Lyapunov functions and topological stability, J. Diff. Eqs., 38 (1980), 192– 209. [M] Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422–429. [NPT] Newhouse, Palis and Takens, Bifurcations and stability of families of diffeomor- phisms, Publ. Math. I.H.E.S., 57 (1983), 5–71. [NP] Newhouse and Palis, Hyperbolic nonwandering sets on two-manifolds, in “Dynamical Systems,” (ed. M. Peixoto), Acad. Press, (1973), 293–301. [NP1] Newhouse and Palis, Bifurcations of Morse-Smale dynamical systems, in “Dynamical systems,” (ed. M. Peixoto), Acad. Press, (1973), 303–366. [PT] Palis and Takens, Hyperbolicity and the creation of homoclinic orbits, Annals of Math., 125 (1987), 337–374. [W] Walters, Anosov diffeomorphisms are topologically stable, Topology, vol. 9 (1970), 71–78. [Wj] Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dyn. Sys., 5 (1985), 145–161.

Received July 1996; revised September 1997.