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First Homoclinic Tangencies in the Boundary of Anosov Diffeomorphisms

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First Homoclinic Tangencies in the Boundary of Anosov Diffeomorphisms DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 4, Number 4, October 1998 pp. 765–782 FIRST HOMOCLINIC TANGENCIES IN THE BOUNDARY OF ANOSOV DIFFEOMORPHISMS 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 orbit, 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 stable manifold 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.
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