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Structural Stability Revisited: Finest Dominated Splittings and Equidimensional Cycles

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Structural Stability Revisited: Finest Dominated Splittings and Equidimensional Cycles STRUCTURAL STABILITY REVISITED: FINEST DOMINATED SPLITTINGS AND EQUIDIMENSIONAL CYCLES M. BESSA, J. ROCHA, AND P. VARANDAS Abstract. In this paper we revisit the structural stability questions and es- tablish a sharper characterization of the open set SS1 of C1-structurally stable diffeomorphisms on a compact Riemmanian manifold. We prove that there are 1 1 1 two open subsets SSc and SS1d whose union is dense in SS and dense sub- set of diffeomorphisms in SS1 so that either the finest dominated splitting is in one-dimensional subspaces or there exists an equidimensional cycle dis- playing robust distinct signatures. In the conservative setting, the space of one-dimensional finest dominated splitting consists on the continuity points of the Lyapunov exponents. These results lie ultimately in the analysis of the eigenvalues and finest dominated splitting over the periodic points. Finally, 1 we prove that the conjugacy of a pair of diffeomorphisms, one in SSc and the 1 other in SS1d, cannot be taken Lipschitz continuous. In consequence, smooth conjugacy classes are always strictly contained in any of these sets. 1. Introduction A global view of dynamical systems has been one of the leading problems con- sidered by the dynamical systems community. Based on the pioneering works of Peixoto and Smale, a conjecture proposed by Palis in the nineties has constituted a route guide for a global description of the space of dynamical systems. This program, that roughly describes complement of uniform hyperbolicity as the space of diffeomorphisms that are approximated by those exhibiting either homoclinic tangencies or heteroclinic cycles, has been completed with much success in the C1- topology, where perturbation tools like the Pugh closing lemma, Franks' lemma, Hayashi's connecting lemma or Ma~n´eergodic closing lemma developed for the char- acterization of structural stability are available (see e.g. [13, 12, 18, 10, 11, 17] and references therein). The characterization of struturally stable diffeomorphisms has been a problem that drawn the attention of many researchers and contributed in a large sense to the development of hyperbolic dynamical systems. One of our purposes in the present work is to establish a dichotomy similarly flavored to the Palis conjecture in the context of structurally stable diffeomorphisms. Recall that a C1-diffeomorphism f is called structurally stable if there exists a neighborhood U of f in the C1- topology such that for any g 2 U there exists a homeomorphism hg : M ! M such that hg conjugates the dynamics, that is, hg ◦ f = g ◦ hg. Ma~n´e[13] proved that structural stability is equivalent to Axiom A and strong transversality condition. In particular, if f is structurally stable, then there exists a Df-invariant decomposition Date: May 22, 2015. Key words and phrases. Structural stability; Lipschitz stability classes; Finest hyperbolic split- ting; Oseledets splitting; Lyapunov exponents; Periodic points. 1 s u TΩ(f)M = E ⊕ E and constants C > 0 and λ 2 (0; 1) such that n n −n n kDf (x) j s k ≤ Cλ and kDf (x) j u k ≤ Cλ Ex Ex for every n ≥ 1 and every x in the non-wandering set Ω(f), and Ω(f) = Per(f). Maps displaying previous property are called hyperbolic diffeomorphisms. More recently, An [1] proved that a C2 diffeomorphism f satisfies Axiom A and the strong transversality condition if and only if any C1 diffeomorphism g sufficiently C1 close to f is conjugate to f by a homeomorphism which is H¨oldercontinuous. Although the uniform geometric structures of invariant manifolds present in hyperbolic diffeomorphisms are a basic ingredient in achieving several core results and are quite well established, some important questions on the regularity of finer dynamical properties still remain to be answer. Motivated by the analysis of the regularity of the Lyapunov exponents and a better understanding of the regularity of conjugacy classes one can expect that, similarly to the proof of the stability conjecture, periodic orbits and their eigenvalues should play a key role. We say that a periodic point p of period π(p) ≥ 1 for a diffeomorphism f has simple spectrum if all the eigenvalues of Df π(p)(p) are real and distinct. Since all periodic points for an Axiom A diffeomorphism have the same index and are homoclinically related, then we will attribute them a signature, which consists of an ordered list of the dimensions of their finest dominated splittings. Moreover, we will say that two periodic points p; q have different signatures if the Oseledets splittings for Df π(p)(p) and Df π(q)(q), which consist of the finest dominated spittings at the points, are distinct. In the first part of the paper we study mechanisms to deduce simple Lyapunov spectrum for all invariant measures assuming the same property at periodic points, in the spirit of the previous works of Bonati, Gourmelon, Vivier [7], Cao [8] and Castro [9]. On the one hand we prove that, if all periodic points of a structurally stable diffeomorphism f have simple spectrum and the same property holds in a C1 neighborhood of f (c.f. Definition 2.3) then f has a finest dominated splitting into one-dimensional sub-bundles (see Theorem 1). In this case all Lyapunov exponent functions of f are continuous in the weak∗-topology. On the other hand, if periodic points miss to have simple spectrum robustly, then, by a C1-arbitrarily small per- turbation, one can create an equidimensional cycle associated to a pair of periodic points which robustly exhibit different signatures (see Subsection 2.3). Further- more, in the volume-preserving framework and under additional conditions, we can use Bochi and Viana characterization of continuity points for Lyapunov exponents (see [3]) to prove that the existence of the above mentioned equidimensional cy- cles characterize the discontinuity points of the Lyapunov exponent function with respect to the Lebesgue volume-measure. Finally, we justify that these equidimen- sional cycles associated to periodic points with different signatures play a similar role to the one of tangencies and heterodimensional cycles by its instability charac- ter. In fact, the existence of homoclinic tangencies or heterodimensional cycles is often associated to the so-called Newhouse phenomenon of persistence of infinitely many sources or sinks (see e.g. [14, 15, 4] and references therein). In this setting, the existence of a equidimensional cycle associated to periodic points with different signatures can be used to generate (by perturbation) infinitely many periodic points with any of the signatures of the generating periodic orbits (c.f. Theorem 2). 2 In the second part of the paper we study the regularity of the conjugacies among structurally stable diffeomorphisms. A natural question is whether the conjugacy classes preserve the two open sets of the dichotomy of Theorem 1. While it is clear that smooth conjugacies preserve the finest dominated splitting of an Axiom A dif- feomorphism, we are interested in determining the optimal regularity of conjugacy classes that intersect both open sets in the twofold decomposition of structurally sta- ble diffeomorphisms: finest dominated splitting into one-dimensional sub-bundles 1 (SS1d) or else existence of equidimensional cycles with robust different signatures 1 (SSc ). We prove that conjugacies at boundary points lack to have Lipschitz regu- 1 larity and, consequently, Lipschitz conjugacy classes are contained either in SS1d 1 or SSc . In fact we prove that the conjugacies at boundary points cannot be even Lipschitz continuous at periodic points (c.f. Theorem 4). 2. Preliminaries and statement of the main results 2.1. Hyperbolic, Oseledets and finest dominated splittings. Let f be a structurally stable diffeomorphism on a compact Riemmanian manifold M. Let Per(f) denote the set of periodic points for f and Ω(f) ⊂ M denote the non- wandering set of f. We say that a C1-diffeomorphism f is Axiom A if Per(f) = Ω(f) and Ω(f) is a uniformly hyperbolic set: there is a Df-invariant splitting s u TΩ(f)M = E ⊕ E and constants C > 0 and λ 2 (0; 1) so that n n n −1 n kDf (x) j s k ≤ Cλ and k(Df (x) j u ) k ≤ Cλ Ex Ex for every x 2 Ω(f) and n ≥ 1. We can always change the metric in order to obtain s u C = 1. We refer to TΩ(f)M = E ⊕ E as the hyperbolic splitting associated to f. If µ is an f-invariant probability measure, then it follows from Oseledets' theo- rem [16] that for µ-almost every x there exists a decomposition (Oseledets splitting) 1 2 k(x) TxM = Ex ⊕ Ex ⊕ · · · ⊕ Ex and, for 1 ≤ i ≤ k(x), there are well defined real numbers 1 n i ~ λi(f; x) = lim log kDf (x)vik; 8vi 2 Ex n f0g n→±∞ n called the Lyapunov exponents associated to f and x. It is well known that, if µ is ergodic, then the Lyapunov exponents are almost everywhere constant and k(x) = k is constant. In this case the Lyapunov exponents are denoted simply by λi(f; µ). The Lyapunov spectrum of a probability measure is the collection of all its Lyapunov exponents. Given an f-invariant and ergodic probability measure µ by Poincar´erecurrence theorem one has supp(µ) ⊂ Ω(f) and, consequently, the 1 2 k s u decomposition TxM = Ex ⊕ Ex ⊕ · · · ⊕ Ex is finer than TxM = Ex ⊕ Ex which implies that µ is hyperbolic (i.e. has only non-zero Lyapunov exponents). Definition 2.1. Given a Df-invariant decomposition F = E1 ⊕ E2 ⊂ TM we say E1 is dominated by E2 if there exists C > 0 and λ 2 (0; 1) so that n n −1 n kDf jE1 k:k(Df jE2 ) k ≤ Cλ for every n ≥ 1.
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