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 establish 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 subset of diffeomorphisms in SS1 so that either the finest dominated splitting is in one-dimensional subspaces or there exists an equidimensional cycle displaying 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 Mañéergodic closing lemma developed for the characterization 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 ∈ U there exists a homeomorphism hg : M → M such that hg conjugates the dynamics, that is, hg ◦ f = g ◦ hg. Mañé[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 splitting; Oseledets splitting; Lyapunov exponents; Periodic points. 1 s u TΩ(f)M = E ⊕ E and constants C > 0 and λ ∈ (0, 1) such that

n n −n n kDf (x) | s k ≤ Cλ and kDf (x) | 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 Höldercontinuous. 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 perturbation, 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 cycles characterize the discontinuity points of the Lyapunov exponent function with respect to the Lebesgue volume-measure. Finally, we justify that these equidimensional 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 diffeomorphism, we are interested in determining the optimal regularity of conjugacy classes that intersect both open sets in the twofold decomposition of structurally stable 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 ∈ (0, 1) so that n n n −1 n kDf (x) | s k ≤ Cλ and k(Df (x) | u ) k ≤ Cλ Ex Ex for every x ∈ 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’ theorem [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, ∀vi ∈ Ex \{0} 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 Poincarérecurrence 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 ∈ (0, 1) so that n n −1 n kDf |E1 k.k(Df |E2 ) k ≤ Cλ for every n ≥ 1. We can also change the metric in order to obtain C = 1. It is clear from the definition that any hyperbolic splitting is necessarily a dominated splitting. Definition 2.2. We say that an Axiom A diffeomorphism f has a one-dimensional finest dominated splitting if there exists a continuous Df-invariant decomposition 1 2 dim M TxM = Ex ⊕ Ex ⊕ · · · ⊕ Ex in one-dimensional subspaces at every x ∈ Ω(f) i i+1 such that Ex is dominated by Ex for every i = 1 ... dim M − 1. 3 It is clear from the previous definition that having a one-dimensional finest dominated splitting is a C1-open condition for Cr-diffeomorphisms (r ≥ 1). 2.2. Simple star diffeomorphisms and one-dimensional finest dominated splittings. In this section we provide a criterium over the periodic points for a structurally stable diffeomorphism to admit a one-dimensional finest dominated splitting. Definition 2.3. We say that an Axiom A diffeomorphism f is simple star if there exists an open C1-neighborhood U of f so that, for any g ∈ U all periodic points for g have simple spectrum. In fact if follows from the previous results by Bonatti, Gourmelon, Vivier [7] that periodic points are enough to determine dominated splittings at the closure of periodic points. Since periodic points are clearly dense for Axiom A diffeomorphisms then we obtain the following result. Theorem 1. If f is a simple star structurally stable diffeomorphism, then f admits a one-dimensional finest dominated splitting. Moreover, the Lyapunov spectrum is simple for all f-invariant ergodic probability measures and the map dim M L : M1(f) → R µ 7→ (λ1(f, µ), . . . , λdim M (f, µ)) is continuous, where M1(f) denotes the space of f-invariant probability measures. We observe that if the diffeomorphism f has a one-dimensional finest dominated splitting then every f-invariant probability measure admits d = dim M distinct Lyapunov exponents that differ by a definite amount. Thus, the following is a direct consequence of Theorem 1. Corollary 1. Let f be a structurally stable diffeomorphism. The following sen- tences are equivalent: (a) f is simple star; (b) f admits a one-dimensional finest dominated splitting; (c) there exists a constant c > 0 so that any f-invariant probability measure µ has dim M distinct Lyapunov exponents and λi+1(f, µ) − λi(f, µ) ≥ c > 0 for every i = 1 ... dim M − 1. 2.3. A dichotomy for structurally stable diffeomorphisms. In this subsection we prove a dichotomy of one-dimensional finest dominated splitting versus equidimensional cycles with robust different signatures for structurally stable diffeomorphisms. Let us introduce these notions more precisely. Let f be an Ax- iom A diffeomorphism and p be a hyperbolic periodic point of period π(p) ≥ 1. π(p) u u u Given the finest Df (p)-invariant dominated splitting Ep = E1 ⊕ · · · ⊕ Ek we define the unstable signature at p (and denote it by sgnu(p)) to be the k-uple u u (dim E1 ,..., dim Ek ). Stable signatures at periodic points are defined analogously and denoted by sgns. Unstable (resp. stable) signatures describe the existence of finer splittings in the unstable (resp. stable) bundle of periodic points. Definition 2.4. We say that a C1 Axiom A diffeomorphism f exhibits a equidimensional cycle with robust different signatures if there are two periodic points p, q in the same homoclinic class with different stable or unstable signatures, and this property holds under arbitrarily C1-small perturbations. 4 π(p) u Figure 1. Different linear behavior of Df (p) |Ep in the case of two- dimensional unstable subbundle corresponding to signatures (2), (2), (2) and (1, 1), respectively.

π(p) Action of Df (p) | u on the projective space in the Figure 2. Ep case of two-dimensional unstable sub-bundle ordered as Figure 1.

Figure 3. Equidimensional cycle between to ﬁxed points p1 and u u p2 such that sgn (p1) = (1, 1) and sgn (p2) = (2).

Since transversality is an open condition, it follows from the definition that the existence of an equidimensional cycle is an open property in the space of C1- diffeomorphisms. In dimension three, the robustness of the different signatures is associated to the existence a pair of periodic points so that the corresponding action of the dynamical cocycle on the projective space of one periodic point exhibits a dominated splitting while for the other point it presents a rotational effect (c.f. Figures 1-3 below). The fact that periodic points generating the cycle have different stable and/or unstable signatures correspond to the lack of domination present in the case of homoclinic tangencies. In the space of C1-diffeomorphisms the presence of homoclinic tangencies and heterodimensional cycles appears as a mechanism to create infinitely many sources 5 or sinks (see e.g. the Newhouse phenomenon in [15]). Roughly, if there exists a homoclinic tangency associated to a dissipative periodic point of saddle type then the lack of domination allows to perturb the system to mix the directions and to create a sink/source. The infinitely many sinks/sources arise associated to some phenomenon of persistence of tangencies. Motivated by this parallelism, we obtain the following characterization on the space of C1-structurally stable diffeomorphisms: Theorem 2. If f is a structurally stable diffeomorphism, then either f has a one- dimensional finest dominated splitting, or f can be C1-approximated by diffeomorphisms that exhibit a equidimensional cycle associated points with robust different 1 1 1 1 1 signatures. In other words, SS = SS1d ∪ SSc where both sets SS1d and SSc are C1-open sets. In what follows we explain the conceptual similarity between equidimensional cycles with robust different signatures and the notion of heterodimensional cycles for dynamics beyond uniform hyperbolicity. The following result asserts that either all periodic points have the same signature or (at least residually) there are infinitely many periodic points with distinct behavior. Theorem 3. If f is a structurally stable diffeomorphism that admits two periodic points p, q with robust different unstable signatures in a basic (hence transitive) piece Λ ⊂ Ω(f) then there exists a C1-open neighborhood U of f and a residual subset R ⊂ U such that for any g ∈ R we have that: u (1) there exists a countable infinite set of points P1 ⊂ P er(g)∩Λg with sgn (p1) = u sgn (p) for all p1 ∈ P1 and u (2) there exists a countable infinite set of points P2 ⊂ P er(g)∩Λg with sgn (p2) = u sgn (q) for all p2 ∈ P2, where Λg stands for the hyperbolic continuation of the set Λ. An application for structurally stable flows. In what follows we apply our results to the case of C1-structurally stable flows. By Hayashi [12], structurally stable flows are Axiom A and have the strong transversality condition. By Bowen and Ruelle [6] Axiom A flows have a finite Markov partition and are semiconjugate to suspension flows over a subshift of finite type. Moreover, the Poincarémap inherits the information of the Lyapunov exponents for both periodic points and all invariant measures: the Lyapunov exponents for the flow and for the Poincaré map differ from the integral of the return time function. The definitions from the discrete time setting extend to this setting in an analogous way. Since the C1- perturbation tools used in the proofs of Theorems 1 and 2 are available for flows then the following result is a consequence of the above mentioned theorems..

Corollary 2. If (Xt)t is a simple star structurally stable flow then (Xt)t admits a one-dimensional finest dominated splitting and the Lyapunov exponent functions vary continuously with the invariant probability measure. Moreover, we have the following dichotomy: if (Xt)t is a structurally stable flow then either (Xt)t admits 1 a one-dimensional finest dominated splitting, or (Xt)t can be C -approximated by flows that exhibit a equidimensional cycle associated points with robust different signatures. 2.4. Regularity of conjugacy classes. In the current subsection we describe the conjugacy classes (continuous, Hölderand Lipschitz) of structurally stable diffeomorphisms. In view of the result of An mentioned above, one is led to ask which 6 implications the dichotomy proven in Theorem 2 has for more regular conjugacy classes. We focus firstly on Lipschitz conjugacy classes. 1 1 It is not hard to check from the definition that the two C -open sets SS1d and 1 1 1 1 SSc are preserved by C conjugacies. In other words, if f ∈ SS1d is C -conjugate 1 1 to g then g ∈ SS1d (similarly to SSc ). Our interest is to prove a partial converse, that is, to show that smooth conjugacy classes cannot cross the boundary of the two open sets described in Theorem 2. First, we recall some preliminary notions. Given a periodic point p we say that f is C1-linearizable at p if there is an open 1 neighborhood Up of p ∈ M and an open neighborhood U˜p of 0 ∈ TpM and a C - π(p) π(p) diffeomorphism ψ : U˜p → Up so that ψ ◦ Df (p) = f ◦ ψ. This roughly means that, up to iteration by the period, the dynamical system is locally smoothly conjugate to its linear part in the neighborhood of the periodic point. While conditions for smooth linearization around periodic points have been much studied this will not be needed in full strength due to the Hölderregularity on the conjugacy given by Hartman-Grobman theorem (see e.g. Barreira, Valls [2]). Actually, a global bi- Hölderconjugacy class also preserves the proportion of growth/contraction given by invariant topological sub-manifolds that arise from Hartman-Grobman theorem. We are now in position to state our last main result of the subsection. Theorem 4. Let f ∈ Diff 1(M) be a structurally stable diffeomorphism that does 1 1 not belong to SS1d∪SSc . Then, the Lipschitz conjugacy class of f does not intersect 1 1 both open sets SS1d and SSc simultaneously. 3. Proofs This section is devoted to the proof of our main results. We shall use the following result from Bonatti, Gourmelon and Vivier: Theorem 3.1. [7] Let f : M → M be a diffeomorphism of a compact Riemannian manifold M. Then for any > 0 there are two integers m and n such that, for any periodic point x of period π(x) ≥ n: (1) either f admits an m-dominated splitting along the orbit of x or else; (2) for any neighborhood U of the orbit of x, there exists an -perturbation g of f in the C1-topology, coinciding with f outside U and on the orbit of x, π(x) and such that x is a source or a sink of g for which the differential Dgx has all eigenvalues real and with the same modulus. 3.1. Proof of Theorem 1. If f is a structurally stable diffeomorphism then it is an Axiom A diffeomorphism and, consequently, all periodic points are hyperbolic and P er(f) = Ω(f). We explore the hyperbolicity at periodic points. S Observe that `≥n P er`(f) = Ω(f) for any n ≥ 1 (here P er`(f) denotes the space of periodic points of period `). Using that a dominated splitting extends to the closure of a set and that f is a simple star diffeomorphism (thus cannot be C1-approximated by a diffeomorphism so that the Lyapunov spectrum at some periodic point is not simple) then, by Theorem 3.1, there exists m1, n1 ≥ 1 so that there exists an m1-dominated splitting T M = TS M = E ⊕ F . Ω(f) P er`(f) `≥n1 If E has dimension one we take E1 = E. Otherwise, repeating the procedure with Df |E there exists n2 ≥ 1 and m2 ≥ 1 (multiple of m1) and a m2-dominated splitting 1 2 T M = TS M = (Eˆ ⊕ Eˆ ) ⊕ F Ω(f) P er`(f) `≥n2 7 with 0 < dim Eˆ1, dim Eˆ2 < dim E. If Eˆ1 has dimension one we take E1 = Eˆ1. Proceeding recursively with E and F it follows that there exist m ≥ 1 and an 1 dim M m-dominated splitting TΩ(f)M = E ⊕ · · · ⊕ E . Thus f has a one-dimensional finest dominated splitting, which proves the first claim in the theorem. Now, we shall prove that the Lyapunov exponent function associated to f is 1 dim M continuous. In fact, let TΩ(f)M = E ⊕ · · · ⊕ E be the one-dimensional finest dominated splitting associated to f. Hence, it is clear from Birkhoff’s ergodic theorem that for any f-invariant and ergodic probability measure µ the Lyapunov exponents are given by the integrals Z λ (f, µ) = log kDf(x) | i k dµ, i Ex where i = 1,..., dim(M). By, the ergodic decomposition theorem the same property holds among the space of all f-invariant measures. Consequently, the map dim M L : M1(f) → R µ 7→ (λ1(f, µ), . . . , λdim M (f, µ)) is continuous, where M1(f) denotes the space of f-invariant probability measures endowed with the weak∗ topology. By the dominated splitting property it follows that the Lyapunov spectrum of every invariant measure µ is simple. This finishes the proof of the theorem.

1 3.2. Proof of Theorem 2. To prove the first part of the theorem, since SS1d and 1 1 SSc are open sets, we are reduced to prove that if f∈ / SS1d is a structurally stable diffeomorphism then f can be C1-approximated by diffeomorphisms that exhibit a equidimensional cycle associated points with robust different signatures. Let f be a structurally stable diffeomorphism that does not have a one-dimen- 1 1 sional finest dominated splitting nor can be C -approximated by elements of SS1d. Fix an arbitrary > 0. Assume for simplicity that f |Ω(f) is transitive (otherwise we shall consider each basic piece of the Axiom A diffeomorphism separately) and 1 k let TΩ(f)M = E ⊕ · · · ⊕ E be the finest dominated splitting for f where k < dim(M). By assumption there exists 1 ≤ i ≤ k so that dim Ei ≥ 2 and Ei is indecomposable, that is, it cannot be splitting into invariant subspaces exhibiting a dominated splitting. We shall assume that dim M = 3, Eu = E1, Es = E2, dim E1 = 2 and dim E2 = 1 since this case encloses all the difficulties. The general case where dim M ≥ 3 is analogous to this setting and, although combinatorically more subtle, it has the advantage that changing the restrictions to change the stable or unstable signature of periodic points are smaller. Since all periodic points are homoclinically related, we are reduced to prove first that there are (up to a C1-perturbation) two periodic points with robust different signatures. We organize our argument in three cases. π(p) Case 1: Df (p) | 1 has a complex eigenvalue for every p ∈ Per(f). Ep If this is the case then sgnu(p) = (2, 0) for every p ∈ Per(f). By Theorem 3.1, item (2), there is a diffeomorphism g, C1-arbitrarily close to f, and a periodic point π(p) p for which Dg (p) |E1 has two real eigenvalues with the same modulus. Since 1 this is a non-generic property, by a C -small perturbationg ˜ of g, the continuation pg˜ π(pg˜) of the periodic point p is so that Dg˜ (pg˜) |E1 has two real and simple spectrum. Sinceg ˜ can be obtained displaying some periodic point q in the basic piece such 8 π(q) u that Dg˜ (q) | 1 has a complex eigenvalue (hence sgn (q) = (2, 0) robustly), then Eq 1 g˜ ∈ SSc . π(p) Case 2: There are p, q ∈ P er(f) such that Df (p) | 1 has a complex eigenvalue Ep π(q) and Df (q) | 1 has two real eigenvalues. Eq π(q) If Df (q) | 1 has simple real spectrum we are done. Otherwise, since there are Eq periodic points whose diffeomorphism at the period has a complex eigenvalue in this case a simple and arbitrary small C1-perturbation g supported in a neighborhood π(q) of q in such a way that Dg (q) | 1 has real simple spectrum and, consequently, Eq an equidimensional cycle with different signatures robustly. π(p) Case 3: for all p ∈ Per(f) the linear map Df (p) | 1 has only real eigenvalues. Ep In this case there are three situations to consider, depending on the number of periodic points with two real equal eigenvalues (presenting a nilpotent part or not).

π(p) (a) all periodic points p ∈ Per(f) are so that Df (p) | 1 has real simple Ep u 1 spectrum (or, in other words, sgn (p) = (1, 1)). Since f∈ / SS1d then it follows from Theorem 1 that f is not a simple star diffeomorphism. Hence, it can be arbitrarily C1-approximated by a diffeomorphism g so that either there exists a π(p) periodic point p whose eigenvalues of Dg (p) |E1 has one complex eigenvalue or two real equal eigenvalues. Such perturbation can be performed in order to maintain a periodic point with real simple eigenvalues. Moreover, the period of p can be taken as larger as necessary. We are left to prove that a perturbation can be done in order to obtain complex eigenvalues (along E1) for some periodic point π(p) in the case when we have two real equal eigenvalues for Dg (p) |E1 . We will deal with this situation in the next two cases (b) and (c). π(p) (b) if there exists a periodic point p so that dim Ker(Df (p) | 1 −I) = 2 Ep π(p) then Df (p) | 1 has two real eigenvalues larger than one with no nilpotent part. Ep Hence, by a small C1-perturbation whose support is contained in a neighborhood 1 π(p) of p one can obtain a C -diffeomorphism g so that Dg (p) | 1 has a complex Ep eigenvalue (this can be obtained via Franks’ lemma and which can be written locally 1 by Dg = Rθ ◦ Df, where Rθ represents the rotation of angle θ on the plane E and 2 1 the identity in E ). We notice that g ∈ SSc . (c) if for all periodic points p have two real equal eigenvalues and are such that π(p) dim Ker(Df (p) | 1 −I) = 1 we shall borrow an idea from [5, Section 9] and Ep proceed as follows. In this case all of these periodic points have a nilpotent part and, in a suitable coordinate system, can be written by π(p) λp n(p) Df (p) |E1 = p 0 λp where n(p) ∈ R denotes the nilpotent part and λp > 1. Actually, since f is Axiom A then it admits a finite Markov partition P = {Pi}i. Let ni,j ≥ 1 be given such ni,j that f (Pi) ∩ Pj 6= ∅. Fix p ∈ P er(f) as above and assume, for simplicity, that p ∈ P1 is a fixed point. j Given n ≥ 1, let xn ∈ Λ be a periodic point so that f (xn) ∈ P1 for every 1 ≤ j ≤ n, n1,2+n n2,1+n1,2+n f (xn) ∈ P2 and xn = f (xn) ∈ P1. For any m 1 the f-invariant S S j compact set Ym = f (xn) inherits the behavior of Df(p). For periodic n≥m j∈Z 9 points q ∈ Ym with period π(q) ≥ 1 one has π(q)−` ! π(q) λp N(q) Df (q) | 1 ≈ · A(q, `), Eq π(q)−` 0 λp where A(q, `) is a 2 × 2 matrix corresponding to the ` = n2,1 + n1,2 iterates of transitions between P1 and P2, and the nilpotent part N(q) grows linearly with π(q). π(q) In particular, N(q)/λp → 0 as π(q) → ∞. Given ε > 0 choose q ∈ Ym with large π(q)−` 1 period so that N(q) ≤ λp ε. First we perform a ε-C -small perturbation along a piece of orbit of q so that the resulting diffeomorphism g satisfies, in appropriate coordinates,

π(q)−` ! π(q) λp 0 π(q)−` Dg (q) | 1 ≈ · A(q, `) = λ A(q, `). Eq π(q)−` p 0 λp If A(q, `) have complex eigenvalues we are done. Otherwise we notice that A(q, `) = Df `(f π(q)−`(q)) and that q 7→ A(q, `) is continuous. There exists an appropriate base, given by the Jordan canonical form, so that the matrix A(q, `) can be written of the form a 0 a b A := or A := 1 0 b 2 0 a for some real numbers a, b (depending on the ` iterates {f j(f π(q)−`(q)): j = 0 . . . ` − 1} of the orbit of the periodic point q). By the continuity of the matrix A(q, `) with the periodic point q the real numbers a, b can be taken bounded from above and below by uniform constants independent on the periodic point q and period π(q). The strategy is to perturb the derivative of the diffeomorphism g along a piece of orbit {f j(q): j = 0 ...N} (N ≥ 1 to be determined below) in such a way that the 1 π(q) resulting C -diffeomorphismg ˜ is so that Dg˜ (q) | 1 has complex eigenvalues. Eq 1 1 b Given ε > 0 take N(ε) ≈ ε (more precisely, N(ε) ≥ 1+ε log a in case of matrix b A1 and N(ε) ≥ ε in case of matrix A2) and take q ∈ Ym with period π(q) ≥ N(ε)+`. 1 By Franks’ lemma, in the case of a matrix of the form A1 then we perturb the C - diffeomorphism over the piece of orbit {f j(q): j = 0 . . . j = 0 ...N(ε) − 1} by concatenations of perturbations of the derivative to get a C1-diffeomorphism g whose derivative is obtained from the one of f by perturbations (C0-close to the identity) of the form 1 + ε 0 0 1 in an appropriate base. For the resulting diffeomorphism we get that N(ε) π(q) π(q)−` (1 + ε) a 0 π(q)−` b 0 Dg (q) |E1 ≈ λ ≈ λ . q p 0 b p 0 b

In the case of a matrix of the form A2 we proceed analogously perturbing the derivative of the diﬀeomorphism f over the piece of orbit {f j(q): j = 0 ...N(ε)−1} by perturbations of the form ε 1 − a 0 1 10 in an appropriate base. Thus we get π(q) π(q)−` a b − εN(ε) π(q)−` a 0 Dg (q) |E1 ≈ λ ≈ λ . q p 0 a p 0 a

π(˜q) Hence there exists a periodic pointq ˜ such that Dg˜ (˜q) | 1 has simple real spec- Eq˜ trum. This ﬁnishes the proof of the theorem.

1 Remark 3.1. It is not hard to construct examples that lie in the boundary of SS1d 1 but not on the closure of SSc due to the existence of nilpotent part. In particular, there are examples of C1 diffeomorphisms exhibiting periodic points (of low period) with equal eigenvalues that cannot be perturbed to create obtain a complex eigenvalue. 3.3. Proof of Theorem 3. Let U be a C1-open neighborhood of f so that for any g ∈ U the analytic continuations pg and qg of p and q, respectively, are well-defined, u u u u sgn (pg) = sgn (p) and sgn (qg) = sgn (q). 1 For any n ∈ N we define Rn to be the set of C diffeomorphisms g ∈ U such that g has n distinct periodic points with robust unstable signature equal to the one of pg and has n distinct periodic points with robust unstable signature equal to the one of qg. 1 Observe that, by hypothesis, R1 = U and so R1 is C -open and dense in U. For 1 1 any n ≥ 1 the set Rn is clearly C -open. If one shows that each Rn is C -dense then the residual set R = ∩nRn satisfies the statement of Theorem 3. The proof follows by induction and for simplicity reasons we will present the proof in dimension 3 assuming dim(Es) = 1 and dim(Eu) = 2. The difficulties arising in the general case are of combinatorial nature being the proof essentially 1 the same. Assume that Rk is C -dense for any k = 1, ..., n and fix > 0 and 1 any g ∈ Rn. We claim that there exists g1 ∈ Rn+1 which is -C -close to g. 1 The diffeomorphism g1 will be obtained from g by C -small perturbations at two periodic points in order to obtain one more point of each robust signature. By Theorem 3.1 we know that there exists `, m ∈ N such that, for any periodic point x of period π(x) ≥ ` either there exists an m-dominated splitting along the orbit of x or else for any neighborhood U of the orbit of x, there exists an /4-perturbation g1 of g in the C1-topology, coinciding with g outside U and on the orbit of x, and for π(x) which the tangent map (Dg ) | u has a real eigenvalue with multiplicity two. 1 x Ex u u Assume that without generality that sgn (pg) = (1, 1) and sgn (qg) = (2). The existence of qg implies that the set of periodic points x of period larger that ` and with an m-dominated splitting along the unstable fiber is not dense in Λg. Indeed, if this was not the case, then qg would have an m-dominated splitting on Eu . By the previous dichotomy that follows that q is accumulated by open sets qg g without periodic points displaying an m-dominated splitting on Eu. Thus we can pick a periodic point x distinct from the 2n marked periodic point for g in one of these open sets and with arbitrarily large period π(x). Using the dichotomy there 1 exists an /4-perturbation g1 of g in the C -topology, coinciding with g outside π(x) U and on the orbit of x, and for which the tangent map (Dg ) | u has a real 1 x Ex eigenvalue with multiplicity two. Finally, since the periodic point x can be chosen with an arbitrarily large period we can proceed as in Subsection 3.2 and perform 1 π(x) an /4-perturbation g of g in the C -topology so that (Dg ) | u has a complex 2 1 2 x Ex eigenvalue. Clearly, x has an unstable robust signature sgnu(x) = (2). 11 If g2 already have n + 1 distinct periodic points with robust unstable signature equal to the unstable signature of pg we are done. Otherwise, we are left to show 1 that g2 can be /4-approximated in the C -topology by g3 exhibiting one more distinct periodic point with robust unstable signature equal to the unstable signature of pg. Indeed, if all but n periodic points have unstable signature (2) and since there is no dominated splitting restricted to the unstable fiber and along these orbits we can perform an /4-approximated in the C1-topology in order to obtain a distinct periodic point x with robust unstable signature equal to sgnu(x) = (1, 1). 1 This completes the proof of the C -denseness of Rn in U and finishes the proof of the theorem. Remark 3.2. This construction to obtain periodic points of a certain signature has se same flavor of the construction of infinite sinks and sources for diffeomorphisms with homoclinic tangencies: the perturbation are localized in regions where there is a lack of domination among the stable/unstable fibers. Remark 3.3. The structural stability assumption in the statement of Theorem 3 is not used in its full strength. In fact we just have to assume that Λ is a isolated basic set. Moreover, a similar statement holds in the C1-residual subset of Kupka- Smale diffeomorphisms with dense periodic points in the non-wandering set (i.e. considering also the C1-residual subset given by the general density theorem). 3.4. Proof of Theorem 4. Let f ∈ Diff1(M) be a structurally stable diffeomor- 1 1 phism that does not belong to the open and dense set SS1d ∪ SSc . Let Cf denote the Lipschitz conjugacy class of f, which is clearly open. Assume, by contradic- 1 1 tion, that Cf intersects both open sets SS1d and SSc . In particular, there exists 1 a C -open neighborhood U of f so that for any g ∈ U ∩ Cf there exists a bi- Lipschitz homeomorphism hg such that hg ◦ f = g ◦ hg. In consequence, for any 1 1 given f1 ∈ SS1d ∩Cf and f2 ∈ SSc ∩Cf there exists a bi-Lipschitz homeomorphisms h : M → M so that h ◦ f1 = f2 ◦ h. 1 Using that f2 ∈ SSc we deduce that there exists p2 ∈ Per(f2) with period π π := π(p2) ≥ 1 for which Df2 (p2) has two eigenvalues with the same absolute −1 value. Clearly p1 = h (p2) ∈ Per(f1) is a periodic point of period π and the 1 C -diffeomorphisms f1 and f2 are locally conjugate (by h) in neighborhoods U1 and U2 for the points p1 and p2, respectively. Assume, without loss of generality, that Ui ⊂ M is an open neighborhood of pi (i = 1, 2) and that h(U1) = U2. Up to reducing the neighborhoods of the periodic points if necessary, it follows from the Hölderregularity of the linearization in Hartman-Grobman theorem (c.f.

[2]) that there are open neihbourhoods Vi ⊂ Tpi M of 0, constants αi > 0 and −1 αi homeomorphisms hi : Vi ⊂ Tpi M → Ui so that both hi, hi belong to C (M, R) for i = 1, 2. For any Df π(p )-invariant subspace F ⊂ Es it holds that h (F ) ⊂ U is a f π- i i pi i i i invariant Höldercontinuous submanifold, and a completely similar statement holds for F ⊂ Eu using (f π)−1. In consequence, if F ⊂ Es is the eigenspace associated pi i λ pi to some eigenvalue λ ∈ (0, 1) then there exists a uniform constant C > 0 so that 1 αiπn αi πn αiπn αi λi kvk ≤ dW i (fi (hi(v), pi) ≤ Cλi kvk C λ i for every v ∈ Tpi M ∩Vi and every n ≥ 1, where Wλ = hi(Fλ). Since we assumed f1 and f2 are bi-Lipschitz conjugate by h, the contraction and expansion rates along 1 2 invariant submanifolds must coincide. However, using that h(Wλ ) = Wλ the latter 12 implies that the norm of eigenvalues is preserved on bi-Lipschitz conjugacy classes. This proves that f1 and f2 cannot be Lipschitz conjugate, which finishes the proof of the theorem. Remark 3.4. (1) It is simple to provide a pair of C1-diffeomorphisms that are Hölderconju- gate but the conjugacy does not preserve the absolute value of eigenvalues at conjugate periodic points. (2) Similarly to the proof of Theorem 4, it is not hard to check that having all eigenvalues of distinct norm is a property which is preserved by bi-Hölder conjugacies. Thus, no bi-Hölderconjugacy class can intersect simultane- 1 1 ously the boundary of the two open sets SSc and SS1d.

Acknowledgements: This work was partially supported by CMUP (UID/MAT/ 00144/2013), which is funded by FCT (Portugal) with national (MEC) and Euro- pean structural funds through the programs FEDER, under the partnership agree- ment PT2020. PV was partially supported by a CNPq-Brazil postdoctoral fellow- ship at Universidade do Porto.

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Mario´ Bessa, Departamento de Matematica,´ Universidade da Beira Interior, Rua Marquesˆ d’Avila´ e Bolama, 6201-001 Covilha˜ Portugal. E-mail address: [email protected]

Departamento de Matematica,´ Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal E-mail address: [email protected]

Paulo Varandas, Departamento de Matematica,´ Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil & CMUP, University of Porto - Portugal E-mail address: [email protected]