Robustly non-hyperbolic transitive symplectic dynamics

We construct symplectomorphisms in dimension $d\geq 4$ having a semi-local robustly transitive partially hyperbolic set containing $C^2$-robust homoclinic tangencies of any codimension $c$ with $0


Introduction
According to KAM theory, for symplectomorphisms close to an integrable one, an orbit, with large probability, belongs to an invariant torus and thus stays bounded for all time. Furthermore, orbits which are close, take a large time to escape from a neighborhood of these tori. However, in higher dimensional systems the action variables may, a priori, exhibit considerable change showing unbounded orbits [4]. Such behavior was coined with the term of Arnold diffusion or instability. Diffusion orbits can be constructed by using normally hyperbolic invariant laminations [10]. But, this kind of invariant laminations also allow the construction of robustly transitive sets which give a more complex mechanism of diffusion [21]. In this paper we continue the work of [21] and the study of these large (semi-local) transitive sets and provide new examples in this context. [21] were robustly non-hyperbolic. One of the classical tools to create robustly non-hyperbolic dynamics is via a heterodimensional cycle [8]. However in the symplectic case this idea fails since all hyperbolic periodic points have the same stability index and thus, there are no heterodimensional cycles. The other classical approach to destroy hyperbolicity is the construction of robust homoclinic tangencies [22]. In the symplectic setting known results on persistence of tangencies are restricted to area-preserving diffeomorphisms [12,13,20]. Motivated by this problem, we have developed in [7] a new method to construct robust homoclinic tangencies in higher dimensions. These techniques can be applied in the symplectic framework as we show in this paper. As a consequence, we extend the results of [21] showing that the semi-local transitive sets can be made robustly non-hyperbolic.

It was unknown if the examples constructed in
Assume that N and M are symplectic connected manifolds (not necessarily compact). Let F : N → N be a C r -symplectomorphism having a hyperbolic set Λ ⊂ N conjugated to a full shift with a big enough set of symbols (depending only on the dimension of M). In order to state our main theorem we also need the following notion: A diffeomorphism f has a homoclinic tangency of codimension c > 0 if there is a pair of points P and Q belonging to the same transitive hyperbolic set so that the unstable invariant manifold of P and the stable invariant manifold of Q have a non-transverse intersection Y of codimension c. That is, Y ∈ W u (P) ∩ W s (Q) and c = dim T Y W u (P) ∩ T Y W s (Q). Now we are ready to state the main result: Theorem A. There is an arc { f ǫ } ǫ≥0 of C r -symplectomorphisms of N × M such that f 0 = F × id and for ε > 0, any small enough C 2 -perturbation g of f ε has -a transitive set ∆ g homeomorphic to Λ × M, -a homoclinic tangency (in ∆ g ) of codimension c > 0.
The codimension c of the tangency can be chosen to be any integer 0 < c ≤ dim M/2.
As far as we know, the theorem above gives the first direct construction (i.e., not based on a dimension reduction argument using normally-hyperbolic manifolds) of robust tangencies for symplectomorphisms in higher dimensions.
Notice that f 0 cannot be a complete integrable system since the fiber map on M is the identity function. Even so, we construct nearby symplectomorphisms with a transitive set which projets onto M. Large transitive sets with this property had been previously constructed in [21] only close to integrable systems. The integrability could be a restriction on the manifold M since, as far as we know, it is unknown if a given symplectic structure admits integrable systems. Thus, Theorem A covers, a priori, new examples where diffusion orbits can be obtained. To prove this result we will use a different method to get orbits drifting along M. The approach of [21] consists in constructing transversal invariant tori for the fiber dynamics on M. Also the geometrical mechanism of diffusion developed in [11] is based on a similar idea. Here we introduce a different mechanism of propagation (drift) called globalization, which will be obtained by means of small translations in Darboux local charts, compatible with the symplectic structure.
The paper is organized as follows. First, in §2 we introduce blenders in a framework of normally hyperbolic invariant laminations. These invariant laminations give rise to natural skew-shifts over the space of sequences in a finite number of symbols, called symbolic skew-products. These kind of systems have also been studied in [9] and are related to the formation of stochastic diffusive behaviour for the generalized Arnold example. Next, in §3 we provide a criterium to construct robustly transitive symbolic skew-products. Finally, in §4 the main result is proven by combining the criteria for robust transitivity in §3 and for robust tangencies from [7].

Symbolic skew-products and blenders
Let A be a finite set (with at least two points), that we call an alphabet of symbols, and fix 0 < ν < 1 and 0 < α ≤ 1. Consider the product space Σ ≡ Σ(A , ν) def = A Z of the bi-sequences ξ = (ξ i ) i∈Z of symbols in A endowed with the metric In what follows M will denote a differentiable manifold (not necessarily compact and not necessarily boundaryless) of dimension c ≥ 1.
2.1. Symbolic skew-products. Given a compact set K in M, we consider the pseudometric in the set C 0 (M) of continuous functions of M given by Since M is σ-compact, there is a sequence of relatively compact subsets K n whose union is M and then we can endow C 0 (M) with the weak topology (also called compact-open topology) induced by the family of pseudometrics (1). That is, The set of skew-products. We consider skew-product homeomorphisms of the form where the base map τ : Σ → Σ is the lateral shift map and the fiber maps φ ξ : M → M are homeomorphisms of M. In order to emphasize the role of the fiber maps we write Φ = τ ⋉ φ ξ and call it a symbolic skew-product. When no confusion arises we also write M = Σ × M.
We introduce the set of symbolic skew-products with which we will work: the set of α-Hölder continuous symbolic skew-products of M = Σ × M. This is, the set of symbolic skew-products Φ = τ ⋉ φ ξ as in (2) such that • φ ξ depend α-Hölder with respect to ξ: there is a non-negative constant C 0 ≡ C 0 (Φ) ≥ 0 such that We will denote by S 0 (M) the set S(M) with C 1 -diffeomorphisms for fiber maps.
We define in S(M) the metric where the symbolic skew-products Φ = τ ⋉ φ ξ and Ψ = τ ⋉ ψ ξ belong to S(M) and An important class of α-Hölder continuous symbolic skew-products is the following: where γ andγ are given in (3). We denote by PHS(M) ≡ PHS α A ,ν (M) the set of partially hyperbolic symbolic skew-products. In addition, PHS 0 (M) = PHS(M) ∩ S 0 (M).

Stable and unstable sets for skew-products.
We define the local stable and unstable set of the lateral shift map τ : Σ → Σ at ξ ∈ Σ respectively as The (global) stable set of the skew-product map Φ : M → M at P ∈ M is defined as We define the (global) stable set of a compact Φ-invariant set, i.e. so that Φ(Γ) = Γ, by or equivalently as the set of the points of M so that its ω-limit is contained in Γ. The set Γ is called isolated (or maximal invariant set) if there is a compact neighborhood U of Γ, called the isolating neighborhood for Γ, such that every invariant subset of U lies in Γ. In such a case, we introduce the local stable set of Γ as the forward invariant set of Φ in the isolating neighborhood U, that is, Similarly W u loc (Γ) ≡ W u loc (Γ; Φ) and W u (Γ) ≡ W u (Γ; Φ) are, respectively, the local unstable set and the global unstable set of Γ. We have that Finally, given an S-perturbation of Φ, that is a symbolic skew-product Ψ close to Φ in the metric given in (5), we denote by Γ Ψ the maximal invariant set in U of Ψ. Although isolated sets vary, a priori, just upper semicontinuously by an abuse of terminology, we call Γ Ψ the continuation of Γ for Ψ. Each leaf of the partition W ss is called the local strong stable set. We define the (global) strong stable set of Φ at P as

Blenders.
In this subsection, we will first introduce the notion of hyperbolic set for symbolic skew-products homeomorphisms. After that we give the formal definition of blenders and finally we provide a criterion to obtain these local tools.
2.2.1. Hyperbolic sets. Fix ε > 0 small enough. We introduce the local stable set (of size ε) of Φ at P = (ξ, x) as The local unstable set (of size ε), denoted by W u ε (P), is defined analogously.
, Φ n (Q)) ≤ Kθ n for all P ∈ Γ, Q ∈ W s ε (P) and n ≥ 0; d(Φ −n (P), Φ −n (Q)) ≤ Kθ n for all P ∈ Γ, Q ∈ W u ε (P) and n ≥ 0; and there exists δ > 0 such that Every isolated hyperbolic set Γ for Φ is topologically stable [2]; i.e., there is an isolating neighborhood U of Γ such that for any homeomorphism Ψ which is C 0 near Φ, the restriction of Ψ to the maximal invariant set in U, is semiconjugate to the restriction of Φ to Γ.
We will now introduce the notion of index of an isolated transitive hyperbolic set Γ in our context. In the sequel we will assume that the topological dimension (in the sense of the Lebesgue covering dimension) of M cs From this assumption and being Γ transitive, the dimensions of M cs ε (P) and M cu ε (P) remain constant for any P ∈ Γ. Thus, we may define the cs-index and cu-index of Γ, denoted by ind cs (Γ) and ind cu (Γ) as these dimensions respectively. Notice that dim M = ind cs (Γ) + ind cu (Γ) and from the topological stability, the cs-index remains constant under small S-perturbations of Φ.
We say that two s-discs, D s 1 , D s 2 ⊂ W s loc (ξ) × M are close if they are the graphs of close α-Hölder functions. This proximity between discs allows us to introduce the following:

Definition 2.7 (open set of s-discs). We say that a collection of discs D s is an open set of s-discs in
Example of s-discs are the almost horizontal discs defined as follows: given δ > 0 and a Following [21,6,7], we introduce symbolic cs, cu and double-blenders.

The open set B is called a superposition domain and the open sets of discs D s and D u are called the superposition regions of the blender. Finally, the cs-blender (resp. cu-blender) with cs-index (resp. cu-index) is equal to dim M is called a contracting-blender (resp. expanding-blender).
Blenders are actually a power tool in partially hyperbolic dynamics when the superposition region contains the local strong stable/unstable set in the superposition domain. For this reason, without loss of generality, we will assume the following blender properties:

Similar conditions are also assumed for cu-blenders of partially hyperbolic skew-products.
We must show that property (B2) follows from the definition of a blender.
Proof. First of all, notice that the assumption (B1) and Definition 2.8 imply that A priori, the neighborhood of the S-perturbation of Φ where (6) holds depends on the s-disc W ss loc (P; Φ). However, this can be taken independent of the disc assuming that the disc belongs to a superposition subdomain is an open set whose closure is contained in B. For this reason, without loss of generality, we can assume that (B2) holds.

Blenders from one-step maps.
In order to provide a criterion to construct blenders we need the following definition.  We call cs-index (resp. cu-index) of the blending region B the cs-index (resp. cu-index) of Γ. As in the case of the blender, if its cs-index (resp. cu-index) is equal to dimension of M the blending region is called contracting (resp. expanding).
With the above terminology, the following result showed in [7, Corolory 5.3] gives a criterion to construct a blender. Then the maximal invariant set Γ in S Z × D is a cs/cu/double-blender of Φ whose superposition region contains the family of almost horizontal discs in Σ + S × B or/and almost vertical discs in Σ − S × B. Moreover, it also contains the family of local strong stable/unstable sets, i.e., (B1) holds.
Blending regions which cover an open ball B around a hyperbolic fixed point of a map φ can be easily constructed from a sufficient number of sets of the form φ i (B), where φ i is a translation of φ. This idea was developed in [21] and [7, Proposition 5.6], obtaining a blending region in local coordinates: Moreover, the cs-index of the blending region is equal to the s-index of the hyperbolic fixed point x.

Robust transitivity
We explain how blenders can be used to yield a S 0 -robust topologically mixing symbolic skew-product Φ of M = Σ × M. That is, for any S 0 -perturbation Ψ of Φ and for every pair of open sets U, V of M, there is n 0 > 0 such that Ψ n (U) ∩ V ∅ for all n ≥ n 0 . In particular topologically mixing implies transitivity.
3.1. Criterion to yield robust transitivity. One of the classical ways to create robustly transitive diffeomorphisms is to construct a map that robustly has a hyperbolic periodic point with dense stable and unstable manifolds. Then, using the inclination lemma (or λ-lemma) one concludes that the diffeomorphism is topologically mixing. In the symbolic setting an analogous result was proved in [6, Theorem 5.7] for fiber attracting/repelling hyperbolic fixed points. Here we extend this criterion to any hyperbolic periodic point. Later on we will use this result to construct robustly topologically mixing skew-products.
Then S 0 -robustly it holds that M = W u (P) for any periodic point P ∈ Γ. If Γ is a cu-blender satisfying (RT) for Φ −1 , then the stable set of any periodic point of Γ is S 0 -robustly dense in M.
Moreover, if Γ is either a double-blender, a contracting-blender or an expanding-blender satisfying (RT) for both Φ and Φ −1 , then the global stable and unstable sets of any periodic point of Γ are both S 0 -robustly dense. In this case, some power Φ k is S 0 -robustly topologically mixing which in particular implies that Φ is S 0 -robustly transitive.
Notice that the globalization property (7) is a necessary condition for robust transitivity. In order to prove Theorem 3.1 we need the following lemmas.

Lemma 3.2.
Consider a symbolic skew-product Φ and let Γ be an isolated hyperbolic transitive set of Φ ℓ for some ℓ ∈ N. Then for every periodic point P ∈ Γ, Combining the above facts we get the density of the unstable set, and an analogous argument concludes the density of the stable set. Proof. Without loss of generality, we can assume that P = (ξ, x) is a fixed point of Φ. Taking an integer m > 0 large enough, we get that Hence, Lemma 3.2 implies W u (P) ∩ Φ −n (U) ∅ for all periodic points P ∈ Γ. Consequently, S 0 -robustly it holds W u (P) ∩ U ∅ . From the invariance of the local strong stable partition, we get that W u (P) meets U, S 0 -robustly.
A similar argument holds assuming that Γ is a cu-blender and (RT) holds for Φ −1 . Thus according to Lemma 3.3 we conclude the theorem when Γ is a double-blender. If Γ is a contracting-blender (resp. expanding-blender), then the local stable (resp. unstable) set of any periodic point in Γ has dimension dim M. Hence, any u-disc (resp. s-disc) in the superposition region transversally meets the local stable (resp. unstable) set of these periodic points and thus (8) immediately holds for Φ −1 (resp. for Φ). Therefore, in this case, the same argument works assuming the globalization property (RT) for Φ and Φ −1 . This shows the density of both W s (P) and W u (P) and again, according to Lemma 3.3, we conclude the theorem.

Remark 3.4.
We actually get that Φ is S 0 -robustly topologically mixing if Γ has fixed points.

3.2.
Robust transitivity from one-step maps. Theorem 3.1 provides conditions to yield robustly transitive symbolic skew-products. In what follows, we will first translate these conditions to the particular case of one-step maps, Φ = τ ⋉ (φ 1 , . . . , φ d ). Afterwards, we will construct arcs of IFSs unfolding from the identity and satisfying these conditions. It is not difficult to see that (7) is equivalent to where B 0 is an open set in B such that The robustness in (9) means that for every compact set K ⊂ M there is a S 0 -neighborhood U of Φ such that K is contained in the projection on M of the closure of the union of forward Ψ-iterated of B 0 for all Ψ ∈ U . Observe that (9) holds if there exists n ∈ N such that This finite cover requires the compactness of M. In the non-compact case, (9) holds if there exists an increasing sequence of compact sets K i ⊂ M such that their union is M and for all i where recall that φ 1 , . . . , φ d + denotes the semigroup generated by the maps φ 1 , . . . , φ d .   Next, we will give another proof of the above statement, which will be useful later on for the symplectic setting. The following construction uses local tools in contrast to the global nature of Morse-Smale diffeomorphisms. The local perturbations will be translations of the identity map, compatible with symplectomorphisms. To show the result, we need the following lemma and notation. Given δ > 0 and a subset A of R c , we write Indeed, there is ρ > ε so that If B ρ ⊂ U 1 , as the translation directions u i only depend on the dimension c then Repeating the above procedure, since the radius of the covered ball B ρ is strictly increasing, by changing the center we can reach the boundary of U 1 and thus cover ∂U 0 .
Since v i (ε) tends continuously to zero (i.e. δ < ε goes to zero continuously), we get that T i (ε) tends continuously to the identity and conclude the proposition.
Proof of Proposition 3.7. By means of the well-known procedure of Milnor (see [ be a refinement of A . Relabeling the atlas if necessary, we assume that x ∈ U 0 10 . Lemma 3.8 provides (in local coordinates) C r -diffeomorphisms T 1 (ε 10 ), . . . , T m (ε 10 ) of U 10 , and an open ball B(ε 10 ) centered at x such that T ℓ (ε 10 )| ∂U 10 = id for all ℓ = 1, . . . , m and Observe that T ℓ j is well defined since for each j, U ij , i ∈ N are pairwise disjoint open sets and T ℓ (ε ij ) restricted to ∂U ij is equal to the identity for all ℓ = 1, . . . , m. Moreover, T ℓ j : ℓ = 1, . . . , m, j = 0, . . . , dim M + has forward globalization of the neighborhood B(ε 10 ) of any x. Similarly by the same procedure, we can assume that this semigroup also has backward globalization of B(ε 10 ). Finally, if ε ij → 0 then T ℓ j tends continuously to the identity and thus relabeling the maps, we have arcs of C r -diffeomorphisms T 1 = T 1 (ε), . . . , T s (ε), ε ≥ 0, s = (dim M + 1)m, satisfying the required properties.

3.2.3.
Arcs of robustly transitive one-step maps. The previous proposition allows us to construct an arc of robustly transitive symbolic skew-products. Proof. Consider x ∈ M. By means of an arbitrarily small perturbation of the identity map [14,15] we can create a map φ for which x is a hyperbolic fixed point. Hence, applying Proposition 2.12 we can get arcs of C r -diffeomorphisms φ 1 ≡ φ 1 (ε), . . . , φ k ≡ φ k (ε) homotopic to the identity as ε → 0 + , where k ≥ 2 and only depends on c and a cs-blending regions B in B 2ε (x). Without loss of generality, assume that B is a contracting/double-blending region. According to Proposition 3.7, there exist s ≡ s(c) ≥ 3 arcs of C r -diffeomorphisms T 1 ≡ T 1 (ε), . . . , T s ≡ T s (ε) homotopic to the identity as ε → 0 + so that T 1 , . . . , T s + has globalization of a small open ball B 0 ⊂ B. Adding these diffeomorphisms to the previous maps if necessary, B is a globalized contracting/double-blending region. Take d = s + k, which only depends on c, and set Hence Φ 0 = τ×id and according to Theorem 3.6, for any ε > 0, Φ ε is S 0 -robustly topologically mixing for any ε > 0. This completes the proof of the theorem.

Symplectic skew-products
By a symplectic manifold M we mean a manifold equipped with a closed non-degenerate differential two-form which is called the symplectic form. The nondegeneracy of the form implies that the space must be even-dimensional. We will create robust homoclinic tangencies inside semi-local transitive partially hyperbolic sets for symplectomorphisms, that is diffeomorphisms preserving the symplectic form. We will first show that the robust transitivity of Theorem 3.9 and robust tangencies constructed in [7] hold for symplectic symbolic skew-products, i.e., for symbolic skew-products where the fiber maps are symplectomorphisms. To do this, let us first recall the method to construct robust tangencies in symbolic skew-products. 4.1. Tangencies in symbolic skew-products. Following [7] we will introduce the notion of tangencies in symbolic skew-products and afterwards give a criterion for their construction. 4.1.1. The set of smooth symbolic skew-products. Since we will need to work with differentiable fiber maps, it will be useful to extend the set of symbolic skew-products to this setting.

Definition 4.1. For an integer r
In addition, PHS r (M) ≡ PHS r+α A ,ν (M) def = PHS(M) ∩ S r (M) for r ≥ 1. Finally, a partially hyperbolic skew-product is said to be fiber bunched if ν α < γγ.
We endow S r (M) with the metric Hence

Tangencies.
To define the notion of a tangency for symbolic skew-products we first need to introduce the notion of a tangent direction.

The maximum number of independent tangent directions at (ξ, x) is denoted by d T ≡ d T (ξ, x).
Now we are ready to give the definition of a tangency.
If Γ 1 = Γ 2 , the tangency is called homoclinic, and otherwise heteroclinic. The tangency (of dimension ℓ) is said to be S r -robust if for any small enough S r -perturbation Ψ of Φ has a tangency (of dimension ℓ) between the unstable set W u (Γ 1 Ψ ) and the stable set W s (Γ 2 Ψ ). The codimension of the tangency is defined as For the rest of this section, we will work in local coordinates and thus may assume that M = R c with c ≥ 2.

Cone fields in symbolic skew-products.
Consider an integer 1 ≤ ℓ ≤ c. An ℓ-dimensional vector subspace of R c is called a ℓ-plane. The Grassmannian manifold G(ℓ, c) is defined as the set of ℓ-planes in R c . A standard ℓ-cone in R c is a set of the form More generally, a ℓ-cone is the image of a standard ℓ-cone under an invertible linear map. In fact, any ℓ-cone C in R c induces an open set in G(ℓ, c), which we will continue denoting by C.

Definition 4.4 (stable and unstable cones). Let
and consider an open set B of M = Σ × R c . An ℓ-cone C uu in R c is said to be unstable for Φ on B if there is 0 < λ < 1 such that Similarly we define the stable ℓ-cone C ss for Φ on B.

Tangencies in one-step maps.
Let Φ = τ ⋉ (φ 1 , . . . , φ d ) ∈ PHS 1 (R c ) be a fiber bunched one-step map. That is, We fix 0 < ℓ < c and denotê In order to provide a criterion to get robust tangencies we need the following definitions: Here C uu is an unstable ℓ-cone for Φ = τ⋉(φ 1 , . . . , φ d ) on Σ×B. We say that the setB is a ℓ-tangency (of the blending region B). Similarly, we define a cu-blending region with a tangency of dimension ℓ.
Let A 1 and A 2 be two subsets ofM. Definition 4.6 (transition). We say that the semigroup φ 1 , . . . ,φ d + has a transition from A 1 to Using these terminologies, the following result from [7, Corollary 5.15] provides a criterion to construct robust tangencies: Then Φ has a S 1 -robust tangency of dimension ℓ between W u (Γ 1 ) and W s (Γ 2 ) where Γ 1 and Γ 2 are the maximal invariant sets in Σ × D 1 and Σ × D 2 .

Symplectic perturbations.
We would need the perturbative tools stated below. The following two remarks deal with local perturbations and are done in local Darboux charts, which are coordinates in which the symplectic form is written in the canonical way.

Remark 4.8 (Pasting lemma).
A symplectic pasting lemma [3, Lemma 3.9] states that given a C r -symplectic map with a periodic point, one can locally around the periodic point, glue any other C r -close symplectic map. Thus for example, one can modify the identity map to obtain locally any linear symplectic matrix.
The fact described in the next remark (see also [21, Proof of Theorem 3.16: pertubations]), says that one can locally perturb any symplectomorphism by translations. Let ω N and ω M be the symplectic two-forms on the symplectic manifolds N and M. Call π N and π M the projections of N × M onto N and M. Then a symplectic form on N × M is given by ω = π * N ω N + π * M ω M (π * indicating the pull-back). The following lemma allows for semi-local perturbations, and will use the fact that we are dealing with skew-product symplectomorphisms defined on a product manifold N × M. Lemma 4.10 (semi-local perturbations). Consider φ, a C r -symplectomorphism of M, with the additional hypothesis that φ is C r -symplectically isotopic to the identity. Let F be a C r -symplectomorphism of N and U 1 , U 2 be small enough neighborhoods contained in some Darboux chart such that U 1 ⊂ U 2 . Then there exists a C r -symplectomorphism f of N ×M such that f | U 1 ×M = F×φ and f | ∂U 2 ×M = F×id Proof. Let V be the Darboux chart of N so that U 1 , U 2 ⊂ V. We can suppose that V is, in fact, an open ball around the origin in R n where n is the dimension of N. We take open balls of radius r 1 < r 2 < r 3 such that Let φ : M → M be C r -symplectically isotopic to the identity, and assume that the isotopy is defined in the interval [0, r 3 ], writing φ(s, y), for s ∈ [0, r 3 ]. Moreover we can reparametrize so that φ(s, ·) = φ(·) for s ∈ [0, r 1 ] and φ(s, ·) = id for s ∈ [r 2 , r 3 ]. Consider a map f defined by Notice that f | U 1 ×M = F × φ and f | ∂U 2 ×M = F × id. This map can be extended C r -globally to N × M simply by defining it as f = F × id outside U 2 × M.
It is left to see that f preserves the form ω, and it is enough to show this locally in product Darboux coordinates V × W ⊂ N × M. We can write the form ω N in V as dx i ∧ dx j and the form ω M in W as dy i ∧ dy j , and then ω = dx i ∧ dx j + dy k ∧ dy ℓ where (x, y) = (x 1 , . . . , x n , y n+1 , . . . , y n+c ) are the coordinates in V × W. Then Since F is symplectic then det( Since for any x fixed, the map φ(||x||, ·) preserves the form dy k ∧ dy ℓ , this means that writing Dφ = (∂ x φ, ∂ y φ), the submatrix ∂ y φ is symplectic. Going back this implies that det( ∂ f k ∂y l ) = 1, when k > n, and thus f * ω = ω.

4.3.
Tangencies and transitivity for symplectic symbolic skew-products. In this subsection we want to get the following result: In order to prove the above theorem, according to Theorem 4.7 and Corollary 3.6 we need to create arcs of IFSs generated by symplectomorphisms having blending regions with tangency, transitions and globalization. Namely, fixing an integer 0 < ℓ ≤ c/2, we need to prove the following result: and having the following properties: for any ε > 0, there are bounded open sets B 1 ≡ B 1 (ε) and B 2 ≡ B 2 (ε) of M such that with respect to {φ 1 , . . . , φ d }, i) B 1 is a double-blending region with cs-index ℓ and having a ℓ-tangencyB 1 ; ii) B 2 is a cu-blending region with cu-index ℓ and having a ℓ-tangencyB 2 ; iii) φ 1 , . . . ,φ d + has transition fromB 1 toB 2 and/or transition fromB 1 toB 1 ; iv) B 1 is globalized by φ 1 , . . . , φ d + .
Now let us show the existence of such symplectic arcs. We will always work locally inside Darboux charts. To create local blending regions B 1 and B 2 , one can take a symplectic map with a local fixed hyperbolic point and compose the map with the necessary local translations as described in the Remark 4.9 where the directions of the translations come from Proposition 2.12. The hardest work is to obtain a ℓ-tangency for these local blending regions. First we need some basic facts from symplectic geometry: A symplectic vector subspace is one on which the symplectic form still restricts to a nondegenerate two-form. By a coisotropic vector subspace is understood a subspace of R c which contains its symplectic complement. Lemma 4.13. Let φ be a hyperbolic linear map of R c that preserves the symplectic form, i.e., a symplectic matrix, having a splitting in eigenspaces at the fixed point p = 0 of the form E s ⊕ E cu ⊕ E uu where the unstable direction is E u = E cu ⊕ E uu and 0 < ℓ = dim E uu < c. If ℓ is odd we ask that E uu is a coisotropic subspace and a symplectic subspace if ℓ is even. On the other hand, Dφ(p) induces a map A on G(ℓ, c) given by A(E) = Dφ(p)E. Since φ is C 2 , this map has a hyperbolic attracting fixed point E uu . Then, as in Proposition 2.12, there are arcs T 1 ≡ T 1 (ε), . . . , T k 2 ≡ T k 2 (ε) of translations on G(ℓ, c) homotopic to the identity and a neighborhood G ≡ G(ε) of E uu in G(k, c), such that {F j (G) : j = 1, . . . , k 2 } is an open cover of G where F j = T j • A. Each translation map F j in the Grassmannian corresponds to a map of the form A j · Dφ(p) in the tangent bundle where A j is an orthogonal matrix and A j − id → 0 as ε → 0 for all j = 1, . . . , k 2 . Moreover, we can take G small enough so that G ⊂ C uu where C uu is a unstable ℓ-cone around E uu for φ. Taking A j − id < ǫ/ φ and φ ij = A j · φ + c i then φ ij − f i < ǫ and so The translations c i preserve the symplectic form, and so we only need to show the matrix A j can be taken symplectic. Suppose for now that ℓ is even and so E uu is a symplectic subspace. Observe that the symplectic subspaces form an open and dense subset in the set of ℓ-planes. Then we can take the ℓ-cone C uu around E uu in G(ℓ, c) as an open set of symplectic subspaces. Since A j is arbitrarily close to the identity, we may assume that E = A j E uu is in C uu and so is also a symplectic subspace. As symplectic matrices act transitively on symplectic vector subspaces [24, Cap. V], we can take a symplectic matrixÃ j , close to the identity, such thatÃ j E uu = E. Without loss of generality we assumeÃ j = A j . If ℓ is an odd number, we proceed similarly. Since coisotropic subspaces are an open an dense set of G(ℓ, c) where the symplectic matrices also act transitively [19,Exer. 2.33], then in the same manner as before, we can take A j as a symplectic matrix.
Finally, by construction, the maps φ ij induce a set of mapsφ ij onM so that {φ ij (B)} is an open cover of the closure ofB = B × G. Moreover, since A j is an orthogonal matrix then the hyperbolicity of the blending regions still holds. That is, B is also a cs-blending region with respect to {φ ij : i = 1, . . . , k 1 , j = 1, . . . , k 2 } with cs-index equal to dim E s and as well is a ℓ-tangency (the cs-blending regionB onM). This completes the proof.
Notice that, by means of an arbitrarily small perturbation of the identity map, we can create a map φ for which x is a hyperbolic fixed point. Thus the above construction can be done homotopic to the identity.
Proof of Proposition 4.12. To make the transition map fromB 1 toB 2 , the blending regions can be chosen inside the same Darboux chart in such a manner so that the transition map is simply a translation. On the other hand the transition map fromB 1 toB 1 , can be chosen as a local linear symplectic map that has a fixed point inside the double-blending region and takes the unstable cone into the stable one over several iterates, or for example maps E uu into E ss . Recall that these subspaces can be chosen to be symplectic or coisotropic on which the symplectic matrices act transitively [ To globalize the blending region B 1 , observe that the C r -diffeomorphisms constructed to prove Theorem A are locally small translations. Thus again using Darboux coordinates and a Hamiltonian bump function as in the Remark 4.9, one can glue the maps required for the Lemma 3.8 and Proposition 3.7. The Milnor atlas required for the globalization has to be taken subordinate to the Darboux atlas.
The proof of Theorem 4.11 is now complete.

Symplectic realization: proof of Theorem A.
Recall that our goal is to construct arcs of C r -symplectomorphisms f ε of N × M isotopic to f 0 = F × id having robust homoclinic (or equidimensional) tangencies of codimension ℓ ≤ dim(M)/2. From Theorem 4.11, suppose Φ ε = τ ⋉ (φ 1 , . . . , φ d ) ∈ PHS 1 (M) is an arc of robustly transitive symplectic symbolic skewproducts with a robust homoclinic tangency of dimension ℓ. It is desirable to obtain a one-parameter family of diffeomorphisms f ε satisfying f ε | R i ×M = F × φ i for i = 1, . . . , d where R i is the Markov partition of the set Λ. The Markov partition must be small enough so that each rectangle R i is inside the Darboux chart U i . If the initial set Λ is conjugated to a shift of d symbols, the pieces of the new partition would correspond to cylinders C i in the shift. A cylinder consists of all sequences with a given block of indices fixed around the zeroth coordinate, the length of the cylinder being the size of the given block. With each U i we associate the map φ k i where the index k i corresponds to the zeroth index of the cylinder C i .
Since U i are disjoint, we may apply Lemma 4.10 inductively with respect to the maps φ k i , and thus obtain a symplectomorphism f ε in N × M satisfying Then by construction the dynamics over Λ × M will be conjugated to Φ ε . Using this conjugation one can obtain robust tangencies and transitive sets for the map f ε , exactly as was proven in [7, Sec. 6.1]. This concludes the proof of Theorem A.