$C^{r}-$prevalence of stable ergodicity for a class of partially hyperbolic systems

We prove that for $r \in \mathbb{N}_{\geq 2} \cup \{\infty\}$, for any dynamically coherent, center bunched and strongly pinched volume preserving $C^r$ partially hyperbolic diffeomorphism $f \colon X \to X$, if either (1) its center foliation is uniformly compact, or (2) its center-stable and center-unstable foliations are of class $C^1$, then there exists a $C^1$-open neighbourhood of $f$ in ${\rm Diff}^r(X,\mathrm{Vol})$, in which stable ergodicity is $C^r$-prevalent in Kolmogorov's sense. In particular, we verify Pugh-Shub's stable ergodicity conjecture in this region. This also provides the first result that verifies the prevalence of stable ergodicity in the measure-theoretical sense. Our theorem applies to a large class of algebraic systems. As applications, we give affirmative answers in the strongly pinched region to: 1. an open question of Pugh-Shub in \cite{PS}; 2. a generic version of an open question of Hirsch-Pugh-Shub in \cite{HPS}; and 3. a generic version of an open question of Pugh-Shub in \cite{HPS}.

Smooth ergodic theory, that is, the study of statistical and geometric properties of measures invariant under a smooth transformation or flow, is a much studied subject in the modern dynamical systems. It has its root in Boltzmann's Ergodic Hypothesis in the study of gas particles back in the 19 th century. Ever since Birkhoff's proof of his ergodic theorem, there has been a constant interest in understanding the genericity of ergodic systems. The pioneering work of A. Kolmogorov (Anosov diffeomorphisms). Given a compact Riemannian manifold X, a diffeomorphism f ∈ Diff 1 (X) is called uniformly hyperbolic or Anosov if there exists a continuous splitting T X = E s f ⊕E u f of the tangent bundle into Df -invariant subbundles and constantsχ u ,χ s > 0 such that for any x ∈ X, we have For the next nearly thirty years after Anosov-Sinai's work, uniformly hyperbolic systems remained the only examples where ergodicity was known to appear robustly, a property which is also called "stable ergodicity": we say that a C 2 volume preserving diffeomorphism f is stably ergodic if any volume preserving g sufficiently close to f in the C 2 topology is also ergodic. A breakthrough came when M. Grayson, C. Pugh and M. Shub [22] gave the first non-hyperbolic example of a stably ergodic system, i.e., the time-one map of the geodesic flow on the unit tangent bundle of a surface of constant negative curvature. Such systems are special cases of partially hyperbolic systems, which are defined as follows.
Definition 1.2 (Partially hyperbolic diffeomorphisms). Given a smooth Riemannian manifold X, a C 1 diffeomorphism f : X → X is called partially hyperbolic if its C 1 norm is uniformly bounded and there exist a nontrivial continuous splitting of the tangent bundle into Df -invariant subbundles, T X = E s f ⊕ E c f ⊕ E u f , and continuous functionsχ u ,χ s : X → R >0 ,χ c ,χ c : X → R, such that (1.3) −χ s <χ c ≤χ c <χ u , and for any x ∈ X, We set E cs f := E c f ⊕ E s f and E cu f := E c f ⊕ E u f . Partially hyperbolic systems have served as the principal source for finding stably ergodic systems (for other examples, see [9,12,41]). They also appear in the study of SRB measures, statistical mechanics, rigidity theory and homogeneous dynamics. Based on [22] and other results, Pugh and Shub formulated the following fundamental conjecture: Conjecture 1.3 (Pugh-Shub's Stable Ergodicity Conjecture, [32]). Stable ergodicity is C r -dense among the set of C r volume preserving partially hyperbolic diffeomorphisms on a compact connected manifold, for any integer r ≥ 2.
Since its introduction, this conjecture and related questions on stable ergodicity have been extensively studied, for instance in the following series of works [33,15,20,39,38,37,14,4,5]. We will later elaborate on the connections between them in Subsection 1.1. We mention that Conjecture 1.3 is far from being solved: results directly related to Conjecture 1.3 are only known for dim E c f = 1. 1 Our main result (Theorem E), will be given in Section 2; as we will see, we actually obtained prevalence in Kolmogorov's sense, a notion which is much stronger than density. In the next section, we will state Theorem A, a corollary of Theorem E, to help the reader understand the main features of our result.
1.1. Stable ergodicity and accessibility. In [33], the authors proposed a route to prove the Stable Ergodicity Conjecture. They divided the conjecture into two parts, using a geometric notion originating in an argument due to E. Hopf [24].
Let f be C r partially hyperbolic diffeomorphism of a smooth compact Riemannian manifold X, r ∈ N ≥1 ∪{∞}. It is well-known (see [25]) that E s and E u uniquely integrate to continuous foliations W s f and W u f respectively, called the stable and unstable foliations. For any x ∈ X and * = s, u, the leaf of W * f through x, denoted by W * f (x), is an immersed C r -manifold, and f (W * f (x)) = W * f (f (x)). If f ∈ PH 2 (X), the transverse regularity of W s f and W u f is Hölder (see [35]).
Definition 1.4 (Accessibility). An su-path of f is a path obtained by concatenating finitely many subpaths, each of which lies entirely in a single leaf of W s f or W u f . The map f is said to be accessible if any two points in X can be connected by some su-path. We say that f is (C 1 -)stably accessible if there exists U, a C 1 -open neighbourhood of f , such that any g ∈ U is accessible. Conjecture 1.5 (Accessibility implies ergodicity). Essential accessibility implies ergodicity among C 2 volume preserving partially hyperbolic diffeomorphisms.
In comparison, there is a paucity of progress towards Conjecture 1.6. When the center dimension is one, Conjecture 1.6 was proved by F. Rodriguez-Hertz, M.A. Rodriguez-Hertz and R. Ures in [38]. It is still open for any dim E c > 1. To describe the current state of Conjecture 1.6, we mention several related results, which were obtained among certain classes of systems.
• K. Burns and A. Wilkinson [15] proved a version of Conjecture 1.6 for compact group extensions over Anosov systems. • In a recent paper [26], V. Horita and M. Sambarino obtained some C rdensity result for a class of partially hyperbolic systems with dim E c = 2 and uniformly compact center foliations (see Definition 1.10). • Another C r -density result for partially hyperbolic systems with dim E c = 2 was obtained recently by A. Avila and M. Viana [6] using a very different method. • Z. Zhang [43] recently proved C r -density of C 2 -stable ergodicity for a class of skew products over Anosov maps, satisfying pinching, bunching conditions with certain type of dominated splitting in the center subspace. The difficulty of Conjecture 1.6 is mainly due to the C 2 -smallness of the perturbation. In fact, the C 1 -density of stable accessibility was already proved by D. Dolgopyat and A. Wilkinson [20] in 2003. There was a line of research focused on the C 1 version of Conjecture 1.3. In the case where dim E c = 1, 2, this was proved in [11] and [37]. Recently, the C 1 -version of Conjecture 1.3 was completely solved by A. Avila, S. Crovisier and A. Wilkinson [5]. 2 As the main result of this paper, we will verify C r -density of stable ergodicity in C 1 -neighbourhoods of two classes of partially hyperbolic systems, defined by some technical conditions. Let us first recall some notions needed to state our result.
A related notion is the following. Definition 1.9 (Dynamical coherence, plaque expansiveness). We say that a partially hyperbolic system f is: • dynamically coherent (see [25]) if E cs f , resp. E cu f , integrates to a f -invariant foliation W cs f , resp. W cu f , called the center-stable foliation, resp. the centerunstable foliation. In this case, for any x ∈ X, we let W c f (x) := W cs f (x) ∩ W cu f (x); the collection of all such leaves forms a foliation W c f , called the center foliation, which integrates E c f , and subfoliates both W cs f and W cu f (see [16]); • plaque expansive (see [25,Section 7]) if f is dynamically coherent and there exists ε > 0 with the following property: if (p n ) n≥0 and (q n ) n≥0 are εpseudo orbits which respect W c f and if d(p n , q n ) ≤ ε for all n ≥ 0, then q n ∈ W c f (p n ). It is known that plaque expansiveness is a C 1 -open condition (see Theorem 7.4 in [25]). Definition 1.10 (Uniformly compact foliation). A foliation is uniformly compact if all the leaves are compact, and the leaf volume of the leaves is uniformly bounded.
We can now state our main result.
Theorem A. Let X be a compact smooth Riemannian manifold. Let r ∈ N ≥2 ∪ {∞}, and assume that f ∈ Diff r (X) is a dynamically coherent, center bunched partially hyperbolic diffeomorphism. Let c := dim E c f . If either c = 1 3 and f is plaque expansive, or c > 1 and f satisfies at least one of the following assertions: • f is ( c−1 c ) 1 7 -pinched and has uniformly compact center foliation; • f is ( c−1 c ) 1 9 -pinched and the maps x → E cs f (x), E cu f (x) are of class C 1 , then there exists a C 1 -open neighbourhood of f in Diff r (X), denoted by U, such that C 1 -stable accessibility is prevalent in the C r − J−Kolmogorov sense in U, for any J ≥ J 0 (see Subsection 3.1 below for a precise definition). Here J 0 is an integer depending only on dim X.
Moreover, let Vol be a smooth volume form on X, and assume that f ∈ Diff r (X, Vol) satisfies one of the previous conditions. Then the above conclusion is true for U 0 , a C 1 -open neighbourhood of f in Diff r (X, Vol), in place of U. In particular, C 1 -stable ergodicity is C r -dense in U 0 .
Theorem A generalizes all the results on stable ergodicity from [15,26,6,43] to arbitrary center dimension in the strongly pinched region. Compared to the previous works, our result has two significant novelties: (1) this is the first time that C r -density of stable ergodicity is proved for fully nonlinear systems with arbitrary center dimension, for r ≥ 2 4 ; (2) this is the first result that shows that stable ergodicity and accessibility are prevalent in the measure-theoretical sense.
To provide motivations for (2), let us mention that there are two ways to approach the question of genericity: topological and metric. These notions are sometimes 3 Any partially hyperbolic diffeomorphism with center dimension 1 is automatically center bunched. 4 Among the set of compact group extensions, C r -density of stable ergodicity was proved in [15]. These systems also have higher center dimension, but they are simpler as the action on the fiber is by group translation, hence is characterised by finitely many parameters.
conflicting 5 . The study of prevalence properties goes back to Kolmogorov [29]. We say that a property is prevalent if it holds for a typical dynamical system in Kolmogorov's sense (see Definition 3.3) as in [7,29,34] (see [27,30] for different notions). Even when dim E c f = 1, we have strengthened the result of [38] as we show that stable ergodicity is not only C r -dense but also prevalent among center bunched partially hyperbolic diffeomorphisms with one-dimensional center, assuming plaque expansiveness. In [34,Conjecture 3], the authors conjectured that: for the generic finite dimensional submanifold V in Diff r (M ) and almost every f ∈ V , the equivalence classes of points in the chain recurrent set of f are open in the chain recurrent set. They also mentioned that the validity of such conjecture would give a "finite spectral decomposition for f where each piece of the decomposition has something akin to the accessibility property". P. Berger [7] constructed a counterexample to this conjecture in the Newhouse domain. Our result strongly suggests that the accessibility property could be prevalent among partially hyperbolic diffeomorphisms.

1.2.
Further illustrations of our result. Conjecture 1.3 has its origin in several concrete models. For instance, given an integer n ≥ 1, the linear automorphism of T n := R n /Z n associated to a matrix A ∈ SL(n, Z) is defined as the unique diffeomorphism f A : T n → T n such that the following diagram commutes, where π : R n → T n denotes the natural projection: Back in the 1970's, Hirsch-Pugh-Shub [25] already asked whether any ergodic linear automorphism of T n stably ergodic, for n ≥ 2.
Positive answer to this question is known when the map is Anosov by [2]. This question was solved by F. Rodriguez-Hertz in [39] for any n ≤ 5. This in particular answered a special case of the question, asked in [22] for an explicit 4 × 4 matrix. More generally, in [39], the author investigated the case of pseudo-Anosov maps with 2 eigenvalues of modulus 1. In [32], the authors mentioned that the validity of Conjecture 1.3 would give a positive answer to the following weaker version of the previous question 6 , namely, given two integers n, r ≥ 2, whether the C r -generic volume preserving perturbation of any ergodic automorphism of T n ergodic. To the best of our knowledge, the question remains open for any dim E c > 1.
Let us now consider M = SL(n, R)/Γ for some uniform discrete subgroup Γ of SL(n, R). Let A ∈ SL(n, R) with at least one eigenvalue of modulus different from 1, and let L A : M → M be the left translation by A. In [32], the authors ask whether L A is stably ergodic among C 2 volume preserving diffeomorphisms of M .
Unlike the case of T n , the topological complexity of the homogeneous manifold M has so far prevented a generalization of [39]. One can also consider the generic version of the previous question, namely, whether the C ∞ generic volume preserving perturbation of L A is stably ergodic. This is true when the map L A is Anosov, by 5 For instance, among circle diffeomorphisms, a topological generic map has rational rotation number, while those with irrational rotation number occupy positive measures in many oneparameter families. 6 See the remark below [32, Conjecture 1]. [2], or when the center dimension dim E c is equal to one, but the question remains open for any dim E c > 1.
As corollaries of the main result of this paper, we answer the previous questions in any dimension in the strongly pinched region, namely for maps with pinching exponents close to 1.
Theorem B. Let n ≥ 2 be some integer. For any r ∈ N ≥2 ∪ {∞}, any linear partially hyperbolic automorphism f A : T n → T n , ergodic or not, that is ( c−1 c ) 1 5pinched, where c is the number of eigenvalues of A ∈ SL(n, Z) of modulus 1, there exists U, a C 1 -open neighbourhood of f A in PH r (T n , Vol), such that for some C rdense subset U of U, any map in U is a stably ergodic diffeomorphism.
Theorem C. Let Γ be a uniform discrete subgroup of SL(n, R), let M := SL(n, R)/Γ and let L A : M → M be the left translation by A ∈ SL(n, R), assuming that A has an eigenvalue with modulus different from 1. Then, the C ∞ generic volume preserving perturbation of L A is stably ergodic for any θ-pinched L A , where θ ∈ (0, 1) depends only on the integer n.
The partially hyperbolic splittings for Theorems B and C are the canonical ones: the center spaces are the neutral subspaces of the affine actions. Note that even for linear automorphisms with two-dimensional center, Theorem B partially improves and generalizes the main result in [39], in the following sense: (1) we removed the pseudo-Anosov condition; (2) our result also applies to non-ergodic maps; (3) we weakened the regularity condition (for the perturbations) from Another open question in [32] is the following. 7 Given two compact Riemannian manifolds M, N , where M supports a C r volume preserving Anosov diffeomorphism g : M → M , r ∈ N ≥2 ∪ {∞}, is the C r -generic volume preserving perturbation of g × Id : M × N → M × N ergodic? Again, this question has a positive answer when the map in question satisfies dim N = 1. A recent result of A. Avila and M. Viana [6] also gives an affirmative answer to this question when dim N = 2 (see also [26] for a related result). This question remains open for any N of dimension at least 3. As a corollary of our main result, we can answer this question in any dimension in the strongly pinched region.
Theorem D. Let M, N be two compact Riemannian manifolds, where M supports a C r volume preserving Anosov diffeomorphism g : M → M , r ∈ N ≥2 ∪ {∞}, and let h : M → Isom(M ) be a C r map. Assume that g is n−1 n 1 7 -pinched where n := dim N . Then a C r -generic volume preserving perturbation of the map (x, y) → (g(x), h(x, y)) is stably ergodic.
Theorems B, C and D are immediate consequences of a more general result, Theorem E, stated in Section 2.
1.3. Idea of the proof. We follow closely the method in [43]. In [43], the author studied a class of skew products over Anosov systems, and divided the problem into: I. showing that the property of having a stably open accessibility class is C r -generic; II. showing that the property of having an open accessibility class with intermediate volume is C r -meager. In Step I, we prove the existence of open accessibility classes using a quantitative version of a theorem of M. Bonk and B. Kleiner in [10] (the details about this quantitative statement can be found in Section 7).
In our setting, this boils down to destroying common intersections between many different holonomy loops. This is done by a parameter exclusion within a family of random perturbations.
The main new observation in this paper (à la Avila) is that: by letting the number of loops be sufficiently large compared to the dimension of the manifold, we are left with enough room to create open accessibility class at every point, due to the fact that the random perturbations we use are not very sensitive to the map and the point. The details about this argument can be found in Section 10. This allows us to bypass Step II in [43] which only works under restrictive assumptions (for a different application of the method in Step II, see [44]). Our construction is also suitable to study the measure-theoretical prevalence. Our method suggests that under a strong pinching condition, the failure of accessibility should be a phenomenon of infinite codimension.
On the technical level, in Section 6, we use [20] to construct families of center disks to connect different regions of the space. Small complications arise in the study of prevalence, since we need to organize several families for different maps. For each disk we consider a parametrized family of random perturbations generated by vector fields with disjoint supports, and parametrize a part of the accessibility class of any given point x in a slightly smaller disk by [0, 1] c , where c is the center dimension. We then apply Bonk-Kleiner's criterion to show the openness of the accessibility class of x for most parameters in the family. The regularity results in [35,36] are used to reduce the problem to a finite set of points and loops.
Convention. Given a compact smooth Riemannian manifold X with a smooth volume form Vol, for any r ∈ N ≥1 ∪{∞}, we denote by PH r (X) (resp. PH r (X, Vol)) the set of all C r (resp. C r volume preserving) partially hyperbolic diffeomorphisms on X with bounded C r norms.
In the course of the paper, we will often use constants depending on a diffeomorphism f (and that may or may not depend on other things). We say that a constant C depending on a C r diffeomorphism f is C r -uniform if it works for all diffeomorphisms in a C r -open neighbourhood of f . We introduce several constants related to a diffeomorphism in Notations 3.12, 5.2 and Construction 9.1.
Given l ≥ 0 and diffeomorphisms f 1 , f 2 , . . . , f l , we use the notation l i=1 f i to denote f l • · · · • f 1 , where by convention j i=j+1 f i := Id for any j = 1, . . . , l − 1. χ s ,χ u ,χ c ,χ c as in (1.3)-(1.6). For any real number ∈ R, we set We will focus on the case where f is dynamically coherent (recall Definition 1.9) and satisfies one of the following properties: Moreover, let f ∈ Diff r (X, Vol) satisfy the above condition. Then the above conclusion is true for U 0 , a C 1 -open neighbourhood of f in Diff r (X, Vol), in place of U. In particular, C 1 -stable ergodicity is C r -dense in U 0 .

Preliminaries
In the following section, we recall some general notions about parameter families, prevalence and partially hyperbolic diffeomorphisms.
3.1. Prevalence. Let X be a smooth Riemannian manifold with a smooth volume form Vol. We refer the reader to [21, Chapter II, §3] and [23]for more details about the notions recalled in the following. Definition 3.1 (C r topology). Let m, n ≥ 1 be integers. Given k ∈ N and For r ∈ N ∪ {∞}, the C r topology on C r (R m , R n ) is the topology induced by d C r . Given smooth Riemannian manifolds M, N , we define accordingly the C r topology for maps between M and N , and we denote by C r b (M, N ) the subset of maps in C r (M, N ) with bounded distance to the constant maps. C r (R m , R n ) such that for every 0 ≤ i, j ≤ r the derivative ∂ i ω ∂ j x f ω (x) is a multilinear map which depends continuously on (ω, x) ∈ [0, 1] J × R m . The C r topology on the space C r ([0, 1] J , C r b (R m , R n )) is the topology induced by the norm · C r : is defined by analogy. Let r ∈ N∪{∞}. Given smooth Riemannian manifolds M, N and We define the C r topology on C r ([0, 1] J , U) analogously and denote by d C r the associated metric. We introduce the following notion for technical reasons. Proposition 3.5. There exists J 0 = J 0 (dim X) > 0, which we fix in the rest of this paper, with the following property. For any r ∈ N ≥2 ∪ {∞} and any integer J ≥ J 0 , the set of good C r − J−families is dense in the set of C r − J−families with respect to the C r topology.
Proof. This is essentially contained in the proof of [28,Theorem 2.2]. In [28], the author showed the prevalence of Kupka-Smale diffeomorphisms in Diff r (X). In contrast to the Kupka-Smale property, our notion of good family is only a transversality condition on the level of 0-jets. Next lemma suffices for our purpose. lemma 3.6. For any integers p, q ≥ 3, any r ∈ N ≥2 ∪ {∞}, for any C r map f : (−1, 1) p → R q , there exist an integer L ≥ 1 and C ∞ divergence-free vector fields V 1 , . . . , V L on R q , supported in (−1, 1) q , such that the following is true. Let us denote by F bi Vi : R q → R q the time-b i map of the flow generated by V i , and let F : 3.2. Partially hyperbolic diffeomorphisms. Fix an integer d ≥ 1. We let X be a smooth d-dimensional Riemannian manifold with a smooth volume form Vol.
Let f : X → X be a partially hyperbolic diffeomorphism or an Anosov map. In the following, we will call a leaf of W c f , W cu f , etc. a center leaf, center-unstable leaf, etc. Definition 3.7 (Holonomies). Let f ∈ PH 1 (X) be dynamically coherent.
• Let x 1 ∈ X and x 2 ∈ W s f (x 1 ). By transversality, for i = 1, 2, there exists a neighbourhood C i of x i within W cu f,loc (x i ) such that for any x ∈ C 1 , the local s-leaf through x intersects C 2 at a unique point, denoted by H s f,x1,x2 (x) = H s f,C1,C2 (x). We thus get a well-defined local homeomorphic embedding H s f,C1,C2 : C 1 → C 2 , called the (local) stable holonomy map between C 1 and C 2 . For i = 1, 2, setC i := C i ∩ W c f,loc (x i ); by restriction, H s f,C1,C2 induces a local homeomorphism H s f,C1,C2 :C 1 → C 2 . Unstable holonomies are defined accordingly.
• Let x 1 ∈ X and x 2 ∈ W c f (x 1 ) be two sufficiently close points in the same center leaf. Let * ∈ {u, s}. Then, the (local) center holonomy map H c f,x1,x2 along local leaves in W c f is a well-defined local homeomorphism from a neighbourhood of . The following result [35, Theorem A] relates the pinching condition in Definition 1.7 with the regularity of u, s-holonomy maps.
The following result is contained in the proof of [35,Theorem B]. It relates the center bunching condition in Definition 1.8 to the regularity of u, s-holonomies.
Proposition 3.9. If f ∈ PH 2 (X) is dynamically coherent and center bunched, then local stable/unstable holonomy maps between center leaves are C 1 when restricted to some center-stable/center-unstable leaf and have uniformly continuous derivatives.
In Proposition 3.9, uniformity of the continuity is a simple consequence of the invariant section theorem and the uniform C 2 bound.

3.3.
On leaf conjugacy. Later on, we will focus on dynamically coherent systems satisfying one of the conditions (a) or (b) in Section 2. The following result is due to Hirsch-Pugh-Shub.
Proposition 3.10 (Theorem 7.1, [25], see also Theorem 1 in [36]). Let f be a dynamically coherent partially hyperbolic diffeomorphism satisfying (a) or (b). Then any g ∈ PH 1 (X) which is sufficiently C 1 -close to f is also dynamically coherent. Moreover, there exists a homeomorphism h = h g : X → X, called a leaf conjugacy, such that: (1) h maps a f -center leaf to a g-center leaf; (2) both h and h −1 tend to Id in the uniform norm as d C 1 (f, g) tends to 0.
Proof. It suffices to see that any f in the proposition is plaque expansive (recall Definition 1.9). The plaque expansiveness is proved in [17] (see also [36,Proposition 13]) under (a), respectively in [25] under (b).
The following result, due to Pugh-Shub-Wilkinson [36], ensures that the leaf conjugacy h in Theorem 3.10 has Hölder regularity under (a) or (b).
We will later use Propositions 3.8, 3.9 and 3.11 while keeping track of the uniformity of various quantities. We summarise these statements as follows.
the associated leafwise distances. For any x ∈ X, σ > 0, and * = s, u, c, cs, cu, we set f,x,y is uniformly continuous and has norm bounded by Λ f . in place of θ, and the following is true: (5.1) any g ∈ PH 1 (X) with d C 1 (f, g) < ε f is plaque expansive and for any x ∈ X, any y ∈ W c g (x, Here h gi is given by Proposition 3.11 for g i in place of g. Moreover, we assume that Properties 2, 3, 4 above are also satisfied when we exchange the roles of u and s. Definition 3.13. Let f ∈ PH 2 (X) be dynamically coherent and center bunched.
We say that f satisfies Similarly, we say that f satisfies Remark 3.14. If Theorem E(1) holds for f , then by Definition 1.7, Propositions 3.8, 3.11, we can choose θ f , θ f such that (ae) holds. Similarly, if Theorem E(2) holds for f , then we can choose θ f , θ f such that (be) holds.
Standing hypotheses for the rest of the paper We denote by X a d-dimensional compact smooth Riemannian manifold with a smooth volume form Vol; r belongs to N ≥1 ∪ {∞}; and f ∈ Diff r (X). Whenever f is declaimed to be partially hyperbolic, we denote c :

Random perturbations
In this section, we will establish some estimates for certain perturbations of the holonomy maps of a dynamically coherent plaque expansive partially hyperbolic diffeomorphism.

Basic notions and constructions.
We start with the following more general situation. The following suspension construction will be used repeatedly.
We associate with suchf the suspension map T (f ) defined by If in additionf (b, ·) ∈ Diff r (X, Vol) for all b ∈ U , then we say thatf is volume preserving.
Definition 4.2 (Infinitesimal C r deformation). Given an integer I > 0, a C r map V : Construction 4.3. Given I > 0, a ∈ R I , and V , an infinitesimal C r deformation with I-parameters, then for any sufficiently small > 0, we associate with V a C r deformation at (a, f ) with I-parameters, denoted byf , which is defined bŷ where U = B(a, ) ⊂ R I and for any B ∈ R I , F V (B,·) : R × X → X denotes the C r flow generated by the vector field V (B, ·). In this case, we say thatf is generated by V . If in addition for each B ∈ R I , V (B, ·) is divergence-free, thenf is volume preserving as in Definition 4.1, and we say that V is volume preserving.
For any V as in Definition 4.2, we use · X to denote the uniform norm of the derivatives of V restricted to {0} × X. The following lemma gives bounds on the norms of deformations induced by infinitesimal deformations. lemma 4.4. Assume that r ≥ 2. Let I ∈ N ≥1 and letf : U × X → X be a C r deformation at (0, f ) generated by some infinitesimal C r deformation with Iparameters V . Take T = T (f ) as in Definition 4.1. Then there exists a C 2 -uniform constant C 0 = C 0 (f ) > 0, such that by possibly taking U smaller, it holds: Proof. We defer the proof to Appendix B.
Some of the estimates will depend on the support of a deformation or of an infinitesimal deformation, which we now define.
Given a ∈ R I , an open neighbourhood U of a, and a C r deformation at (a, f ) with

It is clear from Definitions 4.2 and 4.5 that for any infinitesimal
4.2. c-disk and c-family. We first introduce the following notion.
Definition 4.6 (Accessibility class). Let f : X → X be a partially hyperbolic diffeomorphism. For any x ∈ X, any > 0 and any integer k ≥ 1, we let Acc f (x, , k) be the set of all points y ∈ X that can be attained from x through a k-legged accessibility sequence x = z 0 , z 1 , . . . , z k = y, where for each 0 . We let the accessibility class of f at x be Acc f (x) := ∪ >0,k≥1 Acc f (x, , k).
For any f ∈ PH 1 (X), accessibility classes of f form a partition of X. By Definition 1.4, f is accessible if and only this partition consists of a single class.
In the rest of Section 4, we assume f satisfies (H1).
For each x ∈ X and σ > 0, we call C = W c f (x, σ) the center disk of f (or c-disk of f for short) centered at x with radius σ, and we set (C) := σ. In addition, for any θ ∈ (0, 1], we also define θC := W c f (x, θσ). Given θ ∈ (0, 1) and k ∈ N ≥1 , we say that D is a (θ, k)-spanning c-family for f if Given any subset C ⊂ X, and σ ≥ 0, we set (C, σ) A collection D of subsets of X is called σ-sparse if for any two distinct C, C ∈ D, (C, σ), (C , σ) are disjoint. Any c-family for f is σ-sparse for some σ > 0.
The next lemma is a consequence of the continuity of the invariant foliations with respect to the dynamics. Roughly speaking, it says that any c-family can be slightly perturbed into a c-family for a given nearby map.
Proof. By letting be sufficiently small, we clearly have that for any g sufficiently . By Proposition 3.10, W u g , W cs g (resp. W s g , W cu g ) exist and are uniformly transverse for all g sufficiently C 1 -close to f . Then (4.3) follows by letting σ and d C 1 (f, g) be sufficiently small compared to ρ m , ρ M and θ − θ.

Extended map and center subspaces.
Recall that in this subsection, (H1) holds. Letχ c ,χ c ,χ s ,χ u be as in Definition 1.2 so that (1.3) to (1.6) are satisfied. Let ξ > 0 be a constant such that lemma 4.11. Let I ∈ N ≥1 , a ∈ R I and let U ⊂ R I be an open neighbourhood of a.
Letf : U × X → X be a C r deformation at (a, f ) with I-parameters. If U is chosen sufficiently small, then the map T = T (f ) is a C r dynamically coherent partially hyperbolic system for some T -invariant splitting . If in addition (H2) holds, then, after reducing the size of U , u, s-holonomy maps between center leaves of T (within distance 1) are C 1 when restricted to some centerunstable/center-stable leaf, with uniformly continuous, uniformly bounded derivatives.
Proof. For small enough U , the map T is a dynamically coherent partially hyperbolic diffeomorphism (it is C 1 -close to (b, x) → (b, f (x))). A detailed treatment for this statement can be found in [36,Section 7].
In the following, let * = u or s, and let U be small. Then for all is close to E * f , and the expansion/contraction rate off Then Now, if r ≥ 2 and f is center bunched, by C 1 -openness of center bunching, for sufficiently small U , we can verify that T n is also center bunched for some n ∈ N. The smoothness of s, u-holonomy maps of T follows from Proposition 3.9.
Let U ,f and T be as in Lemma 4.11. In the following, for any (b, x) ∈ U × X, we will tacitly use the inclusions By a slight abuse of notation, we let π X (b, x) := x.
We introduce the following definitions, motivated by the need to control return times of a map to the support of a deformation.
Definition 4.12. For any subsets A, B ⊂ X, we define A, B)).
In the following, for * = s, u, and for any p ∈ M , we set We defineχ c k (p) andχ c k (p) in a similar way. The following lemma collects some basic properties of the center bundle E c T .
Arguing similarly for the stable part, we conclude the proof.

Holonomy maps.
Recall that in this section, (H1) and (H2) are satisfied. In the following, we fix an integer I > 0, and letf : U × X → X be a C 2 deformation at (0, f ) with I-parameters. We set T = T (f ). In the following, we will always take U conveniently small so that by Lemma 4.11, the stable and unstable holonomy maps for T between close center leaves of T are C 1 . We need bounds for the derivatives of holonomy maps with respect to parameters. The following lemma is proved by combining the construction in [ The following proposition provides fine control of the derivatives of holonomy maps with respect to parameters when we are given certain recurrence condition. We give some illustration in Figure 1.
Then there exists a C 1 -uniform constant ξ > 0 such that we have the following: . Note that the terms on the RHS of the above inequalities are independent of σ. Figure 1. The point f −j (x) for any integer 1 ≤ j ≤ R 0 , and the point f −k (z) for any integer 2 ≤ k ≤ R 0 lie outside of supp(f ). We apply Lemma 4.14 to f −n (x) and f −n (z) where n is a small fraction of R 0 .
Proof. We first prove (1). Without loss of generality, we assume that R 0 is sufficiently large.
Let 1 ≤ n ≤ R 0 − 1. Successive application of Lemma 4.13 (1) gives Then the invariance of the foliations under the dynamics yields Proof. By y ∈ W cu f (x, σ) and by distortion estimates, for some C 1 -uniform constant ). On the other hand, by distortion estimates, we have |RHS of (4.9) − RHS of (4.10)| < C for some C 2 -uniform constant C > 0. Thus the claim follows from Lemma 4.14(2).

Submersion from parameter space to phase space
In this section, we will estimate the measure of parameters in a C r deformation corresponding to certain "unlikely coincidences". First, we need to estimate the derivatives (with respect to parameters) of holonomy maps along certain su-paths.
, and let C, R 0 > 0 satisfy that: Letγ be the lift of γ for T . Then, the holonomy map H T,γ is C 1 in an open neighbourhood of (0, x) in W c f (x), and for any B ∈ T 0 U , we have We can apply Proposition 4.15 to obtain . Combining this with (5.3) and (5.4), we see that there exists a C 2 -uniform constant C 5 = C 5 (f ) > 0 such that The following definition is motivated by Lemma 5.3 and Lemma 5.4. The next proposition roughly says that if we have enough control on the magnitude of the deformation and on the return times to the support of the perturbation, then we can obtain a lower bound on the determinant of the differential of a certain map from parameter space to phase space (see Figure 2 for an illustration). This will be important in the parameter exclusion which appears in Section 10.
Thus for some C 2 -uniform constant c 5 > 0 depending only on f, L, c, C, for any sufficiently large R 0 depending only on f, L, c, C, κ, we have det(Ξ H ) > c 5 κ L .
Moreover, it is easy to see that R 0 is C 2 -uniform with respect to f . Then κ 0 := c 5 κ L depends only on f, L, c, C, κ and is C 2 -uniform in f . This concludes the proof. 6. Finding suitable spanning c-families Since we will study a parametrised family of diffeomorphisms, we need a bit more work to find suitable c-families. Let us start by recalling a result presented in [20,Lemma 1.2]. We use here the notations introduced in Subsection 4.2.  . This does not introduce any new difficulty into the proof. The following is a consequence of the above lemma.
Corollary 6.2. Assume that f ∈ PH 1 (X) is dynamically coherent, and the fixed points of f k are isolated for all k ≥ 1. Then for every such that the following is true. For all g sufficiently C 1 -close to f , there exists D g , a ( 1 40 , 4)-spanning c-family for g such that Lemma 6.1(1) is satisfied for (D g , g) in place of (D, f ). Moreover, we have (1) r(D g ) > ρ, (2) n(D g ) < N, (3) D g is σ-sparse, (4) R ± (g, (D g , σ)) > R.
While working with a family of diffeomorphisms, we will need to consider several c-families. For that purpose, we use a superposition of a collection of perturbations which are localized in parameter-phase space. The following proposition will allow us to arrange their support in such a way that the interferences between them are very weak; it will serve as a key step in the inductive construction of these localized perturbations in Proposition 10.2. Proposition 6.3. Let r ∈ N ≥2 ∪ {∞}, J ≥ 1, and let {f a } a∈[0,1] J be a good (see Definition 3.4) C r − J−family in the space of dynamically coherent, C r partially hyperbolic diffeomorphisms. Then for any integers K, R 0 ≥ 1, any real numbers ϑ > 0, h 0 > 0, there exists a set Ω 1 compactly contained in [0, 1] J with Leb([0, 1] J \Ω 1 ) < ϑ, an integer N 0 > 1, and real numbers ρ 0 ∈ (0, h 0 ), ρ 1 ∈ (0, ρ 0 ), σ 0 , λ 0 > 0 such that the following is true.
Take any a ∈ Ω 1 , and any integer 0 ≤ l ≤ K − 1. For any collection of points Then there exists a ( 1 20 , 6)-spanning c-family for f a , denoted by D l+1 , such that (1), (2) above are satisfied for i = l + 1, and moreover, Proof. We choose Ω 1 to be any compact set contained in int([0, 1] J ) such that Leb([0, 1] J \Ω 1 ) < ϑ, and for any a ∈ Ω 1 , the fixed points of f k a are isolated for any integer k ≥ 1. The existence of Ω 1 is guaranteed by our hypothesis that {f a } a∈[0,1] J is a good family.
By the compactness of [0, 1] J , there exists ρ 2 ∈ (0, h 0 ) such that for any a ∈ [0, 1] J , x ∈ X, the tangent space of W c fa (x, 4ρ 2 ) is sufficiently close to E c fa (x) so that for any y ∈ B(x, ρ 2 ), W c fa (y, 4ρ 2 ) intersects B(x, ρ 2 ) in a single local center manifold.
Take σ 2 > 0 such that for any a ∈ [0, 1] J , any x ∈ X, any collection of 3R . Take a small constant λ 0 > 0 such that for any a ∈ [0, 1] J , a ∈ B(a, λ 0 ) ∩ [0, 1] J , x ∈ X, and any −R 0 ≤ p ≤ R 0 , we have Fix any a ∈ Ω 1 . We denote D := D(a) = { C 1 , . . . , C N1 } for some N 1 < N 0 . We take l, {a i } l i=1 and {D i } l i=1 as in the proposition. We will modify D to obtain D l+1 that satisfies the conclusion of the proposition. 8 As in the case of (4.3), here (6.3) follows from the uniform transversality of W cs fa and W u fa , respectively W cu fa and W s fa .

A criterion for stable values.
In this section we state a topological lemma that is at the core of our construction of open accessibility classes. First we borrow a few definitions from [10].
Definition 7.1. If f : X → Y is a continuous map between metric spaces X and Y , then y ∈ Y is a stable value of f if there is > 0 such that y ∈ Im(g) for every continuous map g : X → Y such that d C 0 (f, g) < .
Definition 7.2. Given a constant > 0, a continuous map f : X → Y between metric spaces X and Y is called -light if for every y ∈ Y , every connected component of f −1 (y) has diameter strictly smaller than .  (2) x ∈ U α for all α ∈ I.
Then f has a stable value.
Proof. By Theorem F, it suffices to check that f is (c)-light. Given any x ∈ [0, 1] c , take I ⊂ A satisfying (1), (2). In particular, there exists α ∈ I such that f (x) / ∈ f (∂U α ). We denote by P x the connected component of f −1 (f (x)) containing x. We claim that P x is contained in U α . Indeed, by the continuity of f , f −1 (f (x)) has no accumulating point in ∂U α . If P x ∩ (U α ) c = ∅, then we can find two disjoint open sets U, V s.t. P x ⊂ U ∪ V and P x ∩ U , P x ∩ V are both nonempty. This contradicts the connectedness of P x , hence the claim is true. In particular, the diameter of P x is not larger than the diameter of U α which by hypothesis is strictly smaller than (c). Since x is an arbitrary point in [0, 1] c , we deduce that f is (c)-light. Given an integer c ≥ 1, a positive constant θ ∈ ( c−1 c , 1), we set In the following, we will fix c, θ and abbreviate K i (c, θ) as K i , i = 0, 1. By a direct construction, we can fix a cover {U α } α∈A of [0, 1] c by open sets in R c , which satisfies: (1) A is a finite set and for all α ∈ A, there exist constants {p α,i , q α,i } i=1,...,c ⊂ [−1, 2] such that U α = (p α,1 , q α,1 ) × · · · × (p α,c , q α,c ); (2) for any α ∈ A, diam(U α ) < (c), where (c) is given by Theorem F; (3) for each x ∈ [0, 1] c , there exists a subset I ⊂ A with more than K 1 elements satisfying that x ∈ U α for all α ∈ I, and {∂U α } α∈I are mutually disjoint; (4) for each i ∈ {1, . . . , c}, the points {p α,i , q α,i } α∈A are mutually distinct. For each integer i ∈ {1, . . . , c}, we let B i := {p α,i , q α,i } α∈A , and for each α ∈ A, Given any s ∈ [−1, 2], we introduce the normalized coordinate Note that for any i < i and any s, s ∈ [−1, 2], ϕ(i, s) < ϕ(i , s ). We also set

Holonomy maps associated to a family of loops
In this section, (H1) holds.

Continuous and regular family of loops.
Definition 8.1. Given x ∈ X, a one-parameter family {γ(s) = (x 1 (s), x 2 (s), x 3 (s))} s∈[0,1] of f -loops at x is said to be continuous if for any i = 1, 2, 3, the map s → x i (s) is continuous. We define (γ) := sup s∈[0,1] (γ(s)). lemma 8.2 (Continuation of f -loops). There exist U, a C 1 -open neighbourhood of f , as well as ς f > 0 such that the following is true. Let γ be a continuous family of f -loops at x ∈ X satisfying (γ) < ς f 2 . Then for any g ∈ U and y ∈ B(x, ς f ), we can define γ g,y , a continuous family of g-loops at y, such that γ f,x = γ and each coordinate of γ g,y (s) depends continuously on (g, y, s).
The following property combines a global property, based on the notion of "accessibility modulo central disks" which appears in [20] (see also Section 6), and a local one, based on the notion of -light maps in Section 7, which together imply the accessibility property. Definition 8.3 (Property (P)). We say that f satisfies property (P) if there exist 0 < θ < θ < 1, an integer k ≥ 1, and D, a (θ, k)-spanning c-family for f , such that for any C ∈ D, any x ∈ θ C, there exists a continuous family of f -loops at x, denoted by {γ x (s) = (x 1 (s), x 2 (s), x 3 (s))} s , with (γ x ) < σ f , such that the following is true: let ψ x := ψ(f, x, γ x ) be given by (8.1). Then for any (i, B, {s t } t∈B ) ∈ Γ (defined in (7.4)), there exist t, t ∈ B such that ψ x (s t ) = ψ x (s t ).
Proposition 8.4. If f satisfies property (P), then f is C 1 -stably accessible.
Proof. Assume that f satisfies (P) for 0 < θ < θ < 1, k ≥ 1, some (θ, k)-spanning c-family for f , denoted by D, and a set of families of f -loops {γ x } x∈θ C, C∈D . Take C ∈ D, x ∈ θC, and set ψ To see this, take an arbitrary s ∈ [0, 1] c . By property (3) of the open cover {U α } α∈A in Subsection 7.2, there exists a subset I ⊂ A with |I| ≥ K 1 , such that s ∈ U α for all α ∈ I, and {∂U α } α∈I are mutually disjoint. Let us show that ∩ α∈I ψ x (∂U α ) = ∅. Assume it is not true. By (7.1) and the pigeonhole principle, we may choose i ∈ {1, . . . , c} and I ⊂ I with |I | = K 0 , such that ∩ α∈I ψ x (∂ i U α ) = ∅. Thus there exists (i, B, {s t } t ) ∈ Γ such that ψ x (s t ) = ψ x (s t ), ∀ t, t ∈ B, which contradicts (P). Therefore, ∩ α∈I ψ x (∂U α ) = ∅. Since s can be taken arbitrary in [0, 1] c , Corollary 7.5 implies that ψ x has a stable value y, and thus, Im(ψ x ) contains an open neighbourhood of {y} in W c f (x). But ψ x takes values in W c f (x) ∩ Acc f (x), hence the latter has non-empty interior. Saturating by local stable and unstable leaves, we deduce that Acc f (x) has non-empty interior. Then the accessibility class Acc f (x) is open, and the claim is proved.
Since D is (θ, k)-spanning, the previous claim implies that for any x ∈ X, Acc f (x) is open. This shows that f is accessible, since X is connected. Now it suffices to show that (P) is a C 1 -open condition. Let ς f > 0 be as in Lemma 8.2 and let σ ∈ (0, ς f ) be a small constant to be determined. Let θ := θ+θ 2 . By Lemma 4.9 and Remark 4.10, for any g sufficiently C 1 -close to f , there exists D g , a (θ , k + 2)-spanning c-family for g, such that for each C g ∈ D g , there exists C ∈ D so that θ C g ∈ (θ C, σ). For each y ∈ θ C g , take x ∈ θ C with y ∈ B(x, σ) ⊂ B(x, ς f ). Applying Lemma 8.2 to γ x , we obtain γ g,y , a continuous family of g-loops at y which is close to γ x . By choosing σ sufficiently small, we can ensure that for any g sufficiently close to f in C 1 topology, any C g ∈ D g , y ∈ θ C g , any (i, B, {s t } t∈B ) ∈ Γ, there exists t, t ∈ B such thatψ y (s t ) =ψ y (s t ), wherẽ ψ y := ψ(g, y, γ g,y ). Thus (P) is a C 1 -open condition.

8.3.
Parametrising an accessibility set using a family of loops. To optimize the pinching exponents in our theorems, we will mainly consider the class of continuous families of loops as follows.
Definition 8.5 (Regular family). Given x ∈ X and constants σ ∈ 0, 1]. In this case, we say that γ is determined by x and (x 1 (s)) s∈[0,1] . Indeed, for any 9 We now restrict our attention to maps in the region defined as follows.
Notation 8.6. Assume that f 0 ∈ PH 2 (X) is dynamically coherent and center bunched. We consider the following cases: (1) If dim E c f0 = 1 and f is plaque expansive, then we let U(f 0 ) be a C 1 -open neighbourhood of f 0 in which all maps are plaque expansive; (2) If dim E c f0 ≥ 2 and satisfies (ae) or (be), then we let (5)). Moreover, we assume that U(f 0 ) is small enough so that any g ∈ U(f 0 ) is θ f0pinched and center bunched; and the constants h f0 , σ f0 , C f0 , Λ f0 in Notation 3.12 work for any g ∈ U(f 0 ). By points (1), (2), (4), (5) in Notation 3.12, such U(f 0 ) exists. We stress that we do not require Λ f to be uniformly bounded for f ∈ U(f 0 ).

Constructing charts and vector fields
In order to construct infinitesimal deformations with required properties, we will first introduce coordinates in a neighbourhood of each c-disk. In this section, we assume that (H1) holds, and r ≥ 2.
In the following, our goal is to define certain vector fields in order to perturb the dynamics and induce a displacement of the holonomies. More precisely, given a small center disk, we define a vector field localized close to the disk. These vector fields will be rich enough for us to apply Proposition 5.6.
Construction 9.1. There exist C 2 -uniform constants h f ∈ (0, h f ), C f > 1 such that the following is true. For any c-disk of f , denoted by C = W c f (x, h), with x ∈ X and h ∈ (0, h f ), there exists a C r volume preserving map φ = φ(C) : : E su f (y) → R c has norm smaller than ζ.
Note that for any s ∈ [0, 1], x 2 (s), x 3 (s) ∈ W cu g (x , C f σ ) for sufficiently large C f . We get (i) by the continuity of E cu f , and by letting h f ,ε 0 ,σ 0 be sufficiently small. We get (ii) by the continuity of E cs f and E c f ; by C f σ < h < h f ; by letting C f be sufficiently large, and then letting h f ,ε 0 ,σ 0 be sufficiently small. Construction 9.3. For any c-disk C such that (C) =: h ∈ (0, h f ), with h f as in Construction 9.1, for any σ ∈ (0, h), we define a collection of vector fields as follows.
• For each 1 ≤ j ≤ c, let U j : (−2/3, 2/3) c × (−1, 1) du × (−1/3, 1/3) ds → R d be a compactly supported C ∞ divergence-free vector field such that U j restricted to (−1/2, 1/2) c+du × (−1/5, 1/5) ds is equal to the constant vector, denoted by E j , that has 1 at j-th coordinate and 0 at the others. Such U j always exists since d ≥ 3. Moreover, we can assume that U j satisfies U j C 1 < C * for some constant For any x c ∈ R c , x u ∈ R du , x s ∈ R ds , a c , a u , a s > 0, we denote for every z c ∈ R c , z u ∈ R du , z s ∈ R ds : P xc,ac,xu,au,xs,as (z c , z u , z s ) = (x c + a c z c , x u + a u z u , x s + a s z s ). Now, for any i, j ∈ {1, . . . , c}, any t ∈ B i , we let U σ C,i,t,j : (−h, h) d → R d be the vector field . By Construction 9.1, and the C 1 -bound on U j above, we see that the vector field V σ C,i,t,j is divergence-free and satisfies: Remark 9.4. By construction, it is clear that . Thus for any σ 0 > 0, there exists σ > 0 such that for any C, i, j, t in Construction 9.3, supp X (V σ C,i,t,j ) ⊂ (C, σ 0 ). The following lemma describes the values taken by V σ C,i,t,j at the corners of the loops that we constructed in Lemma 9.2. For any C ∈ D , we have C ⊂ (C, σ) for some C ∈ D, and for each x ∈ ..,c | > κ . Proof. Take ε 1 :=ε 0 (f, ρ 1 , σ) and σ 1 :=σ 0 (f, ρ 1 , σ) ∈ (0, σ) given by Lemma 9.2.

On the prevalence of the accessibility property
In this section, we fix r ∈ N ≥2 ∪ {∞} and an integer J ≥ 1. In order to avoid repetition, we consider only the volume preserving case in the following. The more general case is handled by repeating exactly the same proof after replacing Diff r (X, Vol) by Diff r (X), PH r (X, Vol) by PH r (X), etc.
Let us first give an outline of the construction in this section with an illustration in Figure 5. Given a good C r − J−family f := {f ω } ω∈[0,1] J , we will find a family {f θ } θ∈U1⊂R I of C r − J−families which are perturbations of f , in which mostf θ contain a large proportion of accessible maps. More precisely, we will construct a We constructf in the following way. We apply Proposition 6.3 repeatedly to produce a well-distributed finite subset A in [0, 1] J such that for each parameter a ∈ A we get a ( 1 20 , 6)-spanning c-family D a for f a , and produce by Lemma 9.5 a vector field V σ C,i,t,j for each C ∈ D a , 1 ≤ i, j ≤ c, t ∈ B i and some small σ > 0. We construct a C r map V : [0, 1] J × R I × X → T X by gluing together the above data in a careful way, and definef = {f (ω,θ) } (ω,θ)∈[0,1] J ×U1 so that for each ω ∈ [0, 1] J ,f ω := {f (ω,θ) } θ∈U1 is a C r deformation at (0, f ω ) with I-parameters generated by V (ω, ·, ·). By choosing V carefully, we may ensure that for a typical parameter ω ∈ [0, 1] J , the C r deformationf ω exhibits approximately independent perturbations for many different su-paths. Together with a Fubini's argument, this will enable us to verify property (P) for maps at all but extremely small amount of parameters within a typical familyf θ := {f (ω,θ) } ω∈[0,1] J . Notice that the maps inf have uniformly bounded C r -norms. Thus, by studying carefully the proofs of this section, we can see that throughout this paper we only need to use the fact that h f , σ f , C f and Λ f are C 2 -uniform constants.
10.1. Constructing perturbations for a family of diffeomorphisms. In this subsection, we fix a C r −J−family {f ω } ω∈[0,1] J in the space of dynamically coherent, center bunched C r partially hyperbolic diffeomorphisms on X.
Let Ω 0 be an open set compactly supported in (0, 1) J , let U 1 be an open neighbourhood of the origin in R I for some integer I ≥ 1, and letf : Ω 0 ×U 1 ×X → X be a C r map such thatf (a, b, ·) ∈ Diff r (X), for all (a, b) ∈ Ω 0 × U 1 , andf (a, 0, ·) = f a for all a ∈ Ω 0 . In particular, for any a ∈ Ω 0 , the mapf (a, ·) : U 1 × X → X is a C rdeformation at (0, f a ). We set T a := T (f (a, ·)). Moreover, by applying Lemma 4.11 tof (a, ·) in place off , after taking U 1 sufficiently small, for any (b, x) ∈ U 1 × X, we will denote by ν a b (x, ·) : . Given an element of a C r − J−family as above, the following notion combines a global property (through spanning c-families) and a local one (existence of deformations which induce an infinitesimal displacement of the holonomies in many directions) which together will be useful to verify Property (P) in Definition 8.3.
Definition 10.1 (Removability). Letf be as above. Then for ρ m , ρ M , σ, C, κ > 0, a ∈ Ω 0 , we say thatf is (ρ m , ρ M , σ, C, κ)-Removable at a if the following is true. There exists D, a ( 1 10 , 8)-spanning c-family for f a with [r(D), r(D)] ⊂ (ρ m , ρ M ), such that for each C ∈ D, for each x ∈ 1 5 C, there exists a (σ, C)-regular continuous family γ of f a -loops at x with the following properties. Let K 0 , Γ be taken as in Subsection 7.2. For any (i, B, {s t } t∈B ) ∈ Γ and (t, j) ∈ B × {1, . . . , c}, we set γ t,j := γ(ϕ(j, s t,j )), z t := ( c j=1 H fa,γt,j )(x). Letγ t,j be the lift of γ t, for T a , and Then there exists a linear subspace H ⊂ R I of dimension K 0 c such that The main goal of this subsection is the following.
Proof. By compactness, we can choose C 1 , C > 0 so that for all a ∈ [0, 1] J , C 1 > C fa , C > C fa , where C fa is given by Construction 9.3; C fa is given by Lemma 9.2.
Given any C r map V : [0, 1] J ×R I ×X → T X such that V (a, ·) is an infinistesimal C r deformation with I-parameters for any a ∈ [0, 1] J , we associate with V a C r mapf : where U 1 is a sufficiently small neighbourhood of the origin in R I , and for any To prepare for the proof of Proposition 10.2, we first show the following lemma.
For each 1 ≤ i ≤ M 0 , we will inductively define a ( 1 20 , 6)-spanning c-family for f ai , denoted by D i , in the following way. Assume that for some k ∈ {1, . . . , M 0 }, and for all 1 ≤ i ≤ k − 1, we have defined D i satisfying: (1) D i is a σ 0 -sparse ( 1 20 , 6)-spanning c-family for f ai ; This assumption is always true for k = 1.
Let {i 1 , . . . , i l } be the set of all indices p ∈ {1, . . . , k − 1} such that W p ⊂ B(W k , 5c T ). By the choice of {a i }, we have l < K. Then we can apply Proposition 6.3 to obtain a spanning c-family for f a k , denoted by D k , such that (1), (2), (3) above are true for i = k. Moreover, for any 1 ≤ j ≤ l, (D ij , σ 0 ) is disjoint from (D k , σ 0 ), and for all a ∈ W k ⊂ B(a k , λ 0 ), we have R(f a , (D k , σ 0 ), ({D ij } l j=1 , σ 0 )) > R 1 and R ± (f a , (D k , σ 0 )) > R 1 . Having fa k σ, and let V (k) : R I k × X → T X be the infinitesimal C r deformation defined as follows: By Remark 9.4, for all sufficiently small σ > 0, we have supp X (V (k) ) ⊂ (D k , σ 0 ). Let I := M0 k=1 I k . For any B = (B k ) M0 k=1 ∈ R I , where B k ∈ R I k for each 1 ≤ k ≤ M 0 , we define a C r map V : [0, 1] J × R I × X → T X as follows: By definition, the map V is linear in B. For each a ∈ [0, 1] J , let {i 1 , . . . , i l } be the set of indices p such that Θ p (a) = 0. Note that l ≤ K. Moreover, by construction, we see that the sets (D ij , σ 0 ) are mutually disjoint for j ∈ {1, . . . , l}, and By the choice of C 1 above, (9.1), (10.5), and since D k is σ 0 -sparse for all 1 ≤ k ≤ M 0 , Again, by the above construction, we see that Take any k ∈ {1, . . . , M 0 }. By construction, D k is a σ 0 -sparse 1 20 , 6 -spanning c-family for f a k , and for any a ∈ W k , for each B = (B l ) 1≤l≤M0 ∈ R I , we see that We definef by (10.3) for V given as above. It is clear thatf is C r , and for each a ∈ [0, 1] J ,f (a, ·) : U 1 × X → X is the C r deformation at (0, f a ) generated by V (a, ·). By (10.3), (10.6), and Lemma 4.4(1), we obtain f C 1 < Q for some Q > 0 depending only on {f a } a and C 1 , after possibly reducing the size of U 1 .

10.2.
Getting accessibility by perturbation. In this subsection, we fix a map f 0 ∈ PH r (X, Vol) which is dynamically coherent and center bunched.
Proof. Point (1) follows from the fact that f is C 2 , center bunched, and Lemma 4.11. Point (2) follows from Lemma 8.8.
The main technical result of this section is the following. It provides estimates on the volume of "bad" parameters under some removability condition.
For any Q, C, κ 1 > 0, all sufficiently small h > 0, for any ρ 1 ∈ (0, h), and for all sufficiently small σ > 0, the following is true. Assume that there exist an open set Ω 0 compactly supported in (0, 1) J ; and integer I > 0; an open neighbourhood U 1 of the origin in R I ; and a C r mapf : (iii)f is (ρ, h, σ, C, κ 1 )-Removable at a, for all a ∈ Ω 0 .
Proof. We only detail the case where c = 2. We will sketch the adaptation needed for c = 1 at the end of the proof.
For any a ∈ Ω 0 ,f can be regarded as a C r deformation at ((a, 0), f a ) with J + I parameters. Let ν a,0 (x, ·) : R J+I → E su fa (x) be the (unique) linear map given by Lemma 4.11. Set T a := T (f (a, ·)) and let ν a 0 (x, ·) : R I → E su fa (x) be the unique linear map satisfying (10.1) for b = 0. It is direct to see that ν a 0 (x, B) = ν a,0 (x, {0} J × B) for all x ∈ X, B ∈ R I . In the following we tacitly use the inclusion R I ⊂ {0} J × R I and for any B ∈ R I , we abbreviate ν a,0 (x, {0} J × B) as ν a,0 (x, B).
Then, choose η > 0 small enough such that For any sufficiently small δ > 0, for each i ∈ {1, . . . , c}, for any t ∈ B i , let N t,i be a δ 1+β We denote by Σ the diagonal of (R c ) K0 R K0c , that is, and for any δ > 0, we let Σ δ be the δ-neighbourhood of Σ defined by For any a ∈ K, let D = D(a) be the ( 1 10 , 8)-spanning c-family for f a given above. For any C ∈ D, x ∈ A(a, C) and (i, B, {s t } t∈B ) ∈ Γ, set Ψ := Ψ a,C,x,i,B,{st} . By (10.13), the map DΨ is a submersion from W c T ((a, 0, x), δ) to its image, and is uniformly transverse to Σ, i.e., whenever w = (a , b , y) where for any set S, N (S, δ) is the minimal number of δ-balls required to cover S. For any 0 < δ < δ 1 , let E = E(δ) be the subset of "bad" parameters in Ω 0 × U 1 : Since in the above collection, only the last item, N t,i , depends on δ, there exists a constant D 3 > 0 such that for any 0 < δ < δ 1 , By (10.16), there exists 0 < δ 2 < δ 1 such that D 3 δ υ 2 < 1. We deduce that Leb(E) ≤ (2δ) J+I N (E, δ) < δ for all 0 < δ < δ 2 .
Appendix A.
Proof of Lemma 4.14. We detail the case where * = u. The other one is handled similarly.
To show (2), we need the following lemma.
Proof. We follow the construction in [35, Proof of Theorem 4.1] and refer to [35] for many details.
Appendix B.
Proof of Lemma 4.4. Let V : R I ×X → T X be a C r vector field as in Definition 4.2, and let U ⊂ R I be a small neighbourhood of the origin. We let F : R × U × X → X be the associate flow; it is defined by the following equation: with initial condition F (0, b, x) = f (x). For any (s, b, x) ∈ R × U × X, we have • f (x) = F (s, 0, x).
In the above equality, the first term on the RHS equals V (B, F (t, 0, x)) = V (B, f (x)); and the second term on the RHS equals 0. Thus This concludes the proof of (2).
By a slight abuse of notations, we use · to denote the uniform norm for: (a) derivatives of f , ∂ b V and ∂ b ∂ x V as functions on X; (b) derivatives off and V as functions on U × X; (c) derivatives of F as functions on [0, 1] × U × X.
To prove (1), we need to bound the norms of Df abd D 2 f . Since B → V (B, ·) is linear, it is clear that by reducing the size of U , we can assume that ∂ x V < 1 10 . Then by Grönwall's inequality and possibly reducing the size of U , there exists an absolute constant c 0 > 0 such that Since ∂ 2 b V ≡ 0, by Grönwall's inequality and (B.2), there exists an absolute constant c 1 > 0 such that Note that by possibly reducing the size of U , we can ensure that Thus there exists an absolute constant c 2 > 0 such that We conclude the proof of (1) by noticing that D i T D if for i = 1, 2.
Appendix C.
Proof of Theorem F. We repeat the proof of [10, Proposition 3.2] for -light maps, instead of light maps, for some = (n).
Recall that the order of a cover O = {O k } k∈K is the supremum of all numbers #K such that ∩ k∈K O k = ∅. Let then V 0 be a cover of X := [0, 1] n such that V 0 does not admit an open refinement of order less than or equal to n. Take δ > 0 a Lebesgue number of the cover V 0 and define := δ/2.
Assume that f : X → Y is -light. Let T be some triangulation of Y and denote by U = {U i } i∈I its open star cover. For every i ∈ I, f −1 (U i ) can be written as a disjoint union of connected open sets; they form an open cover of X, denoted by V = {V j } j∈J . For each j ∈ J, we let α(j) ∈ I be such that V j ⊂ f −1 (U α(j) ).
By assumption, f is -light, hence we can choose T fine enough such that each V j has diameter smaller than 2 = δ. Therefore V is an open refinement of V 0 ; in particular, any open refinement of V has order at least n + 1.
We define Ner(U) as the collection of subsets I ⊂ I such that ∩ i∈I U i = ∅. We define Ner(V) in a similar way. Both Ner(U) and Ner(V) are simplicial n-complexes, and we identify them with their geometric realization.
Given a partition of unity {ρ i } subordinate to U, we get a homeomorphism ρ : Y → Ner(U), while α induces a local homeomorphism φ : Ner(V) → Ner(U). Then, the functions ν j := χ Vj · (ρ α(j) • φ), j ∈ J, define a partition of unity subordinate to V. We let ν : X → Ner(V) be the associate map, where ν(x) has barycentric coordinates (ν j (x)) j . By construction, the following diagram commutes: Let us see that some n-simplex σ of Ner(V) has an interior point ξ which is a stable value of ν : X → Ner(V). Assume it is not the case; then we may form a set S by choosing one interior point from each n-simplex of Ner(V) and perturb ν on a small neighbourhood of ν −1 (S) to get ν : X → Ner(V)\S. Denote by p : Ner(V)\S → [Ner(V)] n−1 the radial projection to the (n − 1)-skeleton of Ner(V), s.t. the barycentric coordinates of ν := p • ν : X → [Ner(V)] n−1 are subordinate to V. Pulling back the open star cover of Ner(V) by ν , we get a refinement of V of order at most n, a contradiction. Thus, ρ −1 (φ(ξ)) is a stable value of f , which concludes.