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Physical measures at the boundary of hyperbolic maps

We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a"small"subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastical stability. The physical measures are obtained as zero-noise limits which are shown to satisfy the Entropy Formula.


INTRODUCTION
Let M be a compact and connected Riemannian manifold and Diff 1+α (M) be the space of C 1+α diffeomorphisms of M for a fixed α > 0. We write m for some fixed measure induced by a normalized volume form on M that we call Lebesgue measure, dist for the Riemannian distance on M and · for the induced Riemannian norm on T M.
We say that an invariant probability measure µ for a transformation Let T Ω( f 0 ) = E s ⊕ E u be a hyperbolic D f 0 -invariant decomposition (Whitney sum) of the tangent bundle of the non-wandering set Ω( f 0 ) of f 0 . The classical construction of physical measures involves f 0 -invariant measures which are absolutely continuous with respect to Lebesgue measure along the unstable direction through the points of Ω( f 0 ). These uniformly hyperbolic dynamical systems were the first general class of systems where these measures were shown to exist [11,35,37].
An invariant probability measure is called SRB (Sinai- Ruelle-Bowen) measure, if it admits positive Lyapunov exponents and its conditional measures along the unstable manifolds (in the sense of Pesin theory [34,17]) are absolutely continuous with respect to Lebesgue measure induced on the unstable manifolds. For a class of dynamical systems which includes uniformly hyperbolic systems the notions of physical and SRB measures coincide.
The SRB measures as defined above are related to a class of equilibrium states of a certain potential function. Let φ : M → R be a continuous function. Then a f 0 -invariant probability measure µ is a equilibrium state for the potential φ if where M is the set of all f 0 -invariant probability measures.
For uniformly hyperbolic diffeomorphisms it turns out that physical (or SRB) measures are the equilibrium states for the potential function φ(x) = − log | det D f |E u (x)|. It is a remarkable fact that for uniformly hyperbolic systems these three classes of measures (physical, SRB and equilibrium states) coincide.
We will address the problem of the existence of physical measures on the boundary of uniformly hyperbolic diffeomorphisms. The idea is to add small random noise to a deterministic system f 0 in the boundary of uniformly hyperbolic systems and, for a large class of such maps, we prove that as the level of noise converges to zero, the stationary measures of the random system tend to equilibrium states for f 0 which are physical measures. The stationary measures exist in a very general setting, but the "zero noise" limit measures are not necessarily physical measures. The specific choice of random perturbation is important to obtain equilibrium states as zero noise limits. These equilibrium states satisfy Pesin's Entropy Formula and by the characterization of measures satisfying this formula (whose proof in [27] is well known to be valid also for C 1+α diffeomorphisms) we deduce that such zero noise limits are SRB measures. In the setting of the main theorems every SRB measure is a physical measure. The same general idea has been used in [15] to obtain SRB measures for partially hyperbolic maps under strong asymptotic growth conditions on every point.
Let (θ ε ) ε>0 be a family of Borel probability measures on (Diff 1+α (M), B(Diff 1+α (M))), where we write B(X) the Borel σ−algebra of a topological space X . We will consider random dynamical systems generated by independent and identically distributed diffeomorphisms with θ ε the probability distribution driving the choice of the maps.
We say that a probability measure µ ε on M is stationary for the random system (Diff 1+α (M), θ ε ) if Z Z ϕ( f (x)) dµ ε (x)dθ ε ( f ) = Z ϕ dµ ε for all continuous ϕ : M → R. (1.1) We will assume that supp(θ ε ) → f 0 when ε → 0 in a suitable topology. A result based on classical Markov Chain Theory (see [26] or [4]) ensures that every weak * accumulation point of stationary measures (µ ε ) ε>0 when ε → 0 is a f 0 -invariant probability measure, called a zeronoise limit measure. It is then natural to study the kind of zero noise limits that can arise and to define stochastic stability when the limit map f 0 admits physical measures.
We say that a map f 0 is stochastically stable (under the random perturbation given by (θ ε ) ε>0 ) if every accumulation point µ of the family of stationary measures (µ ε ) ε>0 , when ε → 0, is a linear convex combination of the physical measures of f 0 .
Stochastic stability has been proved for uniformly expanding maps and uniformly hyperbolic systems [24,26,40,43]. For some non-uniformly hyperbolic systems, like quadratic maps, Hénon maps and Viana maps, stochastic stability has been obtained much more recently [3,7,8]. The authors have studied random perturbations of intermittent maps and have proved stochastic stability for these maps for some parameters and certain types of random perturbations [5]. The techniques used were extended to higher dimensional local diffeomorphisms exhibiting expansion except at finitely many points, enabling us to obtain physical measures directly as zero-noise limits of stationary measures for certain types of random perturbations, proving also the stochastic stability of these measures.
Stochastic stability results for maps of the 2-torus which are essentially Anosov except at finitely many points were obtained in [15], the physical probability measures of which were constructed in a series of papers using different techniques [23,22,21]. Similar results for different kinds of bifurcations away from Anosov maps at fixed points were also studied in [14].
Using ideas akin to [5] and [15] we prove the existence of physical probability measures and their stochastic stability for diffeomorphisms which are "almost Anosov" under some geometric and dynamical conditions. 1.1. Statement of the results. We assume that f 0 : U 0 → f 0 (U 0 ) is a C 1+α diffeomorphism on a relatively compact open subset U 0 of a manifold M which is strictly invariant, that is, closure( f (U 0 )) ⊂ U 0 . During the rest of this paper we set Λ = ∩ n≥0 closure f n 0 (U 0 ). Moreover we suppose there exists a continuous dominated splitting We may see E ⊕ F on U 0 as a continuous extension of E ⊕ F on Λ. This assumption ensures the existence of for all x ∈ U and a, b ∈ (0, 1), which are D f 0 -invariant in the following sense ; Continuity enables us to unambiguously denote d E = dim(E) and d F = dim(F), so that d = d E + d F = dim(M). Domination guarantees the absence of tangencies between stable and unstable manifolds, since the angles between the E and F directions are bounded from below away from zero at every point. Let us fix the unit balls of dimensions d E , d F where · 2 is the standard Euclidean norm on the corresponding Euclidean space. We say that a C 1+α embedding ∆ : In what follows we denote by H (A) the Hausdorff dimension of a subset A ⊂ M. We first state the results without mentioning random perturbations.
If in addition f 0 | Λ is transitive, then there exists a unique physical measure supported in Λ, with d F positive Lyapunov exponents along the F-direction and whose basin has full Lebesgue measure in U 0 .
We note that if E 1 ∪ F 1 contains no periodic points and is finite, then some power of f 0 is a uniformly hyperbolic map, in which case the conclusions of Theorem A are known. Moreover from the dominated decomposition assumption (1.2) we easily see that E 1 ∩ F 1 = / 0. We remark also that the conditions on E 1 and F 1 in the statement of Theorem A are automatically satisfied whenever E 1 or F 1 is denumerable.
The restriction on the Hausdorff Dimension is used to show that any curve inside a E-disk (or F-disk) intersects E 1 (or F 1 , respectively) in a zero Lebesgue measure subset. In particular our results can be obtained assuming that E 1 and F 1 do not contain any such curves.
We clearly may specialize this result for a transitive C 1+α -diffeomorphism admitting a dominated splitting on the entire manifold and satisfying items (1)-(3) of Theorem A, up to replacing U 0 and Λ by M. Remark 1. We can adapt the statement of Theorem A to the setting where U 0 has a partially hyperbolic splitting, that is, the strictly forward f 0 -invariant open subset U 0 admits a continuous splitting • E s is uniformly contracting and E u is uniformly expanding: there exists σ > 1 satisfying • the restriction of the splitting to Λ is D f 0 -invariant. If we assume that Λ is transitive and either then there exists a unique absolutely continuous f 0 -invariant probability measure µ 0 with dim E u (case 1) or dim E c + dim E u (case 2) positive Lyapunov exponents, whose support is contained in Λ and with an ergodic basin of full Lebesgue measure in U 0 .
The statement essentially means that if an attractor admits a partially hyperbolic splitting which is volume hyperbolic, does not admit mixed behavior along the central direction and the neutral points along the central direction form a small subset, then there exists a physical measure.
Cowieson-Young [15] have obtained similar results, albeit for the existence of SRB measures and not necessarily for physical ones (see Vásquez [41] where it is shown that in the setting of Remark 1 physical measures will necessarily be SRB measures). Moreover Cowieson-Young obtained a strong result of existence of SRB measure for partially hyperbolic maps with onedimensional central direction using the same strategy of proof. Here we get rid of the dimensional restriction assuming a dynamical restriction: see Example 6 in Section 2 for a partially hyperbolic map with two dimensional center-stable and center-unstable directions E and F in the setting of Theorem A.
These results will be derived from the following more technical one, but also interesting in itself. (1) for any non-degenerate isometric random perturbation (θ ε ) ε>0 of f 0 , every weak * accumulation point µ of a sequence (µ ε ) ε>0 of stationary measures of level ε, when ε → 0, is an equilibrium state for the potential (2) every equilibrium state µ as above is a convex linear combination of (a) at most finitely many ergodic equilibrium states having positive entropy with d F positive Lyapunov exponents, with (b) probability measures having zero entropy whose support has constant unstable Jacobian equal to one, i.e., measures whose Lyapunov exponents are non-positive. Remark 2. We note that if F 1 is denumerable, then necessarily the measures in item (2b) of Theorem B are Dirac measures concentrated on periodic orbits whose tangent map has only non-positive eigenvalues.
The restriction on the random perturbations means the following. We assume that U 0 ⊂ M admits an open subset V ⊂ closure(V ) ⊂ U and an action V → M, where V is a small neighborhood of the identity e of a locally compact Lie group G such that for all x ∈ closure(V ), setting Then we definef and take a probability measure θ ε on V , which translates into a probability measure on the family This kind of families are a special case of non-degenerate random perturbations. For more on non-degenerate random perturbation and for examples of non-degenerate isometric random perturbations, see Section 3. In particular, Theorems A and B apply to a bounded topological attracting set for a diffeomorphism on a domain of any Euclidean space. In Section 3 we show that this is enough to obtain Theorems A and B in full generality through a tubular neighborhood construction. In particular, Theorem B shows that in the setting of Theorem A (or Remark 1) the physical measures obtained are stochastically stable, as explained in Section 7.
This paper is organized as follows. In Section 2 we present some examples in the setting of the main theorems. We outline some general results concerning random maps in Section 3. In Section 4 we derive the main dynamical consequence of our assumptions and then, in Section 5, we prove that equilibrium states for f 0 must be either physical measures or measures with no expansion. Finally we construct equilibrium states using zero-noise limits in Section 6 and put together the results concluding stochastic stability for f 0 and proving Theorem A in Section 7.

EXAMPLES OF MAPS IN THE SETTING OF THE MAIN THEOREMS
Example 1. Let f 0 : T 2 → T 2 be a C 2 diffeomorphism with T = S 1 ×S 1 , obtained from an Anosov linear automorphism of the 2-torus by weakening the expanding direction F of the fixed point p in such a way that D f 0 (p) | F = Id | F. The stable direction E continues to be uniformly contracting throughout and F is still expanded by D f 0 on T 2 \ {p}.
This kind of maps where studied by Hu and Young [21,22,23]. In this setting the only physical probability measure for f 0 is δ p , whose basin contains Lebesgue almost every point of T. Hence Theorem B shows in particular that δ p is stochastically stable.
The construction can be adapted to provide maps with finitely many periodic orbits with neutral behavior along the F direction. We note that E 1 = / 0.

Example 2.
Let us take the product f 0 × E d , where f 0 is given by Example 1 and E d : In this example µ = δ p × λ is the unique physical measure, has positive entropy and only one positive Lyapunov exponent, where λ is Lebesgue measure on S 1 .
We note that Examples 1 and 2 can be seen as "derived from Anosov" (DA) maps [42,13] at the boundary of the set of Anosov diffeomorphisms.
In this case F 1 = {0} × R 2 but every F-disk ∆ intersects F 1 at most finitely many times, since ∆ must be locally a graph over S 1 . Theorem A holds and Theorem B shows that every equilibrium state is a convex linear combination of δ (0,5/9) with ν ((0, 5/9) is the unique fixed point of f 0 ).
, where E d was defined in Example 2, d ∈ N, d ≥ 2 and g : T → S 1 of class C 1+α is an extension of g α from Example 3 to T given by x)| ≥ 1 and equals 1 only at (0, 0). The natural projection π : x)) over the attractor Λ = ∩ n≥1 f n 0 (T × closure(B(0, 1))). We note that each g t is conjugate to E 2 through a homeomorphism h t which depends continuously on t ∈ S 1 in the C 0 topology. Hence This shows that we can apply Remark 1 obtaining the existence of a unique physical measure for f 0 . .
Then the map β : I \ K → I is uniformly continuous and so we can continuously extend it to I setting β | K ≡ 0. Moreover it is easy to see that β | (I \ K ) is Lipschitz (with Lipschitz constant 1) and so is its extension to I. It addition, with respect to Lebesgue measure on I, we get , then g 0 (0) = 0, g 0 (1) = 2 and g 0 induces a C 1 map of the circle onto itself whose derivative is Lipschitz satisfying g ′ The map g 0 : S 1 → S 1 is mixing since σ(J) = |g 0 (J)|/|J| > 1 for every arc J ⊂ S 1 , where | · | denotes length. Indeed the continuity of the map σ on arcs together with the compactness of the family Γ(ℓ) = {J ⊂ S 1 : J is an arc and |J| ≥ ℓ}, for any given bound ℓ > 0 on the length, show that there exists σ(ℓ) > 1 such that |g 0 (J)| ≥ σ(ℓ) · |J| for any given arc / 0 = J ⊂ S 1 . Hence for every nonempty arc J there exists n = n(J) ∈ N such that g n 0 (J) = S 1 . Replacing g α by g 0 in the definition of f 0 within Example 3, we get a C 1+1 map from the solid torus into itself whose topological attractor satisfies the conditions of Theorem A, where F 1 is Cantor set. [10].

Example 6. We present an example of a transitive diffeomorphism with 2-dimensional centerunstable and center-stable directions in the setting of Theorem A. The idea for the construction of this example comes from the construction of stably transitive diffeomorphisms without any uniformly hyperbolic direction in
We start with a linear Anosov diffeomorphism f 0 induced in T 4 by a linear map of R 4 with eigenvalues Up to replacing it by some iterate we may suppose that f 0 has at least two fixed points p and q. For small α > 0 we consider a new diffeomorphism f satisfying the following properties: (1) f has center-unstable cone field C cu and center-stable cone field C cs with width bounded by α > 0, respectively, containing the unstable and stable subbundle of f 0 ; (2) there exists σ > 1 such that | det D f |T D cu | ≥ σ for every disk tangent to the cone field C cu and | det D f |T D cs | ≤ σ −1 for every disk tangent to the cone field C cs ; x outside the union of two small balls V p around p and V q around q, and v cu ∈ C cu and v cs ∈ C cs ; (4) the stable index (the dimension of uniformly contracting subbundle of the tangent space) at p is equal to 1 and the unstable index is equal to 2. For q the indexes are given just exchanging "stable" by "unstable" in the case of p; (5) inside the union of the balls mentioned at item 3 above we have To obtain such f we just modify f 0 in a small neighbourhood along the weaker stable direction of p and the weaker unstable direction of q. So the strong stable and strong unstable directions are preserved and f is partially hyperbolic. Since f is transitive (see [10]), by the special tangent bundle decompositions at p and q we conclude that there cannot exist any two-dimensional invariant sub-bundle with uniformly hyperbolic behavior (either uniformly expanding or uniformly contracting). In this example, E 1 = {p} and F 1 = {q}, where E 1 and F 1 are as in Theorem A, and the tangent bundle admits a dominated decomposition into four invariant one-dimensional subbundles E ss ⊕ G ⊕ H ⊕ E uu , both E = E ss ⊕ G and F = H ⊕ E uu are two-dimensional and f satisfies all the hypothesis of Theorem A.

RANDOM PERTURBATIONS
Let a parameterized family of mapsf : X → Diff 1+α (M),t → f t be given, where X is a connected compact metric space. We identify a sequence f 0 , f 1 , f 2 , . . . from Diff 1+α (M) with a sequence ω 0 , ω 1 , ω 2 , . . . of parameters in X and the probability measure θ ε can be assumed to be supported on X . We set Ω = X N to be the space of sequences ω = (ω i ) i≥0 with elements in X (here we assume that 0 ∈ N). Then we define in Ω the standard infinite product topology, which makes Ω a compact metrizable space. The standard product probability measure θ ε = θ N ε makes (Ω, B, θ ε ) a probability space. We write B = B(Ω) for the σ-algebra generated by cylinder sets: the minimal σ−algebra containing all sets of the form {ω ∈ Ω : ω 0 ∈ A 0 , ω 1 ∈ A 2 , ω 2 ∈ A 2 , · · · , ω l ∈ A l } for any sequence of Borel subsets A i ⊂ X , i = 0, · · · , l and l ≥ 1. We use the following skew-product map where σ is the left shift on sequences: (σ(ω)) n = ω n+1 for all n ≥ 0. It is not difficult to see that µ ε is a stationary measure for the random system (f , θ ε ) (i.e. satisfying (1.1)) if, and only if, If we defineΩ = X Z to be the set of all bi-infinite sequences (ω i ) i∈Z of elements of X , then we can define G to be the invertible natural extension of F to this space:

This map is invertible and
). OnΩ we set the natural product topology and the product σ-algebraB = B(Ω) generated by cylinder sets as above but now with indexes in Z. The product probability measureθ ε = θ Z ε makes (Ω,B,θ ε ) a probability space. We set the following notation for the natural projections For ω ∈Ω and for n ∈ Z we define for all Given x ∈ M and ω ∈Ω the sequence ( f n ω (x)) n≥1 is a random orbit of x. Analogously we set f n ω = π M • F n for n ≥ 0 and ω ∈ Ω.
From now on we assume that the family (θ ε ) ε>0 of probability measures on X is such that their supports have non-empty interior and supp(θ ε ) Remark 3. We note that θ ε cannot have atoms by condition ND2 above.
The following is a finiteness result for non-degenerate random perturbations.
be a non-degenerate random perturbation of f 0 . Then for each ε > 0 there are finitely many absolutely continuous ergodic measures µ ε 1 , . . . µ ε l(ε) , and for each Moreover the interior of the supports of the physical measures are nonempty and pairwise disjoint.
The continuity of the map F is enough to get the forward invariance of supp(µ ε ) for any stationary measure µ ε , i.e. if x ∈ supp(µ ε ) then f t (x) ∈ supp(µ ε ) for all t ∈ supp(θ ε ), since θ ε × µ ε is F-invariant. By non-degeneracy condition ND1 supp(µ ε ) contains a ball of radius δ 1 = δ 1 (ε). Moreover defining the ergodic basin of µ ε by then m(B(µ ε )) > 0, since µ ε (B(µ ε )) = 1 by the Ergodic Theorem applied to (F, θ ε × µ ε ) and µ ε ≪ m. These non-degeneracy conditions are not too restrictive since we can always construct a nondegenerate random perturbation of any differentiable map of a compact manifold of finite dimension, with X the closed ball of radius 1 around the origin of a Euclidean space, see [4] and the following subsection.

Isometric random perturbations.
We present below the two main types of families of maps we will be dealing with, satisfying conditions P1-P3 stated in Subsection 1.1.
where L u (x) = u · x is the left translation associated to u ∈ M. The invariance of the metric means that left (an also right) translations are isometries, hence fixing u ∈ U and taking any , closure( f 0 (U 0 )) ⊂ U 0 , then we can define a non-degenerate isometric random perturbation setting • V a small enough neighborhood of the origin in G. Now we show that we can construct non-degenerate isometric random perturbations in the setting of Examples 7 and 8. We define the family of mapsf as in (1.4). The local compactness of G gives a Haar measure ν on G and the isometry condition ensures that dim(G) = d and that Hence for every probability measure θ ε given by a probability density with respect to ν we have (f x ) * θ ε ≪ m, and this gives condition ND2.
Moreover whenever supp(θ ε ) has nonempty interior in V then condition P2, together with the compactness of closure(V ), ensure that there is δ = δ(ε) > 0 such that condition ND1 is satisfied.
Thus we get conditions ND1 and ND2 choosing θ ε as a probability density in V whose support has nonempty interior, and setting X = V for the definition of Ω,Ω.
3.2.1. Isometric perturbations of maps in arbitrary manifolds. Now we show that for any given map f 0 is the setting of Theorems A or B, we may define a random isometric perturbation of a particular extension of f 0 as in Example 8, which is partially hyperbolic.
We may assume without loss that M is a compact sub-manifold of R N and that · and dist are the ones induced on M by the Euclidean metric of R N , by a result of Nash [31,32] with Let also π : W 0 → M be the associated projection: π(w) is the closest point to w in M for w ∈ W 0 , so that the line through the pair of points w, π(w) is normal to M at π(w), see e.g. [20] or [18]. Now we define for ρ 0 ∈ (0, 1) is normal to T w M at w ∈ M and uniformly contracted by DF 0 , as long as ρ 0 is close enough to zero.
We can now define a random isometric perturbation of F 0 and obtain Theorem A as a corollary of Theorem B. For that it is enough to prove Theorem B for non-degenerate random isometric perturbations on an strictly invariant open subset of the Euclidean space. Then given f 0 we construct F 0 as explained above and note that any F 0 -invariant measure must be concentrated on M ⊂Û 0 , thus the results obtained for F 0 are easily translated for f 0 .
3.2.2. The random invariant set. In this setting, letting U 0 denote the strictly forward f 0 -invariant set from the statements in Section 1.1 and U k = f k 0 (U 0 ) for a given k ≥ 1, we have that for some ε 0 > 0 small enough where we set X = closureB(0, ε 0 ) for the definition ofΩ (and of Ω).
Indeed we have closureU k ⊂ U k−1 and d k = dist(closureU k , M \U k−1 ) > 0. Then we may find 3.3. Metric entropy for random perturbations. We outline some definitions of metric entropy for random dynamical systems which we will use and relate them. Let µ ε be a stationary measure for the random system given by (f , θ ε ) ε>0 . Since we are dealing with randomly chosen invertible maps the following results relating F-and G-invariant measures will be needed.
Here is one possibility of the calculation of the metric entropy.
is finite and is called the entropy of the random dynamical system with respect to ξ and to µ ε . θ ε ), ξ) as the metric entropy of the random dynamical system (f , θ ε ), where the supremum is taken over all measurable partitions.
Let B × M be the minimal σ−algebra containing all products of the form A × M with A ∈ B. We writeB × M for the analogous σ-algebra withB in the place of B. We denote by h B×M The analogous Kolmogorov-Sinai result about generating partitions is also available in this setting. We let A = B(M) be the Borel σ-algebra of M. We say that a finite partition ξ of M is a random generating partition We note that in [25] this result is stated only for one-sided sequences. However we know that the Kolmogorov-Sinai Theorem applied to an invertible transformation like G demands that a partition ζ ofΩ × M be generating in the sense that ∨ i∈Z G i (ζ) equalsB × M,μ ε mod 0. Since we are calculating a conditional entropy, it is enough that (∨ i∈Z G i (ζ)) ∨ (B × M) be the trivial partition in order that hB ×M Hence ζ generates (Ω × A,μ ε ) if, and only if, ξ is generating for A, since (π M ) * μ ε = µ ε .

EXPANDING AND CONTRACTING DISKS
Here we derive the main local dynamical properties of the maps in the setting of Theorem B. We show that F-disks (respectively E-disks) are expanded (resp. contracted) by the action of f 0 , and that the rate of expansion (resp. contraction) is uniform for all isometrically perturbed g in a C 1+α -neighborhood of f 0 , but depends on the size of the disks. We also show that the curvature of such disks remains bounded under iteration. These are consequences of the domination condition (1.2) on the splitting together with non-mixing of expanding/contracting behavior along the E and F directions given by condition (1) in Theorem A.
We note that for g sufficiently C 1 -close to f 0 and for a small ζ ∈ (0, α) and a slightly bigger λ 0 ∈ (λ 0 , 1) we still have for all x ∈ U Dg | E(x) · (Dg | F(x)) −1 1+ζ ≤λ 0 . (4.1) Moreover since closure( f 0 (U )) ⊂ U , then for g sufficiently C 0 -close to f 0 in Diff 1+α (M) we also have closure(g(U )) ⊂ U , see Subsection 3.2.2 for more details. We denote by U a C 1+α neighborhood of f 0 where all of the above is valid for g ∈ U.
4.1. Domination and bounds on expansion/contraction. The domination condition (1.2) ensures that the splitting E(x) ⊕ F(x) varies continuously in Λ and that there are stable and unstable cone fields E a , F b , already defined in Subsection 1.1 for small a, b > 0, which are Dg-invariant for every g sufficiently C 1 -close to f 0 . This is a general property of dominated splittings.
We define a norm on T closure(U) M more adapted to our purposes using the splitting: for every x ∈ U and w ∈ T x M we write We observe that we can make the last expression as close to (D f 0 | F(x)) −1 −1 as we like by choosing b very close to zero. Analogous calculations provide Since the above calculations give approximately the same bounds if we allow small perturbations in the factors involved, then the same conclusion holds for other constants a ′ , b ′ perhaps closer to 0 if we replace f 0 by any sufficiently C 1 -close map g. We collect this in the following lemma, which depends on the domination assumption on the splitting, on the non-contractiveness along F and non-expansiveness along E, and also on the isometric nature (specifically property (3.1)) of the perturbations we are considering.
Uniform bound on the curvature of E, F-disks. The "curvature" of the E-and F-disks defined at the Introduction will be determined by the notion of Hölder variation of the tangent bundle as follows. Let us take δ 0 sufficiently small so that the exponential map exp x : B(x, δ 0 ) → T x M is a diffeomorphism onto its image for all x ∈ closure(U 0 ), where the distance in M is induced by the Riemannian norm · . We write V x = B(x, δ 0 ) in what follows. We are going to identify V x through the local chart exp −1 x with the neighborhood U x = exp x (V x ) of the origin in T x M, and we also identify x with the origin in T x M. In this way we get that E(x) (resp. F(x)) is contained in E a y (resp. F b y ) for all y ∈ U x , reducing δ 0 if needed, and the intersection of F(x) with E a y (and the intersection of E(x) with F b y ) is the zero vector. We write ∆ also for the image of the respective embedding for every E-or F-disk. Hence if ∆ is a E-disk and y = ∆(w) for some w ∈ B E , then the tangent space of ∆ at y is the graph of a ). The same happens locally for a F-disk exchanging the roles of the bundles E and F above.
For ζ ∈ (0, 1) given by (4.1) and some C > 0 we say that the tangent bundle of ∆ is (C, ζ)-Hölder if A x (y) ≤ C dist ∆ (x, y) ζ for all y ∈ U x ∩ ∆ and x ∈ U, (4.3) where dist ∆ (x, y) is the distance along ∆ defined by the length of the shortest smooth curve from x to y inside ∆ calculated with respect to the Riemannian norm · induced on T M.
For a E-or F-disk ∆ ⊂ U we define κ(∆) = inf{C > 0 : T ∆ is (C, ζ)-Hölder}. decomposition properties for f 0 that we have already extended for nearby diffeomorphisms g ∈ U with uniform bounds.

Proposition 4.2.
There is C 1 > 0 and a small neighborhood X of t 0 such that for every sequence ) ≤ C 1 for all n ≥ 0; (c) in particular, if ∆ is as in the previous item, then for every fixed g = f ω with ω ∈ Ω J n : f n ω (∆) ∋ x → log | det(Dg | T x ( f n ω (∆))| is (L 1 , ζ)-Hölder continuous with L 1 > 0 depending only on C 1 and f 0 , for every n ≥ 1.

Locally invariant sub-manifolds, expansion and contraction.
The domination assumption on U 0 , the compactness and f 0 -invariance of Λ together with properties (1)-(2) from Theorem A ensure the existence of families of E-disks (C 1+ζ center-stable manifolds) W cs δ (x) tangent to E(x) at x and F-disks (C 1+ζ center-unstable manifolds) W cu δ (x) tangent to F(x) at x which are locally invariant, for every x ∈ Λ and a small δ > 0, as follows -see Hirsch-Pugh-Shub [19] for details.
There exist continuous families of embeddings φ cs : Λ → Emb 1+ζ (B E , M) and φ cu : Λ → Emb 1+ζ (B F , M), where ζ ∈ (0, α) is given by (4.1) and Emb 1+ζ (B, M) is the space of C 1+ζ embeddings from a ball B in some Euclidean space to M, such that for all we have the local invariance properties: for every η > 0 there exists δ > 0 such that for all

Expansion/contraction of inner radius for E/F-disks.
Up to this point we have used some consequences of the dominated decomposition assumption. Now we use assumptions (1) and analogously for Γ F (υ) with υ > 0. We define the inner radius of a F-disk ∆ (with respect to | · |) to be and likewise for E-disks. We note that R = R(∆) ≥ C dist(∆(0), ∆(∂B F )) > 0 where C > 0 relates the norms · and | · | and thus R(∆) is a minimum over Γ F (υ) for some υ > 0. For fixing ε > 0 small we can find υ > 0 and γ ∈ Γ F (υ) such that R ≤ L(∆ • γ) < R + ε, hence we may reparametrize γ such that (∆ • γ) ′ is a constant in (C(R + ε) −1 ,CR −1 ). Since Γ F (υ) is a compact family in the C 1 topology, we have that R(∆) is assumed at some smooth curve. Now we consider the family of E-disks having strictly positive inner radius, bounded curvature and bounded inner diameter: for fixed r, K, δ > 0 and k ∈ N, and analogously for D F (r, K, δ, k). Proof. We argue for E-disks only since the arguments for F-disks are the same. We note that D E (r, K, δ, k) defines a subset of bundle maps D∆ : The bound on the "curvature" of the disks bounds the Hölder constant of D∆(x) for x ∈ B E . This Hölder control together with the bounded diameter condition Finally, the uniform bound on the Hölder constant of D∆ ensures that D∆ is a equicontinuous family for ∆ ∈ D E (r, K, δ, k). The proof finishes applying Ascoli-Arzela Theorem to {D∆ : ∆ ∈ D E (r, K, δ, k)} and noting that closure(U 0 ) is compact, any limit point must share the same inner radius and diameter bounds, and also that the cone families are continuous.
From now on we fix K = C 1 from Proposition 4.2, k ∈ N big enough, δ > 0 small enough so that every E-and F-disk in the above families be contained in U 0 , and write D E (r) and D F (r) for the families in Lemma 4.3. Let λ be 1-dimensional Lebesgue measure.
Remark 4. These estimates on the inner radius enable us to improve on the local invariance properties from Subsection 4.3 as follows: for every x ∈ Λ and δ > 0 small enough there exists Indeed for any given δ > 0 we may find η > 0 such that R f 0 (W cs δ (x)) = η and so , ). This shows that for any given υ > 0 there must be an integer k ≥ 1 such that R f k 0 (W cs δ (x)) < υ, and analogously for the centerunstable disks.
Remark 5. In particular after Remark 4 we ensure that W cu δ (x) ⊂ Λ for every x ∈ Λ and δ > 0 small enough. For there is a constant C > 0 (see Subsection 4.1) such that, if y ∈ W cu δ (x), then for for all y ∈ V 0 and by definition of φ cu ; • D 2 ψ x (y, 0) : R d E → E(y) is an isomorphism, by definition of φ cs .
Hence | det Dψ x (y, 0)| is bounded away from zero for y ∈ Λ, because both the angle between E(y) and F(y) (by domination), and | det D 2 ψ x (y, 0)| are bounded from below away from zero for y ∈ Λ (by compactness). Also we note that ψ x is just the restriction to W cu δ (x) × B E of a map ψ : Λ × B E → M. This shows that ψ is a local diffeomorphism from a neighborhood of Λ × 0 in Λ × B E to a neighborhood of Λ in M. Since ψ | (Λ × 0) ≡ Id | Λ we may choose a neighborhood V 1 of 0 in B E so that ψ 0 = ψ | (Λ ×V 1 ) is a diffeomorphism onto its image, which we write W 0 . Remark 6. In addition, following the arguments of Remark 5 we get dist( f −n 0 (y), f −n 0 (x)) → 0 and dist( f n 0 (z), f n 0 (x)) → 0 when n → +∞ for all x ∈ Λ, y ∈ W cu δ (x) and z ∈ W cs δ (x). In particular this shows that forward time averages along center-stable disks and backward time averages along center-unstable disks are constant.
The special neighborhood W 0 of Λ together with Remark 6 shows that the stable set W s (Λ) = {z ∈ M : lim n→+∞ dist( f n 0 (z), Λ) = 0} of Λ coincides with the union of the stable sets of each point of Λ: W s (Λ) = ∪ x∈Λ W s (x). Proof. We have B(µ) ⊃ ∪ y∈∆∩B(µ) φ cs (y)(B E ) ⊃ ∪ y∈∆∩B(µ) W cs δ (y) by Remark 6 and definition of center-stable manifolds. We note that both the angle between the tangent space to x ∈ ∆ and E(x), and the inner radius of the center-stable leaves are bounded from below away from zero over Λ by β 0 and r 0 respectively. Hence the Lebesgue measure of B 0 = ∪ y∈∆∩B(µ) W cs δ (y) is bounded from below by h 0 · m ∆ (B(µ)), where h 0 > 0 depends only on β 0 and r 0 . Thus if ∆ ⊂ B(µ), m ∆ − mod 0, then B 0 contains a ball of radius bounded from below by h 1 · R(∆) dependent on β 0 , on r 0 and on the curvature of F-disks, all uniform over Λ. Clearly B 0 ∩ Λ = / 0.

Disks as graphs.
We can apply the results from Subsection 4.3 to any sequence of maps in U using the invariance ofΛ (see Subsection 3.2.2) and the "local product structure" from the previous discussion.
Let K = C 1 be as fixed in Subsection 4.3.1. Let k ∈ N be big enough, δ > 0 and ε 0 small enough be fixed so that every disk in D E (r), D F (r) centered atΛ ⊂ closure(U k ) ⊂ W 0 be contained in W 0 , whereΛ was defined in Subsection 3.2.2 for U small enough (corresponding to ε 0 > 0 very small). We consider the family of E-and F-disks which are local graphs as follows and likewise for G F (s) with s > 0, exchanging the roles of E, F,V 1 and V 0 . We note that since cones are complementary (i.e. any d E -subspace of E x together with any d F -subspace of F x span T x M, x ∈ U 0 ) then every E-or F-disk is a local graph for some s > 0. Let also δ 0 = sup{diam ψ x (V 0 ×V 1 ) : x ∈ Λ} > 0, which is finite by compactness.
by the local expression of f 0 on the "local product coordinates" provided by ψ. The expansion on the inner radius of the domains of the graphs is a consequence of the fact that a ball in W cs δ (x) is a E-disk and any ball in W cu The conclusion for f ω and any ω ∈ Ω holds since f ω is taken C 1 -close to f 0 .

EQUILIBRIUM STATES AND PHYSICAL MEASURES
Here we characterize the equilibrium states µ for f 0 with respect to the potential −ϕ(x) where ϕ(x) = log | det D f 0 | F(x)|, as in (1.3). We start by observing that, in the setting of Theorem B, given any f 0 -invariant measure µ the sum χ + (x) of the positive Lyapunov exponents of µ-a.e. point x equals by the Multiplicative Ergodic Theorem [33]. Indeed by condition (1) of Theorem A every Lyapunov exponent along the E direction is non-positive and every Lyapunov exponent along the F direction is non-negative. (1) finitely many ergodic equilibrium states with positive entropy and which are physical probability measures, with (2) ergodic equilibrium states having zero entropy whose support has constant unstable Jacobian equal to one, i.e., measures whose Lyapunov exponents are non-positive.
Moreover if f 0 | Λ is transitive, then there is at most one ergodic equilibrium state with positive entropy whose basin covers U 0 Lebesgue almost everywhere.
Proof. We first show that we may assume µ ergodic.

Lemma 5.2. Almost every ergodic component of an equilibrium state for ϕ is itself an equilibrium state for the same function.
Proof. Let µ be an f -invariant measure satisfying (1.3). On the one hand, the Ergodic Decomposition Theorem (see e.g Mañé [30]) ensures that On the other hand, Ruelle's inequality guarantees for a µ-generic z that (recall (5.1)) h µ ( f 0 ) > 0: According to the characterization of measures satisfying the Entropy Formula [27], µ must be an SRB measure, i.e., µ admits a disintegration into conditional measures along unstable manifolds which are absolutely continuous with respect to the volume measure naturally induced on these sub-manifolds of M. h µ ( f 0 ) = 0: Since ϕ ≥ 0 on Λ by condition (1) of Theorem A, the equality (1.3) shows that ϕ = 0 for µ-a.e. x. Hence χ + = 0, µ-a.e. and µ has no expansion.
The Ergodic Decomposition then ensures that every equilibrium state will be a convex linear combination of the two types of measures described above. The latter possibility corresponds to item (2) in the statement of Theorem 5.1. The former case with positive entropy needs more detail.
Remark 7. Up until now we have shown that ergodic equilibrium states for −ϕ are either measures with no expansion or SRB measures. This is exactly the same conclusion that Cowieson-Young get [15] in a more general setting.
The Entropy Formula (1.3) and the assumption h µ ( f 0 ) > 0 ensure that there are positive Lyapunov exponents for µ. Hence there exist Pesin's smooth (C 1+α ) unstable manifolds W u (x) through µ-a.e. point x. Moreover, as already mentioned, the disintegration µ u x of µ along these unstable manifolds W u (x) is absolutely continuous with respect to the Lebesgue measure m u x induced by the volume form of M restricted to W u (x), for µ-a.e. x.
We claim that µ(F 1 ) = 0. For otherwise there would be some component with µ u This means that Hence the Lyapunov exponents of µ along every direction in F are strictly positive. Thus dimW u (x) = dim F = d F for µ-generic x, and µ is a Gibbs state along the center-unstable direction F. These manifolds are asymptotically backward exponentially contracted by D f 0 , see [34], hence W u (x) ⊂ Λ, since Λ is a topological attractor (see the arguments in Remark 5).
Fixing a µ-generic x, since W u (x) is an F-disk Lemma 4.5 ensures that we may assume R(W u (x)) ≥ ρ for some ρ > 0 dependent only on dist(Λ, M \ U 0 ). Vásquez shows [41] that the support of any Gibbs cu-state such as µ contains entire unstable leaves W u (y) for µ-a.e. y, so we may also assume that W u (x) ∩ B(µ) has full Lebesgue measure in W u (x). Using Lemma 4.4 we get that B(µ) contains Lebesgue modulo zero a ball of radius uniformly bounded from below by h 1 · ρ > 0.
We have shown that each ergodic equilibrium state µ having positive entropy must be a physical measure. Since the ergodic basins of distinct physical measures are disjoint and have volume uniformly bounded from below away from zero, there are at most finitely many such measures. This concludes the proof of items (1) and (2) of Theorem 5.1.
Let f 0 | Λ have a dense orbit and let us suppose that there two distinct equilibrium states µ 1 , µ 2 with positive entropy. Then by the previous discussion there are two balls B 1 , B 2 contained in the ergodic basins B(µ 1 ) and B(µ 2 ) Lebesgue modulo zero, respectively, and intersecting Λ. Since f 0 | Λ is a transitive diffeomorphism and a regular map, there exists k ≥ 1 such that m( f k (B 1 ) ∩ B 2 ) > 0. Thus µ 1 = µ 2 and transitiveness of Λ is enough to ensure there is only one equilibrium state with positive entropy.
Let µ be the unique equilibrium state with positive entropy and let us take an open set As already explained, we may assume that W u (x) ∩ B(µ) has full Lebesgue measure along W u (x). We set δ = R(W u (x)) > 0.
The "local product structure" neighborhoods B and W cu δ (y) ×V 1 .
We can take z ∈ B ∩ Λ whose forward f 0 -orbit is dense in Λ. We may take z as close to x as we like and there are points (w, u) ∈ W u (x) × V 1 such that z = ψ x (w, u). Then z ∈ W cs δ (w) and W u (w) = W u (x), see Figure 1. Then by Remark 6 the forward f 0 -orbit of w is also dense in Λ.
Let n k be a sequence such that w k = f is the graph of a map from an open neighborhood V of y in W cu δ (y) to V 1 , for big enough k. Then V ∩ π 1 (Z) has positive Lebesgue measure and so, after Remark 6, the Lebesgue measure of W u (w k ) ∩ (W 0 \ B(µ)) is also positive. But this implies that W u (x) ∩ (W 0 \ B(µ)) also has positive Lebesgue measure, contradicting the choice of x.
This shows that B(µ) has full Lebesgue measure in W 0 and hence in U 0 , as in the statement of Theorem 5.1.

ZERO-NOISE LIMITS ARE EQUILIBRIUM MEASURES
Here we prove Theorem B. Let f 0 : M → M,f : X → C 1+α (M, M),t → f t , f t 0 ≡ f for fixed t 0 ∈ X , and (θ ε ) ε>0 be a family of probability measures on X such that (f , (θ ε ) ε>0 ) is a nondegenerate isometric random perturbation of f 0 , as in Subsection 3.2.
The main idea is to find a fixed random generating partition for the system (f , θ ε ) for every small ε > 0 and use the absolute continuity of the stationary measure µ ε , together with the conditions on the splitting to obtain a semi-continuity property for entropy on zero-noise limits. Theorem 6.1. Let us assume that there exists a finite partition ξ of M (Lebesgue modulo zero) which is generating for random orbits, for every small enough ε > 0.

Remark 8. Recently Cowieson-Young obtained
The absolute continuity of µ ε , the conditions on the splitting for f 0 and the isometric perturbations permit us to use a random version of the Entropy Formula Theorem 6.2. If an ergodic stationary measure µ ε for a isometric random perturbation (f , θ ε ) of f 0 , in the setting of Theorem B, is absolutely continuous for any given ε > 0, then

Moreover if condition (3) of the statement of Theorem
A also holds, then in addition to the above there exists c > 0 such that h µ ε (f , θ ε ) ≥ c for all ε > 0 small enough.
Putting Theorems 6.1 and 6.2 together shows that h µ 0 ( f 0 ) ≥ R log | det D f 0 (x)| dµ 0 (x), since θ ε → δ t 0 in the weak * topology when ε → 0, by the assumptions on the support of θ ε in Subsection 3. Since the reverse inequality holds in general (that is Ruelle's inequality [36]) we get the first statement of Theorem B. To conclude the proof we just have to recall Theorem 5.1 from Section 5, which provides the second part of the statement of Theorem B.
6.1. Random Entropy Formula. Now we explain how to obtain Theorem 6.2.
The Entropy Formula for random maps is the content of the following result. Theorem 6.3. Let a random perturbation (f , θ ε ) of a diffeomorphisms f 0 be given and assume that the stationary measure µ ε is such that when n → ∞ for θ ε × µ ε -a.e. (ω, x) by the Ergodic Theorem, if µ ε is ergodic. By the assumptions on f 0 and E ⊕F this ensures that the Lyapunov exponents in the directions of F are non-negative.
In the same way we get for θ ε × µ ε -a.e. (ω, x) and so every Lyapunov exponent in the directions of E is non-positive. Since E and F together span T U 0 M, according to the Multiplicative Ergodic Theorem (Oseledets [33]) the sum χ + of the positive Lyapunov exponents (with multiplicities) equals the following limit θ ε × µ ε -almost everywhere The identity above follows from the Ergodic Theorem, if µ ε is ergodic, since the value of the limit is F-invariant, thus constant. Finally since µ ε is absolutely continuous for random isometric perturbations, the formula in Theorem 6.3 gives the first part of the statement of Theorem 6.2.
Remark 9. The argument above together with conditions (1)-(3) from Theorem A ensure that there exists c 0 > 0 satisfying h µ ε (f , θ ε ) ≥ c 0 for every small enough ε > 0. In fact, condition Hence there is c 0 > 0 such that log | det D f 0 | F(x)| ≥ c 0 for every x in a neighborhood U k as in Subsection 3.2.2, for some fixed big k ≥ 1.
Finally, as shown in Subsection 3.2.2, for any given k ≥ 1 there is ε 0 > 0 for which the random invariant setΛ =Λ ε is contained in U k for all ε ∈ (0, ε 0 ). Then supp µ ε ⊂Λ ε will be in the setting of Remark 9 above if condition (3) of Theorem A holds in addition to conditions (1) and (2). This completes the proof of Theorem 6.2.

Uniform random generating partition.
Here we construct the uniform random generating partition assumed in Theorem 6.1. In what follows we fix a weak * accumulation point µ 0 of µ ε when ε → 0: there exist ε j → 0 when j → ∞ such that µ = lim j→+∞ µ ε j .
This shows that υ 0 can be made as small as we please. Then for υ 0 > 0 small enough there exists ∆ ∈ G E (δ) with x 0 , y 0 ∈ ∆, e.g. take the image by ψ w 0 of any d E -plane intersected with V 0 ×V 1 ⊂ R d through (y u 0 , y s 0 ), (0, y s 0 ) and ψ −1 w 0 (x 0 ). Thus we can always reduce to the first case above. The proof is complete. Lemma 6.4 implies that ξ is a random generating partition Lebesgue modulo zero, hence µ ε modulo zero for all ε > 0, as in the statement of the Random Kolmogorov-Sinai Theorem 3.6. We conclude that h µ ε k ((f , θ ε k ), ξ) = h µ ε k (f , θ ε k ) for all k ≥ 1.
Proof. For the first two items we let C n be a finiteθ ε j mod 0 partition of X such that t 0 ∈ int C n (t 0 ) with diam C n → 0 when n → ∞, for any fixed j ≥ 1. Example: take a cover (B(t, 1/n)) t∈X of X by 1/n-balls and take a sub-cover U 1 , . . . ,U l of X \ B(t 0 , 2/n) together with U 0 = B(t 0 , 3/n); then let C n = {U 0 , M \U 0 } ∨ · · · ∨ {U l , M \U l }.

STOCHASTIC STABILITY
Here we prove Theorem A. Let f 0 : M → M be as in the statement of Theorem B and let µ be an equilibrium state for −ϕ, as in (1.3) (recall the definition of ϕ in Section 5) obtained using the construction described in Section 3.2.1 through non-degenerate random isometric perturbations.
Condition (3) in the statement of Theorem A ensures that the only possibility for the ergodic decomposition of µ is the one given by item (2a) in statement of Theorem B. In fact, after Remark 9, every weak * accumulation point µ of µ ε when ε → 0 will be not only an equilibrium state for −ϕ, as shown in Section 6, but will also have strictly positive entropy h µ ( f 0 ) ≥ c > 0, after the statement of Theorem 6.2. Hence combining the statements in Section 6 with Theorem B we see that every weak * accumulation point µ of µ ε when ε → 0 is a finite convex linear combination of the ergodic equilibrium states for −ϕ, which are physical measures.
This shows that the family of equilibrium states for −ϕ in the setting of Theorem A is stochastically stable.
In addition, if f 0 is transitive, then there is only one equilibrium state µ for −ϕ which is ergodic and whose basin covers U 0 Lebesgue almost everywhere, by the last part of the statement of Theorem 5.1. Then every weak * accumulation point of µ ε when ε → 0 necessarily equals µ. This finishes the proof of Theorem A.