A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies

We prove a forward Ergodic Closing Lemma for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms, claiming that the set of eventually strongly closable points is a total probability set. The "forward" means that the closing perturbation is involved along a finite part of the forward orbit of a point in a total probability set, which is the same perturbation as in Mane's Ergodic Closing Lemma for \begin{document}$ C^1 $\end{document} diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms away from homoclinic tangencies is proved, extending the result for \begin{document}$ C^1 $\end{document} diffeomorphisms by Liao, Viana and Yang.


SHUHEI HAYASHI
and then denote by Σ ev + (f ) the set of x ∈ M belonging to Σ ev + (U, f ) for every neighborhood U of f , which is the set of eventually strongly closable points. That is, if x ∈ Σ ev + (f ) then there exists n 0 ≥ 0 such that f n0 (x) is forwardly (U, ε)strongly closable for every neighborhood U of f and every ε > 0. If {U n } n≥0 is a basis of neighborhoods of f in End 1 (M ), we can write When f ∈ Diff 1 (M ), we can define Σ(U, ε) as the diffeomorphisms version of Σ + (U, ε) which was introduced by Mañé [14] who proved that given a neighborhood U of f ∈ Diff 1 (M ) and ε > 0, µ(Σ(U, ε)) = 1 for every µ ∈ M f (M ). This implies that n≥0 Σ + (V n , ε n ) with V n = U n ∩ Diff 1 (M ) is a total probability set of f , or a set having full µ-measure for every µ ∈ M f (M ). We consider an endomorphisms version of this. Unfortunately, we need to take the union of backward iterates of the set of forwardly strongly closable points to ensure the total probability even for nonsinguar endomorphisms. However, it is still useful to obtain asymptotic properties of the forward orbits of points in a total probability set.
For f ∈ End 1 (M ) and p ∈ M , we say that p is a critical point of f if Df |T p M is not injective. Denote by C(f ) the set of critical points of f , which is a compact subset of M . When C(f ) = ∅ we say that f is nonsingular. Let The "forward" means that the perturbation for the closing is involved along a finite part of the forward orbit of µ-almost every x for every µ ∈ M e (f ). The "backward" Ergodic Closing Lemma for nonsingular C 1 endomorphisms has been proved by Moriyasu [19] (see also [5]). Its perturbation for the closing is based on Wen's C 1 Closing Lemma [30] for nonsingular endomorphisms, making the perturbation along a finite part of the backward orbit of a given nonwandering point. In the endomorphisms case, since prehistories of a given point is not unique, some different feature from the diffeomorphisms case may arise. One of the advantages of the forward Ergodic Closing Lemma is that we can safely understand some results for diffeomorphisms obtained by using Mañé's Ergodic Closing Lemma still hold for nonsingular endomorphisms because the perturbation itself is the same.
For the statement of the next theorem, we need to give several definitions. Let X be a compact metric space and let f : X → X be a continuous map. For x ∈ X, ε > 0 and n ∈ Z + , we denote by B n (x, ε) the dynamical ball (or Bowen ball) at x of radius ε > 0 and length n, that is, the set of y ∈ X such that d(f j (x), f j (y)) ≤ ε for all 0 ≤ j < n. Then, for a subset K of X, an (n, ε)-spanning set E for K is a set satisfying Let r n (K, ε) denote the smallest cardinality of any (n, ε)-spanning set for K. Then we also say that K is (n, ε)-spanned by E. Set r(K, ε) = lim sup n→+∞ 1 n log r n (K, ε).
Then the topological entropy h(f ) of f is defined by h(f ) = h(f, X). Denote by B ∞ (x, ε) the set of y ∈ X such that d(f j (x), f j (y)) ≤ ε for all j ≥ 0. For ε > 0 and a subset A of X, we say that f is ε-entropy expansive around A if h(f, B ∞ (x, ε)) = 0 for every x ∈ A. When A = X, we say that f is ε-entropy expansive. If there exists ε > 0 such that sup x∈A h(f, B ∞ (x, ε)) = 0, then f is said to be entropy expansive around A. As an intermediate notion between one-sided expansiveness and entropy expansiveness, we say that f is linearly entropy expansive around A if there exists ε > 0 such that for every x ∈ A and every δ > 0 we have a constant C(x, δ) > 0 satisfying r n (B ∞ (x, ε), δ) ≤ C(x, δ)n for all n sufficiently large. It is easy to observe that this definition is independent of the choice of equivalent metric and therefore shared by topologically equivalent maps. When ε > 0 is specified for which f satisfies the inequality above, we say that f is linearly ε-entropy expansive around A. We will consider this notion for f ∈ End 1 (M ) with respect to the Riemannian metric d on M .
In dealing with f ∈ End 1 (M ), it is natural to consider the inverse limit space of f (see [23,28] for instance). Define the inverse limit space M f by Moreover, for a compact subset Λ ⊂ M that is f -invariant (i.e., f (Λ) = Λ), define Given β > 1, we endow a metric d β on M f by d β (x,ŷ) = x → (f (x 0 ), x 0 , x −1 , . . . ). Then,f is a homeomorphism and the tangent bundle T Λ f over Λ f is defined by its fibers: A compact f -invariant set Λ with Λ∩C(f ) = ∅ is hyperbolic if there exists a splitting such that Df (E s (x)) = E s (f (x)) and D x0 f (E u (x)) = E u (f (x)), and there are constants C > 0 and 0 < λ < 1 satisfying Df n |E s (x) ≤ Cλ n and m(D x0 f n |E u (x)) ≥ C −1 λ −n for all x ∈ Λ,x ∈ Λ f and n ≥ 0, where m(T ) = min{ T v : v = 1} is the minimum norm for a linear transformation T . Then we say that E s and E u are contracting and expanding, respectively. In particular, it is called an Axiom A basic Then we have continuous families of C 1 embedded discs, local stable manifolds See [23,Theorem 2.1] for this fact and it is also known that if y ∈ W s δ (x, f ) with x ∈ Λ then d(f n (x), f n (y)) → 0 as n → +∞, and if y ∈ W u δ (x, f ) withx ∈ Λ f then y has a unique prehistoryŷ = (y i ) i≤0 such that If there exists a nontransversal homoclinic point associated with some hyperbolic periodic point, we say that f exibits a homoclinic tangency. Denote by HT the set of f ∈ End 1 (M ) exhibiting a homoclinic tangency.
The second theorem concerns the robust entropy expansiveness for nonsingular C 1 endomorphisms away from homoclinic tangencies.
Theorem B. Let f ∈ NEnd 1 (M ) \ HT . Then there exist a neighborhood U of f and ε > 0 such that every g ∈ U is ε-entropy expansive.
Remark. It is well-known ( [18]) that the entropy expansiveness implies the upper semicontinuity of the entropy map µ → h µ (f ) defined on the space of f -invariant probability measures. Consequently, as a corollary of Theorem B, we see that f ∈ NEnd 1 (M ) \ HT has a measure of maximal entropy (see [29] for instance), extending [12, Corollary C] to the nonsingular endomorphisms case.
The last theorem is a contribution to Shub's Entropy Conjecture. We denote by f * ,k : where sp(f * ,k ) is the spectral radius of f * ,k . Shub's Entropy Conjecture ( [26]) claims that the logarithm of sp(f * ) is a lower bound for the topological entropy of f : The full statement of the conjecture is known to be true when dim M ≤ 3 by the result of Manning [16] (together with Poincaré duality); i.e., log sp(f * ,1 ) ≤ h(f ) holds for continuous maps. For the historical background, one can consult [8] or [12]. Here we focus on some recent progress on the Entropy Conjecture for f ∈ Diff 1 (M ). After the classical proofs (Shub and Williams [27], Ruelle and Sullivan [24]) of the Entropy Conjecture for Axiom A diffeomorphisms with no cycles in the early 70's, relatively recently Saghin and Xia [25] proved it for partially hyperbolic diffeomorphisms with one-dimensional center bundle. Shortly afterwards, Liao, Viana and Yang [12] established the conjecture for f ∈ Diff 1 (M ) away from homoclinic tangencies. Since all the diffeomorphisms mentioned above are C 1 away from homoclinic tangencies, their result remarkably includes all of them. However, in the original view of the Entropy Conjecture, the invertibility of f is not essential, so its extension to C 1 endomorphisms should be considered. By Yomdin's influential work [32] that established the Entropy Conjecture for C ∞ maps, the proof of the conjecture for f ∈ End 1 (M ) is reduced to proving the upper semicontinuity of the topological entropy at f . Actually, this approach has been used in [12] and other recent works ( [3,4,13]). However, the C 1 creation of homoclinic tangencies, or the appearance of transversal homoclinic points by arbitrarily small C 1 perturbations can be the obstruction for its upper semicontinuity as observed in [17] by Misiurewicz. In fact, Liao, Viana and Yang [12] proved the upper semicontinuity at C 1 diffeomorphisms away from homoclinic tangencies through the robust entropy expansiveness. Our proof is in line with theirs. We first prove the one-sided version of [12, Proposition 2.5] for a continuous map f : X → X on a compact metric space X. From this it follows that the almost entropy expansiveness is sufficient to guarantee the full entropy expansiveness. Therefore, under a proposition claiming the robust linearly ε-entropy expansiveness around a total probability set we obtain the robust ε-entropy expansiveness as in Theorem B. For the proof of the proposition we use the forward Ergodic Closing Lemma by which similar arguments to the diffeomorphisms case become available.
For the robustness, we need S(g) having the δ-neighborhood U δ (S(g)) of S(g) with some δ > 0 independent of g such that U δ (S(g))∩C(g) = ∅ for all g sufficiently close to f ∈ End 1 (M ) \ HT . However, it may happen that S(f n ) intersects an arbitrarily small neighborhood of C(f n ) as f n → f in the C 1 topology for some f , f n ∈ End 1 (M ) \ HT , n ≥ 1. This is the difficulty of extending Theorem B (and therefore Theorem C below) to f ∈ End 1 (M ) \ HT satisfying only S(f ) ∩ C(f ) = ∅ that is the hypothesis of Theorem A.
As mentioned above, the Entropy Conjecture for f ∈ NEnd 1 (M ) \ HT easily follows from the upper semicontinuity of the topological entropy at f through Yomdin's theorem ( [32]); i.e., the Entropy Conjecture is true for C ∞ maps on M . In fact, choosing a convergent sequence of C ∞ maps f n → f , n ≥ 1, in the C 1 topology from the homotopy class of f , we have sp((f n ) * ) = sp(f * ), and then where the upper semicontinuity is used in the first inequality. Similarly to the proof of diffeomorphisms case in [12], the robust ε-entropy expansiveness implies the upper semicontinuity of the map f → h(f ) for f ∈ NEnd 1 (M ) \ HT (Theorem 3.7), extending [12,Theorem D] and proving the Entropy Conjecture for nonsingular C 1 endomorphisms away from homoclinic tangencies.
Theorem C. If f ∈ NEnd 1 (M ) \ HT , then the Entropy Conjecture holds for f .
Let us give two typical examples of endomorphisms belonging to NEnd 1 (M )\HT . We say that f ∈ End 1 (M ) is Axiom A (or satisfies Axiom A) if the nonwandering set Ω(f ) is hyperbolic and contains a dense periodic points. It is known (see [23,Theorem 2.3] for instance) that every nonsingular Axiom A endomorphism has the spectral decomposition by Axiom A basic sets: Note that the transitivity and continuity imply that the dimensions of the stable and unstable manifolds are constant over each Axiom A basic set Λ i , 1 ≤ i ≤ s. Define a partial ordering on the Axiom A basic sets by declaring A cycle among Axiom A basic sets are a nontrivial chain Λ i1 < Λ i2 · · · < Λ i k = Λ i1 . If f ∈ NEnd 1 (M ) has a cycle, by a local perturbation, f can be C 1 approximated by some g ∈ NEnd 1 (M ) exhibiting a homoclinic tangency similarly to the diffeomorphisms case. Therefore, nonsingular Axiom A endomorphisms with no cycles belong to NEnd 1 (M ) \ HT .
We say that a compact set Λ ⊂ M \ C(f ) with f (Λ) = Λ admits a dominated splitting if there exists a continuous splitting for all x ∈ Λ and allx ∈ Λ f with π(x) = x 0 = x. Then, we say that the splitting More generally, when we have a splitting for any pair of nontrivial subbundles E i and E j with 1 ≤ i < j ≤ 3, we say that the splitting is dominated (resp. m-dominated). If M f admits a dominated splitting with dim E 2 = 1 such that E 1 and E 3 are contracting and expanding, respectively, f is called partially hyperbolic with one-dimensional center, which is obviously C 1 far away from homoclinic tangencies. Thus, we see that nonsingular partially hyperbolic endomorphisms with one-dimensional center belong to NEnd 1 (M ) \ HT . Theorems B and C will be proved in Section 3 after giving the Preliminaries in Section 2. Finally, we will prove Theorem A in Section 4.

2.
Preliminaries. In this section, we give basic definitions and known facts that will be needed in the following sections. The first fact is the existence of locally invariant discs associated with dominated splittings. Let Λ ⊂ M \ C(f ) satisfy f (Λ) = Λ and admit a dominated splitting

A FORWARD ERGODIC CLOSING LEMMA AND THE ENTROPY CONJECTURE 2291
Applying the proof of [11,Theorem 5.5], we have a continuous family of atx ∈ M f with size η 0 > 0. These discs are locally invariant in the sense that there is a constant L 0 ≥ 1 such that if 0 < η ≤ η 0 thenf (D F η/L0 (x)) ⊂D F η (f (x)) for allx ∈ Λ f . Moreover, by the same argument as in the proof of [10, Theorem 6.2], if η 0 > 0 is small enough, E(x), x ∈ Λ, are extended to some E(y) over y ∈ π(D F η0 (x)), having a continuous family of C 1 discs {D E η0 (y) : y ∈ π(D F η0 (x))} in M tangent to E(y) at y with size η 0 > 0 and D E η0 (y) = D E η0 (x) when y = x. These discs are also locally invariant in the sense that if 0 < η ≤ η 0 then for all y ∈ π(D F η0 (x) ∩f −1 (D F η0 (f (x)))) with anyx ∈ Λ f . Next we recall definitions and facts about topological entropy (see [29] for more details). The definition of the topological entropy given in the Introduction coincides with that of topological entropy using open covers. Let α be a finite open cover of X and let |α n | be the smallest cardinality of a subcover of Define, for ε > 0, the ε-local entropy by Bowen gave the following inequality in [2, Theorem 2.4]: given ε > 0, In the proof of this inequality, Bowen used the following lemma, which will be also needed in the next section. . Suppose 0 = t 0 < t 1 < · · · < t r−1 < t r = n and f ti (K) is (t i+1 − t i , ε)-spanned by E i for all 0 ≤ i < r. Then when diam α ≤ ε. Given ε > 0, we say that a continuous map f : X → X is ε-almost entropy expansive if there exists a total probability set A ⊂ X of f such that h(f, B ∞ (x, ε)) = 0 for every x ∈ A. This is the one-sided version of the definition introduced in [12] for homeomorphisms by Liao, Viana and Yang. A direct consequence of (2.2) shows that the upper semicontinuity of the topological entropy follows from the robust ε-entropy expansiveness with respect to the C 0 topology. In fact, when f n → f in the C 0 topology, letting α be a finite open cover of X with diam α ≤ ε, from (2.2) together with the robust ε-entropy expansiveness and the upper semicontinuity of Finally, we recall the so-called Pliss Lemma, which has been one of the basic tools in the study of weak hyperbolicity. Denote by (x, y; f ) with y = f n (x), n ≥ 0, a finite set {f j (x) : 0 ≤ j ≤ n}, which is called a srting. For f ∈ End 1 (M ) and The following lemma is due to Pliss [20] (see also [15,Lemma 11.8] and [12, Lemma 3.5]): (Pliss Lemma). For all 0 < γ <γ < 1 there exist N (γ,γ) > 0 and 0 < c(γ,γ) < 1 such that the following properties hold.
3. Proofs of Theorems B and C. In [12], Liao, Viana and Yang proved that if f ∈ Diff 1 (M ) \ HT , then there exist a neighborhood U of f and ε > 0 such that every g ∈ U is ε-entropy expansive. More precisely, letting HT then there exist a neighborhood U of f , ε > 0 and a constant C(δ) > 0 depending on an arbitrarily small δ > 0 such that for all g ∈ U and n sufficiently large, r n (B ± ∞ (x, ε, g), δ) ≤ C(δ)n holds at every x in a total probability set of g (see the proof of [12, Theorem 3.1]). In this section, we extend this property to g C 1 close to f ∈ NEnd 1 (M )\HT , replacing the property with B ± ∞ (x, ε, g) and C(δ) by B ∞ (x, ε, g) and C(x, δ) at every x in a total probability set of g, where and C(x, δ) is a constant depending on x and δ. The following lemma, the one-sided version of [12, Proposition 2.5], achieves the ε-entropy expansiveness from the εalmost expansiveness as a = 0. The proof is essentially the same as (or even slightly simpler than) that of [12, Proposition 2.5], but we give the proof for completeness.
Proof. Suppose on the contrary that h(f, B ∞ (x 0 , ε)) > a for some x 0 ∈ X. Let δ > 0 be an arbitrarily small positive number. Then, we can take constants c > a 0 > a and a subsequence m i → +∞ of m = 1, 2, . . . such that for all i ≥ 1. Choosing a subsequence of m i , i = 1, 2, . . . , if necessary, we may assume that a sequence of probabilities converges in the weak* topology to some f -invariant probability measure µ on the Borel σ-algebra of X. Take b ∈ (a 0 , c) and define Γ n ⊂ X by the set of for all m ≥ n. Note that Γ n ⊂ Γ n when n ≤ n . From the hypothesis, it follows that µ n≥1 Γ n = 1.
By the regularity of µ, we can take a compact subset Λ n of Γ n with µ(Γ n \ Λ n ) → 0 as n → +∞. Fix some arbitrarily large n ≥ 1 and choose an (n, δ/4)-spanning set E n (y) for B ∞ (y, ε), y ∈ Λ n , with #E n (y) < e bn , which implies

Define a neighborhood
for every u ∈ V n (y). Note that any subset of U n (y) is (n, δ/2)-spanned by E n (y). Choose y 1 , . . . , y s for which the compact set Λ n is covered by V n (y j ), 1 ≤ j ≤ s, and set W n = s j=1 V n (y j ), and N n = max{n, N n (y 1 ), . . . , N n (y s )}.

SHUHEI HAYASHI
Here we may assume that µ(∂W n ) = 0 to have which can be arbitrarily close to 1. Let κ δ ∈ Z + be the cardinality of a δ/2-dense subset of X. Note that any subset of X is (1, δ/2)-spanned by the δ/2-dense subset.
Since n ≥ 1 can be arbitrarily large (depending on δ), we can fix n so large that for all i sufficiently large. Now define 0 = t 0 < t 1 < · · · < t ki = m i (where we can suppose m i > N n ) inductively as: , and then Now apply Lemma 2.1 to obtain Hence, using (3.2), we have In what follows, we consider B ∞ (x, ε) as X = M for the Riemannian metric d on M with dim M = ν. We can suppose ε > 0 of B ∞ (x, ε) is always smaller than ε 0 . The weak hyperbolicity over the forward orbit of x defined below are important to prove Theorem B, which will be obtained through Theorem A. Let We put subbundles E s , E cs , E cu and E u as: Now we introduce a coordinatelike structure in B ∞ (x, ε) and its positive iterates for x ∈ Λ and small ε > 0 by using locally invariant C 1 discs given in Section 2. For small η > 0 letD σ η (x) and D σ η (y) with y ∈ π(D σ η (x)), (σ, σ ) ∈ {(s, u), (s, cu), (cs, u)}, be locally invariant C 1 discs tangent to E σ (x) and E σ (y) atx and y, respectively, with size η > 0 given in Section 2 for a dominated splitting Here we suppose 0 < η ≤ η 0 and 0 < ε ≤ η/(3L 0 ) for η 0 < ε 0 /2 and L 0 given in Now, for σ ∈ {s, cs} we have a continuous locally invariant C 1 disc family x ∈ Λ f , with a pre-lamination structure (see [11]) whose leaves F σ , v σ ≤ θ v σ }. To simplify the notation, when such cones are used, we always identify the εneighborhood of x in M with that of 0 in R ν as x = 0. For fixedx ∈ Λ f with π(x) = x, by the continuity of F cŝ x , it is easy to see that x u ∈F u (x) F cŝ x (x u ) contains a neighborhood of x in M by the same argument as in the proof of [10,Lemma 4.1]. When E 2 is trivial and We consider both cases simultaneously by assuming σ = cs in the former case and σ = s in the latter case. By the uniformity with respect to ). By this uniform structure, makingε > 0 smaller if necessary we can suppose that for all x ∈ Λ,x ∈ Λ f and j ≥ 0. Take 0 < ε < ε θ <ε with small 0 < θ < 1.
Then the properties of the derivatives on E σ over strings (x, f n (x); f ), n ≥ 1, are carried over to those on tangent spaces of a part of locally invariant discs F σ xj (x u j ), 0 ≤ j ≤ n, with any fixedx ∈ Λ f . As n → +∞, we can understand the dynamics of the ε-neighborhood of the forward orbit of x through the coordinatelike structure around f j (x), j ≥ 0. Indeed, for every y ∈ B ∞ (x, ε) if an s-distance and a udistance of y from x are defined, then for any j > 0 those of f j (y) from f j (x) are also defined as the coordinatelike structure around f j (x). Moreover, not only for all j ≥ 0, restricted to which f is a diffeomorphism onto its image keeping the coordinatelike structure as above.
For the proofs of the following three lemmas, we take e −ρ < γ < 1 for ρ > 0 given in the lemmas and let 0 <ε < η/(3L 0 ) be so small that if ε <ε then exp −1 there exist ε > 0 and 1 ∈ Z + such that given δ > 0 arbitrarily small, Proof. In this case, we write if ε > 0 is small enough and n > m 1 then the smallest cardinality of (n − m 1 , δ)-spanning set for F ŝ . Therefore, letting κ ∈ Z + be the cardinality of a δ-dense subset of M and using Lemma 2.1, we obtain for all x u ∈ F u (x) and n > m 1 . On the other hand, since F u (x) is one-dimensional, we have r n F u (x) ∩ B ∞ (x, ε), δ ≤ (3ε/δ)n for all n ≥ 1. Thus, making ε = ε(θ(γ)) > 0 smaller if necessary, the coordinatelike structure and these properties give Suppose that there exist m ∈ Z + and ρ > 0 such that F is weakly (ρ, m, f )-expanding atx ∈ Λ f . Then, there exists ε > 0 such that given δ > 0 arbitrarily small, r n (B ∞ (x, ε), δ) ≤ (3ε/δ)n for all n ≥ 1.
Proof. In this case, we write E s = E and E u = F . Since F is weakly (ρ, m, f )expanding atx, we have lim inf From this and Lemma 2.2 (b), there exist a subsequence n k → +∞ and 1 which is a diffeomorphism onto its image. If 0 < ε < ε θ(γ) <ε has been chosen small enough, by (3.4) with σ = s, we have x, ε)). Then the maximal u-distance of y k = g −1 0 (z k ) from x (ranging over all suchz k ) is arbitrarily small for large k. Since for all n ≥ 1. Combining (3.6) and (3.7), we obtain the required inequality.
Proof. The proof is given combining arguments for the proofs of the previous two lemmas. Let ν 1 = dim E s < ν = dim M . Then, if 0 < ε < ε θ(γ) <ε is small enough then property (a) of the hypothesis and Lemma 2.2 (a) imply that there exists 1 ≥ 1 such that for all x cu ∈ F cu (x) and n > m 1 , which corresponds to (3.5) and the proof is similar.
On the other hand, from property (b) of the hypothesis and Lemma 2.
, then the maximal u-distance of such y k from x is arbitrarily small for large k, which is independent of the dimension of E 2 that is 0 or 1. Then we have for all n ≥ 1, which corresponds to (3.6) and the proof is similar. Since the dimen- for all n ≥ 1. Thus, making ε = ε(θ(γ)) > 0 smaller if necessary, the coordinatelike structure and properties (3.8), (3.9) and (3.10) give for all n > m 1 . 2 When n → +∞ in these lemmas, the only concern is that whether 1 = 1 (x, n), n ≥ 1, in the proofs of Lemmas 3.2 and 3.4 are bounded independent of the choice of n or not. If there is a sequence n j → +∞ for which 1 (x, n j ) is not bounded, setting t j = 1 (x, n j ), we have a sequence of strings (x, f mtj (x); f m ), j ≥ 1, such that (x, f mtj (x); f m ) with large j is not an e −ρ -string over E s . In fact, otherwise applying Lemma 2.2 (a) to the string ( is also a uniform γ-string over E s with k j appeared before t j = 1 (x, n j ), satisfying the property of 1 (x, n j ) for large j and contradicting the smallest choice of 1 (x, n j ). Thus E s in the proofs of Lemma 3.2 and 3.4 is not (ρ, m, f )-contracting atx, which contradicts our hypothesis and we conclude sup n 1 (x, n) < +∞. (3.11) Now let us consider the following cases for a dominated splitting f is (ρ, m, f )-contracting and E 3 f is weakly (ρ, m, f )-expanding atx. By (3.11) and Lemmas 3.2, 3.3 and 3.4, in all cases (i)-(iii) we have 0 < θ(γ) < 1 and ε = ε(θ(γ)) > 0 with e ρ < γ < 1 for which there exists a constant C(x, δ) > 0 such that given δ > 0 arbitrarily small, for all x ∈ Λ 0 and n sufficiently large. From this together with the following lemma, we obtain the robust ε-almost entropy expansiveness.
Apply (3.12) to f = g ∈ U 0 , Λ = S(g), Λ 0 = Λ 0 (g), m = m 0 and ρ = ρ 0 given by Lemma 3.5. By the uniform choice of U 0 , m 0 and ρ 0 in Lemma 3.5, it is easy to see that ε > 0 in (3.12) including the choice of ε <ε < ε 0 (guaranteeing a coordinatelike structure in theε-neighborhood of x around which g is a diffeomorphism onto its image) can be fixed independent of the choice of x ∈ Λ 0 (g) and g ∈ U 0 , where we may suppose U 0 ⊂ NEnd 1 (M )\HT . Thus we have proved the following proposition. Proposition 3.6. Let f ∈ NEnd 1 (M ) \ HT . Then there exist a neighborhood U 0 of f and ε > 0 such that every g ∈ U 0 is linearly ε-entropy expansive around a total probability set of g.
The diffeomorphisms version of Lemma 3.5 is included in Crovisier [6, Corollary 1.3] and Liao, Viana and Yang [12,Proposition 3.4]. These results should be extended to our case just replacing Dg −m |E 3 g (g m(j+1) (x)) , j ≥ 0, by for some m ∈ Z + using Theorem A. In fact, by hypothesis, if f ∈ NEnd 1 (M ) \ HT then there exist a neighborhood U 0 of f and ε 0 > 0 such that for every g ∈ U 0 the restriction of g ∈ U 0 to the ε 0 -neighborhood of any point of M is a diffeomorphism onto its image. Therefore, as long as perturbations are made in a sufficiently small neighborhood of a periodic orbit for g ∈ U 0 in order to obtain properties for the periodic orbit, there is no essential difference between nonsingular endomorphisms and diffeomorphisms. Consequently, the basis of their results for diffeomorphisms such as Franks' Lemma [7] (see also [14 subbundles over periodic orbits to those over the forward orbit of µ-almost every x for every µ ∈ M e (g) to continue. However, for completeness and slight differences from the diffeomorphisms case, we give a proof of the lemma.
Proof of Lemma 3.5. As mentioned above, by the same argument to prove [31, Lemma 3.3 and 3.4] (see also [12,Proposition 4.1]), there exist a neighborhood U 0 of f in NEnd 1 (M ) \ HT , constants γ 0 > 0 and m ∈ Z + such that for every g ∈ U 0 and every p ∈ Per(g), lettingp = (p i ) i≤0 ∈ S(g) g be such that π(p) = p and p i ∈ O + g (p) for all i ≤ 0, we have an m-dominated splitting with dim E 2 g ≤ 1 whose nonzero vectors in E 2 g have Lyapunov exponents in [−γ 0 , γ 0 ], satisfying that if p has period τ ≥ m then Here the angles between any two distinct subbundles of (3.13) are uniformly bounded away from zero by its m-domination (see [1, page 288] for elementary properties of dominated splittings). Considerx = (x i ) i≤0 ∈ S(g) g with π(x) = x ∈ Σ ev + (g) satisfying for some µ ∈ M e (g). Then apply Theorem A to find j 0 ≥ 0,g ∈ U 0 arbitrarily close to g and a periodic pointp ∈ Per(g) with periodτ such that d(g j+j0 (x),g j (p)) is arbitrarily small for all 0 ≤ j ≤τ , where j 0 does not depend on the choice ofg andp by the definition of Σ ev + (g). By the Poincaré Recurrence Theorem, we may suppose x ∈ Σ ev + (g) is recurrent and therefore the periodic orbit O + g (p) can be arbitrarily close to supp (µ) in the Hausdorff metric. Take sequences {g n } n≥1 and {p n } n≥1 of suchg andp ∈ Per(g) so that g n → g in the C 1 topology and (1/τ n ) τn−1 j=0 δ g j n (pn) → µ in the weak* topology, where τ n is the g n -period of p n . By continuity and accumulation, the sequence of m-dominated splittings over O + gn (p n ), n ≥ 1, given by (3.13) attaches to supp (µ) g and hence to S(g) g an m-dominated splitting by ranging over x ∈ Σ ev + (g). Now if (3.14) and (3.15) hold for p ∈ Per(g) with period τ ≥ m, then replacing m by some large multiple m 0 of m if necessary, we see that the property of this lemma holds as m 0 , ρ 0 = γ 0 and Λ 0 (g) = Σ ev + (g). Next, in the case where µ is supported on infinitely many points, we consider the above application of Theorem A through (3.16) similarly to the diffeomorphisms case as in [14, page 523]. Perturb g n with large n slightly to h n by using Franks' Lemma so that O + gn (p n ) = O + hn (p n ) and h n → g for which we have an m-dominated splitting with dim E 2 n ≤ 1, satisfying (3.13), (3.14) and (3.15) with g and E ι g replaced by h n and E ι n = E ι hn , ι ∈ {1, 2, 3}, and such that the properties of the derivatives on E 1 n and E 3 n over O + hn (p n ) are isometrically translated to those on E 1 g and E 3 g over the string (ĝ j0 (x),ĝ τn+j0 (x);ĝ), respectively. Here the translation is achieved through isomorphisms A n,j : T h j n (pn) M → T g j+j 0 (x) M such that E ι hn |{ĥ j n (p n )} = A −1 n,j (E ι g |{ĝ j+j0 (x)}), the restrictions to E ι hn |{ĥ j n (p n )} are isometries for all 0 ≤ j ≤ τ n and A −1 n,0 (E ι g |{ĝ j0 (x)}) = A −1 n,τn (E ι g |{ĝ τn+j0 (x)}). Since j 0 ≥ 0 does not depend on the choice of h n and p n , n ≥ 1, when τ n → +∞ we have lim inf By the ergodicity of µ, the limit infimum of (3.17) can be changed to the limit, which together with (3.18) gives the required property in this case for some large multiple m 0 of j 0 (m + 1)m, ρ 0 = γ 0 and Λ 0 (g) = Σ ev + (g). Finally, let us consider the case where p ∈ Per(g) has period τ < m. According to (3.13), the Lyapunov exponents for g ∈ U 0 of nonzero vectors of E 1 g , E 2 g and E 3 g are in (−∞, −γ 0 ), [−γ 0 , γ 0 ] and (γ 0 , +∞), respectively. Denote by Per l (g) the set of p ∈ Per(g) with the minimum period ≤ l and letp = (p i ) i≤0 ∈ Per m (g) g be such that π(p) = p and p i ∈ O + g (p) for all i ≤ 0. By the uniformity of the constant γ 0 > 0 and angles between any two distinct subbundles of (3.16) for g ∈ U 0 , we cannot find h arbitrarily g |{p} is a nonhyperbolic isomorphism. Then, define uniformly hyperbolic families {ξ (p,g,ι) : g ∈ U 0 , p ∈ Per m (g)}, ι ∈ {1, 3}, of periodic sequence of isomorphisms ξ [14,Lemma II.3,(c)] to obtain that for every {ξ (p,g,ι) : g ∈ U 0 , p ∈ Per m (g)}, ι ∈ {1, 3}, there exists m 0 ∈ Z + such that lim sup and lim sup Here, in the proof of [14, Lemma II.3, (c)], setting inequality (3.19) is proved from that for all g ∈ U 0 , p ∈ Per m (g) and j ∈ Z we have Then, choosing as m 0 a common multiple of m and m, we can find ρ 0 > 0 less than γ 0 such that inequalities (3.19) and (3.20) are actually written as: Thus, if m 0 has been chosen as some large multiple of m, the required property holds including the case where p ∈ Per m (g) as Λ 0 (g) = Σ ev + (g). 2 Applying Lemma 3.1 to a = 0 with Proposition 3.6, we see that if f ∈ NEnd 1 (M )\ HT , there exist a neighborhood U of f and ε > 0 such that every g ∈ U is ε-entropy expansive to obtain Theorem B. Then (2.3) implies the following theorem. 4. Proof of Theorem A. In this section, we deal with f ∈ End 1 (M ) satisfying S(f ) ∩ C(f ) = ∅. Then for every p ∈ S(f ) there is a neighborhood U 0 (p) of p such that f |U 0 (p) is a diffeomorphism onto its image. Therefore the following perturbation given in [14,9] still works if it is made in a small neighborhood of a finite part of the forward orbit of p.
Let p ∈ S(f ) and a neighborhood U of f be given. Recall that Per l (f ) is the set of periodic points of f with the minimum period ≤ l. Let R > 0 be such that the ball B R,p = {x ∈ T p M : x ≤ R} is diffeomorphic to a compact neighborhood of p in U 0 (p) by the exponential map. Without loss of generality we may identify them for simplicity to write also B R (p). Then, given constant 1 < δ < 2 there exist J ∈ Z + , 0 < r 1 = r 1 (p) < r 0 = r 0 (p) < R with r 0 arbitrarily small and a norm · 1 equivalent to · , satisfying the following properties. and Restricting the observation of [9, (2.2)] over M to S(f ), we can take {p 1 , . . . , p k } such that with one perturbation framework in each B r0(pi) (p i ) given as (I) and (II).
Given a neighborhood U of f ∈ End 1 (M ) and ε > 0, we say that a pair of points (z, y) with y = f (z) for some ∈ Z + is a (U, ε)-strongly closable pair of f if y = z or y = z satisfies properties (E-a), (E-b) and (E-c) below. Then, since we can apply perturbation (II) to (z, y) in order to obtain a periodic orbit {g j (y) : 0 ≤ j ≤ − 1} with period for some g ∈ U.
(1) Similarly to the diffeomorphisms case, the norm · 1 above actually depends on the configuration of (z, y) in a uniform way (see [14] or [22]). For simplicity, we omit the dependence here.
(2) Since f |f j (Int B δir ((y + z)/2)) is a diffeomorphism onto its image for all 0 ≤ j ≤ J i , there is no difference in the closing perturbation between diffeomorphisms and endomorphisms as long as (E-b) is satisfied. Ji−1 j=1 f j (Int B δir ((y + z)/2)) without hitting Int B δir ((y + z)/2). This interference is the new difficulty in the endomorphisms case. This motivates us to change (D-b) to (E-b) in order to avoid the interference.
Given ∈ Z + , denote by Σ − (U, ε, ) the set of z by which (z, y) becomes a (U, ε)-strongly closable pair for some y with y = f (z). Then, define Define a Borel set Σ J (U, ε) as the set of points x ∈ M such that there exist g ∈ U, ∈ Z + and y ∈ Per (g) satisfying g = f on M \ B ε (f, x, J) and d(f j (x), g j (y)) ≤ ε By the Ergodic Decomposition Theorem, the proof of Theorem A is reduced to proving that given a neighborhood U of f , for every µ ∈ M e (f ). For the proof, given x in a total probability set we need to have some m ≥ 0 such that f m (x) ∈ Σ + (U, ε) for every ε > 0. The nontrivial case is when µ is supported on infinitely many points and then finding f m (x) as a (U, ε)strongly closable point for every ε > 0 suffices to prove (4.1). As the previous step to find (U, ε)-strongly closable pairs, we define pairs satisfying weaker properties. Given a neighborhood U of f and ε > 0, we say that a pair of points (z, y) with y = f (z) for some ∈ Z + is a (U, ε)-quasi-closable pair of f if y = z or y = z satisfies (E-c) in addition to the following property (D-b), one of the properties required for (U, ε)-strongly closable pairs (y, z) for diffeomorphisms in [9]: In particular, when f ∈ End 1 (M ) is a diffeomorphism, adding property (E-a) to a (U, ε)-quasi-closable pair (z, y) makes it (U, ε)-strongly closable. But, as mentioned in (3) of the Remarks above, it is not sufficient to be (U, ε)-strongly closable for endomorphisms. Denote by Σ J (U, ε) the set of z for which (z, y) becomes a (U, ε)-quasi-closable pair for some y. Then, by the same argument as in the proof of Mañé's Ergodic Closing Lemma ( [14], see also [9]), it is easy to see that Σ J (U, ε) is a Borel set and for every µ ∈ M e (f ).
The following selection lemma is the fundamental procedure for the closing, which goes back to Pugh's Closing Lemma [21]. Mañé investigated Pugh's pointwise selection lemma from a global viewpoint in his proof of the Ergodic Closing Lemma [14], dividing the ambient manifold to countable pieces around which the specified smallness and the shapes of boxes for perturbation (II) are available. In [9], it was observed that the finite choice of shapes of the boxes is enough as long as the sufficient smallness is guaranteed. The finiteness is used in the following selection lemma in order to give ρ > 1 independent of the choice of q around which pairs are selected. However, the selection itself with the norm · 1 is the same as one used in the proof of the Ergodic Closing Lemma. So, we omit the proof (see [14,Lemma I.4]).
Given a neighborhood U of f and ε > 0, there exist ρ > 1 and 0 <r 0 =r 0 (ρ) < min{r 1 (p i )/2 : 1 ≤ i ≤ k} such that if q ∈ B r1(pi)/2 (p i ) and {x, f m (x)} ⊂ B r (q) with some m > 0 and 0 < r ≤r 0 , then there are 0 ≤ m 1 < m 2 ≤ m such that Remark. If r > 0 is much smaller than min{ε, min 1≤i≤k r 1 (p i )/2} then the selection to obtain (f m1 (x), f m2 (x)) as above around q by the norm · 1 always works to get properties (D-b) and (E-c). However, we don't know if r > 0 is so small to satisfy (E-c). This is the reason for assuming (E-c) in the conclusion. On the other hand, to get also property (E-a) we need smallness of pairs determined by ε i (q, J i ), which is not assumed in Lemma 4.1.
Let us give the notation and properties involved in the proof of the Ergodic Closing Lemma [14] (see also [9]). For a neighborhood U of f in End 1 (M ), ε > 0, r > 0 and ρ > 1, define a Borel set Σ(U, ε, r, ρ) as the set of points x ∈ M such that if w ∈ Br(x) and f m (w) ∈ Br(x) for some 0 <r ≤ r and m ∈ Z + , then there exist 0 ≤ m 1 < m 2 ≤ m such that (f m1 (w), f m2 (w)) is a (U, ε)-quasiclosable pair with f m1 (w), f m2 (w) ∈ B ρr (x). Note that this property is weaker than the property required for the set written by the same notation in [14] even for diffeomorphisms. If r n > 0, n ≥ 0, is a monotone decreasing sequence converging to 0, then Σ(U, ε, r n , ρ), n ≥ 0, is an increasing sequence and it is known ( [14,9]) that S(f ) = n≥0 Σ(U, ε, r n , ρ).
The fact that ρ > 1 can be fixed was observed in [9] (while Mañé used a sequence ρ m → +∞). This property corresponds to the uniformity of shapes of perturbation boxes (see [14]) from which J ∈ Z + is fixed after U was fixed first. The advantage is that smallness r n for (U, ε)-quasi-closable pairs with property (E-b) to be (U, ε)strongly closable can be uniformly chosen outside a neighborhood of Per J (f ). For the measure-theoretical argument in the proof of the Ergodic Closing Lemma, Mañé introduced a sequence of partition of M as follows. The ambient manifold M is isometrically embedded in T s = S 1 × s · · · × S 1 and a cube A as an atom of the partition of T s is defined by A = I 1 × · · · × I s where I i , 1 ≤ i ≤ s, are intervals in S 1 with equal lengths. If p i ∈ I i is the middle point of I i then (p 1 , . . . , p s ) is called the center of A. For each k ∈ Z + , let P (k) 1 ≤ P (k) 2 ≤ · · · be a sequence of partitions, where P (k) j is the partition whose each atom A have the side 2π/k j (the length of the interval I i ). Then, denote byÂ andÃ the cubes with the same center as A's with the sides 2π/k j−1 and 6π/k j−1 , respectively. If x ∈ T s , we denote by P In order to consider µ ∈ M(M ) with the partition of T s , we extend µ to a measure on T s by defining µ(B) = µ(B ∩ M ) for every Borel set B of T s . Now let µ ∈ M e (f ) be supported on infinitely many points. Given ρ > 1 and n ≥ 0, there is j(n) ≥ 1 such that for all j ≥ j(n) take 0 < r ≤ r n and an odd integer k ≥ 1 satisfying where B t (w) is the closed ball in T s with radius t and center w. Note that if w ∈ M then B t (w) ∩ M = B t (w). Without loss of generality, we may assume that j (x) = 0 for all k, j and x ∈ M . The following lemma corresponds to [14,Lemma I.6] and its proof is the same as that of [14,Lemma I.6], giving only a sketch of the proof for later arguments.

A FORWARD ERGODIC CLOSING LEMMA AND THE ENTROPY CONJECTURE 2307
Sketch of Proof. Since µ is ergodic, there is x 0 ∈ M such that the hitting frequency in P j (x) as → +∞, respectively. If there are consecutive points in the forward orbit of x 0 hitting P (k) j (x), we can choose a (U, ε)quasi-closable pair in P j (x) minus one is less than or equal to that of those hitting P (k) ε). On the other hand, by the hypothesis of δ-compatibility µ P j (x), the hitting frequency of the forward orbit of x 0 in P j (x), obtaining the required inequality.
2 Now we need to change some of (U, ε)-quasi-closable pairs to (U, ε)-strongly closable ones. The following lemma is the first step.
Now, if 0 < ε <r 0 then, by Lemma 4.1 with x = q = f j0 (z), m = t 0 − j 0 and r = ε, there exist t 0 ≤ t 1 < t 2 ≤ such that, setting (z , y ) = (f t1 (z), f t2 (z)), if (z , y ) satisfies property (E-c) then we have a pair (z , y ) ⊂ B r1(pi) (p i ) with properties (Db) and (E-c), becoming (U, ε)-quasi-closable. If the (U, ε)-quasi-closable pair (z , y ) does not satisfy (E-b) again, this is the same situation for the pair (z, y) above, and we can continue the same process to get a new pair (z , y ) instead of (z , y ), satisfying properties (D-b) and (E-c) when (z , y ) is (U, ε)-quasi-closable. Since such pairs are always found in the finite set {f j (z) : 0 ≤ j ≤ }, we finally obtain the pair that is not (U, ε)-quasi-closable or a (U, ε)-quasi-closable pair satisfying property (E-b) as well as (E-c). Writing the pair as (f s1 (z), f s2 (z)) with 0 ≤ s 1 < s 2 ≤ , we obtain the pair as required. 2 Remark. The hypothesis that (f s1 (z), f s2 (z)) is (U, ε)-quasi-closable seems superfluous at first sight. However, because of the selection process using the norm · 1 , the size of the pair f s1 (z) − f s2 (z) 1 is not clear although pairs necessary to us are only small ones (see the Remark below Lemma 4.1). We will see later that large pairs can be neglected.
The next step for (U, ε)-quasi-closable pairs to be (U, ε)-strongly closable is obtaining property (E-a). Let x 0 / ∈ Per(f ) be µ-almost every point for µ ∈ M e (f ) and let i , i ≥ 1, be a monotone increasing sequence such that { i : i ≥ 1} is the set of j ≥ 1 for which f j (x 0 ) ∈ Σ J (U, ε) with ε <r 0 . Then there are and (E-c) does not satisfy property (E-a). We consider two cases according to the positions of f ti (x 0 ), i ≥ 1. To simplify the notation, put (z , y ) = (f ti (x 0 ), f ui (x 0 )), which will be called a pair chosen from (f i (x 0 ), f ki (x 0 )). For every ξ ∈ Per J (f ) with period ≤ J, we can find a neighborhood V (ξ) of ξ such that f |f j (V (ξ)) is a diffeomorphism onto its image for all 0 ≤ j ≤ J, and if x ∈ V (ξ) then for all 0 ≤ j ≤ . Since Per J (f ) is compact, we can take a finite subcover {V (ξ l ) : 1 ≤ l ≤ τ } of {V (ξ) : ξ ∈ Per J (f )} and let the union be V J , i.e., On the other hand, for every q ∈ S(f ) \ V J , there is a neighborhood U (q) such that f |f j (U (q)) is a diffeomorphism onto its image for all 0 ≤ j ≤ J. Then we can take ε i (q, J i ) > 0 given in (E-a) with q ∈ B r1(pi)/2 (p i ) for some 1 ≤ i ≤ k so small that Int B εi(q,Ji) (q) ⊂ U (q). Then set W (q) = Int B εi(q,Ji)/2 (q).
Since S(f ) \ V J is compact, we can take a finite subcover {W (q l ) : 1 ≤ l ≤ t} of {W (q) : q ∈ S(f ) \ V J } and let the union be W J , i.e., and let η = min 1≤l≤t ε i(l) (q l , J i(l) )/2.
Since Λ 1/m,n is increasing in both n ≥ 0 and m ≥ 1, the proof of (4.1) is reduced to proving that for any a > 0 there exist arbitrarily large m and n such that For the proof of this inequality, we need the following lemma corresponding to [14,Lemma I.7], whose proof is omitted for it is the same as that of [14,Lemma I.7] where Σ(U, ε) c is used instead of Σ ev + (U, f ) c . In fact, we can replace Σ(U, ε) c by any Borel set in the proof of [14,Lemma I.7].
If (4.7) does not hold, there exist a 0 > 0 and m 0 , n 0 ≥ 1 such that for any m ≥ m 0 and n ≥ n 0 we have To lead a contradiction from this inequality, we will find a sufficient quantity of points in Σ ev + (U, f ) around Σ ev + (U, f ) c ∩ Λ 1/m,n . By Lemma 4.5, for fixed such m and n, we have an arbitrarily small neighborhood U of Σ ev where one can recall from (4.2) that µ( Σ J (U, ε)) = 1. Take a neighborhood U N of Σ ev If U has been taken small enough so that it contains a small amount of points in Σ ev + (U, f ) depending on m, we have (4.10) To simplify the notation, we put ji (x i ), i = 1, 2, . . . , N , with x i ∈ Λ 1/m,n . We need the following lemma in order to restrict the argument to U N and use (4.9).