Mohammed Bachir
email: mohammed.bachir@univ-paris1.fr

Limited operators and differentiability

Keywords: Limited operators, Gâteaux differentiability, Fréchet differentiability, convex functions, extreme points. 2010 Mathematics Subject: Primary 46G05, 49J50, 58C20, 46B20, secondary 47B07

Introduction.

A subset A of a Banach space X is called limited, if every weak * null sequence (p n ) n in X * converges uniformly on A, that is,

lim n→+∞ sup x∈A | p n , x | = 0.
We know that every relatively compact subset of X is limited, but the converse is false in general. A bounded linear operator T : Y -→ X between Banach spaces Y and X is called limited, if T takes the closed unit ball B Y of Y to a limited subset of X. It is easy to see that T : Y -→ X is limited if and only if, the adjoint operator T * : X * -→ Y * takes weak * null sequence to norm null sequence. For useful properties of limited sets and limited operators we refer to [START_REF] Th | Limited sets in Banach spaces[END_REF], [START_REF] Bourgain | Limited operators and strict cosingularity[END_REF], [START_REF] Carrión | Banach spaces whose bounded sets are bounding in the bidual[END_REF] and [START_REF] Andrews | Dunford-Pettis sets in the space of Bochner integrable functions[END_REF].

We know that in a finite dimensional Banach space, the notions of Gâteaux and Fréchet differentiability coincide for convex continuous functions. In [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF], Borwein and Fabian proved that a Banach space Y is infinite dimensional if and only if, there exists on Y functions f having points at which f is Gâteaux but not Fréchet differentiable. They also pointed in the introduction of [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF] the observation that if the sup-norm . ∞ on c 0 is Gâteaux differentiable at some point, then it is Fréchet differentiable there. In this article we observe that this phenomenon is not just related to the sup-norm but more generally, for each convex lower semicontinuous function g : l ∞ → R ∪ {+∞}, if g is Gâteaux differentiable at some point a ∈ c 0 which is in the interior of its domain, then the restriction of g to c 0 is Fréchet differentiable at a. This hold in particular when g = (f * ) * is the Fenchel biconjugate of a convex continuous function f : c 0 → R. In fact, this phenomenon is due, (see Corollary 1 in the Appendix and the comment just before), to the fact that the canonical embedding i : c 0 -→ l ∞ is a limited operator (see the reference [START_REF] Carrión | Banach spaces whose bounded sets are bounding in the bidual[END_REF]).

The goal of this paper, is to prove the following characterization of limited operators in terms of the coincidence of Gâteaux and Fréchet differentiability of convex continuous functions.

Theorem 1. Let Y and X be two Banach spaces and T : Y -→ X be a continuous linear operator. Then, T is a limited operator if and only if, for every convex continuous function f : X -→ R and every y ∈ Y , the function

f • T is Fréchet differentiable at y ∈ Y whenever f is Gâteaux differentiable at T (y) ∈ X.
As consequence we give, in Theorem 2 below, new characterizations of infinite dimensional Banach spaces, complementing a result of Borwein and Fabian in [ [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF], Theorem 1.].

A real valued function f on a Banach space will be called a PGNF-function (see [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF]) if there exists a point at which f is Gâteaux but not Fréchet differentiable. A JN-sequence (due to Josefson-Nissenzweig theorem, see [ [START_REF] Diestel | Sequences and series in Banach spaces[END_REF], Chapter XII]) is a sequence (p n ) n in a dual space Y * that is weak * null and inf n p n > 0. We say that a function g on X * has a norm-strong minimum (resp. weak * -strong minimum) at p ∈ X * if g(p) = inf q∈X * g(q) and (p n ) n norm converges (resp. weak * converges) to p whenever g(p n ) -→ g(p). A norm-strong minimum and weak * -strong minimum are in particular unique.

Theorem 2. Let Y be a Banach space. Then the following assertions are equivalent.

(1) Y is infinite dimensional.

(2) There exists a JN-sequence in Y * .

(3) There exists a convex norm separable and weak * compact metrizable subset K of Y * contaning 0 and a continuous seminorm h on X * which is weak * lower semicontinuous and weak * sequentially continuous, such that the restriction h |K has a weak * -strong minimum but not norm-strong minimum at 0.

(4) There exists a Banach space X and a linear continuous non-limited operator T : Y -→ X.

(5) There exists on Y a convex continuous PGNF-function.

In Section 2 we give some preliminary results, specially the key Lemma 2. In Section 3, we give the proof of Theorem 1 (divided into two part, Theorem 3 and Theorem 4) and the proof of Theorem 2. In Section 4 we give some complementary remarks.

Preliminaries.

We recall the following classical result. Lemma 1. Let T 1 and T 2 be topologies on a set K such that

(1) K is Hausdorff with respect T 1 , (2) K is compact with respect to T 2 , ( 3
) T 1 ⊂ T 2 . Then T 1 = T 2 . Proof. Let F ⊂ K be a T 2 -closed set. It follows that F is T 2 -compact, since K is T 2 -compact.
Let {O i : i ∈ I} be any cover of F by T 1 -open sets. Since T 1 ⊂ T 2 , then each of these sets is also T 2 -open. Hence, there exist a finite subcollection that covers F . It follows that F is T 1 -compact and therefore is T 1 -closed since T 1 is Hausdorff. This implies that T 2 ⊂ T 1 . Consequently,

T 1 = T 2 .
Now, we establish the following useful lemma. If B is a subset of a dual Banach space X * , we denote by co w * (B) the weak * closed convex hull of B. Lemma 2. Let X be a Banach space and K be a subset of X * .

(1) Suppose that K is norm separable, then there exists a sequence (x n ) n in the unit sphere S X of X which separate the points of K i.e. for all p, p ′ ∈ K, if p, x n = p ′ , x n for all n ∈ N, then p = p ′ . Consequently, if K is a weak * compact and norm separable set of X * , then the weak * topology of X * restricted to K is metrizable.

(2) Let (p n ) n be a weak * null sequence in X * . Then, the set co w * {p n : n ∈ N} is convex weak * compact and norm separable.

Proof. (1) Since K is norm separable, then K -K := {ab/(a, b) ∈ K × K} is also norm separable and so there exists a sequence (q n ) n of K -K which is dense in K -K. According to the Bishop-Phelps theorem [START_REF] Bishop | A proof that every Banach space is subreflexive[END_REF], the set D = {r ∈ X * | r attains its supremum on the sphere S X } is norm-dense in the dual X * . Thus, for each n ∈ N, there exists r n ∈ D such that q nr n < 1 1+n . For each n ∈ N, let x n ∈ S X be such that r n = r n , x n . We claim that the sequence (x n ) n separate the points of K. Indeed, let q ∈ K -K and suppose that q, x n = 0, for all n ∈ N. There exists a subsequence (q

n k ) k ⊂ K -K such that q n k -q < 1
k for all k ∈ N * and so we have

r n k -q < 1 1+n k + 1 k . It follows that r n k = r n k , x n k = r n k , x n k -q, x n k ≤ r n k -q < 1 1 + n k + 1 k . Hence, for all k ∈ N * , q ≤ q -r n k + r n k < 2( 1 1+n k + 1 k )
, which implies that q = 0, and so that (x n ) n separate the points of K. Now, suppose that K is weak * compact subset of X * . We show that the weak * topology of X * restricted to K is metrizable. Indeed, each x ∈ X determines a seminorm ν x on X * given by

ν x (p) = | p, x |, p ∈ X * .
The family of seminorms (ν x ) x∈X induces the weak * topology σ(X * , X) on X * . The subfamily (ν xn ) n also induces a topology on X * , which we will call T . Since this is a smaller family of seminorms, we have T ⊆ σ(X * , X). Suppose that p, p ′ ∈ K and ν xn (pp ′ ) = 0 for all n ∈ N. Then we have p, x n = p ′ , x n for all n ∈ N and so we have that p = p ′ since (x n ) n separates the points of K. Consequently, K is Hausdorff with respect to the topology T |K (the restriction of T to K). Thus T |K is a Hausdorff topology on K induced from a countable family of seminorms, so this topology is metrizable. More precisely, T |K is induced from the metric

d(p, p ′ ) := +∞ n=0 2 -n ν xn (p -p ′ ) 1 + ν xn (p -p ′ ) .
Then we have that K is Hausdorff with respect to T |K , and is compact with respect to σ(X * , X)

|K . Lemma 1 implies that T |K = σ(X * , X) |K . Hence σ(X * , X) |K is metrizable. (2) Let (p n ) n be a weak * null sequence in X * and set K = co w * {p n : n ∈ N}.
Clearly K is a convex and weak * compact subset of X * . According to Haydon's theorem [ [START_REF] Haydon | An extreme point criterion for separability of a dual Banach space, and a new proof of a theorem of Corson[END_REF], Theorem 3.3] the weak * compact convex set K is the norm closed convex hull of its extreme points whenever ex(K) (the set of extreme points of K) is norm separable. By the Milman theorem [[10], p.9] ex(K) ⊂ {p n : n ∈ N} w * = {p n : n ∈ N} ∪ {0} so that ex(K) is norm separable and, hence, by Haydon's theorem, K itself is weak * compact, convex, and norm separable.

The following proposition will be used in the proof of Theorem 3. Proposition 1. Let X be a Banach space and K be a weak * compact and norm separable subset of X * containing 0. Then, there exists a continuous seminorm h on X * satisfying (1) h is weak * lower semicontinuous and sequentially weak * continuous, (2) the restriction h |K of h to K has a weak * -strong minimum at 0.

Proof. Using Lemma 2, there exists a sequence (x k ) k ⊂ S X which separate the points of K. Define the function h : X * -→ R as follows:

h(x * ) = ( k≥0 2 -k ( x * , x k ) 2 ) 1 2 , ∀x * ∈ X * .
It is clear that h is a seminorm, and since h(x * ) ≤ x * for all x * ∈ X * , it is also continuous. Since h is the supremum of a sequence of weak * continuous functions, it is weak * lower semicontinuous. On the other hand, since the series k≥0 2 -k ( x * , x k ) 2 uniformly converges on bounded sets of X * and since the maps xk : x * → x * , x k are weak * continuous for all k ∈ N, then h is sequentially weak * continuous. If p ∈ K and h(p) = 0, then p, x k = 0 for all k ∈ N which implies that p = 0, since the sequence (x k ) k separate the points of K. Hence, the restriction of h to K has a unique minimum at 0. This minimum is necessarily a weak * -strong minimum since K is weak * metrizable by Lemma 2, this follows from a general fact which say that for every lower semicontinuous function on a compact metric space (K, d), a unique minimum is necessarily a strong minimum for the metric d in question.

3 Limited operators and differentiability.

Recall that the domain of a function f : X -→ R ∪ {+∞}, is the set

dom(f ) := {x ∈ X/f (x) < +∞}.
For a function f with dom(f ) = ∅, the Fenchel transform of f is defined on the dual space for all p ∈ X * by f * (p) := sup

x∈X { p, x -f (x)}.
The second transform (f * ) * is defined on the bidual X * * by the same formula. We denote by f * * , the restriction of (f * ) * to X, where X is identified to a subspace of X * * by the canonical embedding. Recall that the Fenchel theorem state that f = f * * if and only if f is convex lower semicontinuous on X.

The "if" part of Theorem 1 is given by the following theorem.

Theorem 3. Let Y and X be Banach spaces and let T : Y -→ X be a linear continuous operator. Suppose that f • T is Fréchet differentiable at y ∈ Y whenever f : X -→ R is convex continuous and Gâteaux differentiable at T (y) ∈ Y . Then T is a limited operator.

Proof. Let (p n ) n be a weak * null sequence in X * . We want to prove that

T * (p n ) Y * → 0. Set K = co w * {p n : n ∈ N}.
According to Lemma 2, K is convex weak * compact and norm separable. Using Proposition 1, there exists a continuous seminorm which is weak * lower semicontinuous and sequentially weak * continuous h : X * -→ R such that the restriction h |K of h to K has a weak * -strong minimum at 0 and in particular min K h = h(0) = 0. Since the sequence (p n ) n weak * converges to 0, it follows that lim n h(p n ) = h(0) = min K h. Thus, (p n ) n is a minimizing sequence for h |K . Set g = h + δ K , where δ K denotes the indicator function, which is equal to 0 on K and equal to +∞ otherwise. Since K is convex, weak * -closed and norm bounded, then g is a convex and weak * lower semicontinuous function with a norm bounded domain dom(g) = K. Moreover we have, (1) g(p) > 0 = g(0) = min X * (g) for all p ∈ X * \ {0}.

(2) lim n→+∞ g(p n ) = min X * (g).

Hence, there exists a convex and Lipschitz continuous function f : X -→ R such that g = f * (we can take f = g * |X ). The function f is Gâteaux differentiable at 0 with Gâteaux derivative ∇f (0) = 0, this is due to the fact that f * = g has a weak * -strong minimum at 0 (we can see [Corollary 1. [START_REF] Asplund | Gradients of convex functions[END_REF]]). Thus, from our hypothesis, f • T is Fréchet differentiable at 0 with Fréchet derivative equal to 0. It follows that (f • T ) * has a norm-strong minimum at 0 (see [Corollary 2. [START_REF] Asplund | Gradients of convex functions[END_REF]]). Now, we prove that (T * (p n )) n is a minimizing sequence for (f • T ) * , which will implies that T * (p n ) Y * → 0. Indeed, on one hand, we have 0 = min X * (g) = -g * (0) = -f (0). On the other hand we have

0 = -f (0) ≤ sup y∈Y {-f • T (y)} := (f • T ) * (0) ≤ sup x∈X {-f (x)} = f * (0) = g(0) = 0.
It follows that (f • T ) * (0) = 0. Hence, since (f • T ) * has a minimum at 0, we obtain

0 = (f • T ) * (0) ≤ (f • T ) * (T * (p n )) := sup y∈Y { T * (p n ), y -f • T (y)} = sup y∈Y { p n , T (y) -f (T (y))} ≤ sup x∈X { p n , x -f (x)} = f * (p n ) = g(p n ).
Since, g(p n ) → 0, it follows that (f • T ) * (T * (p n )) → 0 = (f • T ) * (0). In other words, (T * (p n )) n is a minimizing sequence for (f • T ) * . Since (f • T ) * has a norm-strong minimum at 0, we obtain that T * (p n ) Y * → 0, which implies that T is a limited operator.

The "only if" part of Theorem 1 is given by the following theorem.

Theorem 4. Let Y and X be two Banach spaces and T : Y -→ X be a limited operator. Let f : X -→ R ∪ {+∞}, be a convex lower semicontinuous function and let a ∈ Y such that T (a) belongs to the interior of dom(f ). Then,

f • T is Fréchet differentiable at a ∈ Y with Fréchet-derivative T * (Q) ∈ Y * , whenever f is Gâteaux differentiable at T (a) ∈ X with Gâteaux-derivative Q ∈ X * .
Proof. Since f is convexe lower semicontinuous and T (a) is in the interior of dom(f ), there exists r a > 0 and L a > 0 such that f is L a -Lipschitz continuous on the closed ball B X (T (a), r y ). It is well known that there exists a convex L a -Lipschitz continuous function fa on X such that fa = f on B X (T (a), r a ) (See for instance Lemma 2.31 [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]). It follows that fa

• T = f • T on B Y (a, ra T ), since T (B X (a, ra T )
) is a subset of B X (T (a), r a ) (we can assume that T = 0). Replacing f by 1 La fa , we can assume without loss of generality that f is convexe 1-Lipschitz continuous on X. It follows that dom(f * ) ⊂ B X * (the closed unit ball of X * ).

Claim. Suppose that f is Gâteaux differentiable at T (a) ∈ X with Gâteaux-derivative Q ∈ X * , then the function q → f * (q)q, T (a) has a weak * -strong minimum on B X * at Q.

Proof of the claim. See [Corollary 1. [START_REF] Asplund | Gradients of convex functions[END_REF]]. Now, suppose by contradiction that T * (Q) is not the the Fréchet derivative of f • T at a. There exist ε > 0, t n -→ 0 + and h n ∈ Y , h n Y = 1 such that for all n ∈ N * ,

f • T (a + t n h n ) -f • T (a) -T * (Q), t n h n > εt n . ( 1
)
Let r n = t n /n for all n ∈ N * and choose p n ∈ B X * such that

f * (p n ) -p n , T (a + t n h n ) < inf p∈B X * {f * (p) -p, T (a + t n h n ) } + r n . (2)
From ( 2) we get

f * (p n ) -p n , T (a) < inf p∈B X * {f * (p) -p, T (a) } + 2t n T + r n .
This implies that the sequence (p n ) n minimize the function q → f * (q)q, T (a) on B X * . Using the claim, the function q → f * (q)q, T (a) has a weak * -strong minimum on B X * at Q, it follows that (p n ) n weak * converges to Q and so (since T is limited) we have

T * (p n -Q) Y * -→ 0. (3)
On the other hand, since

f (T (a + t n h n )) = f * * (T (a + t n h n )) = -inf p∈B X * {f * (p) -p, T (a + t n h n ) }, using (2) we obtain for all y ∈ Y f • T (a + t n h n ) -p n , T (a + t n h n ) < -f * (p n ) + r n ≤ f • T (y) -p n , T (y) + r n .
Replacing y by a in the above inequality we obtain

f • T (a + t n h n ) -p n , T (t n h n ) ≤ f • T (a) + r n . (4)
Combining ( 1) and ( 4) we get

ε < p n , T (h n ) -T * (Q), h n + r n /t n = T * (p n ), h n -T * (Q), h n + 1 n ≤ T * (p n -Q) Y * + 1 n which is a contradiction with (3). Thus f •T is Fréchet differentiable at a with Fréchet derivative T * (Q).
Now, we give the proof of Theorem 2.

Proof of Theorem 2. (2) =⇒ ( 1) is well known.

(2) =⇒ (3) Let (p n ) n be a weak * null sequence in Y * such that inf n p n > 0 and set K = co w * {p n : n ∈ N}. By Lemma 2, the set K is convex norm separable and weak * compact metrizable. On the other hand, from Proposition 1, there exists a continuous seminorm h which is weak * lower semicontinuous and weak * sequentially continuous on Y * such that the restriction of h to K has a weak * -strong minimum at 0. It remains to show that 0 is not a norm-strong minimum for h |K . Indeed, since (p n ) n is weak * null and h is weak * sequentially continuous, then lim n h(p n ) = h(0) = min K h. So (p n ) n is a minimizing sequence for h |K which not converges to 0 since inf n p n > 0. Hence, 0 is not a norm-strong minimum for h |K .

(3) =⇒ (2) Since 0 is not a norm-strong minimum for the restriction h |K , there exists a sequence (p n ) n that minimize h on K but p n 0. Since h |K has a weak * -strong minimum at 0, it follows that (p n ) n weak * converges to 0. Hence, (p n ) n weak * converges to 0 but p n 0. Thus, there exists a JN-sequence in Y * .

(2) =⇒ (4) This part is given by taking X = Y and T = I the identity map. Indeed, there exists a sequence (p n ) n which weak * converges to 0 but inf n I * (p n ) = inf n p n > 0. So I cannot be a limited operator.

(4) =⇒ [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF]. Indeed, if there exists a Banach space X and a non-limited operator T : Y -→ X, by using Theorem 1, there exists a convex continuous function f :

X -→ R and a point y ∈ Y such that f is Gâteaux differentiable at T (y) ∈ X but f • T is not Fréchet differentiable at y. So f • T is Gâteaux but not Fréchet differentiable at y. Hence, f • T is a convex continuous PGNF-function on Y .
(5) =⇒ (2) Let f be a PGNF-function on Y . We can assume without loss of generality that f is Gâteaux differentiable at 0 with Gâteaux-derivative equal to 0, but f is not Fréchet differentiable at 0. It follows from classical duality result (see Corollary 1. and Corollary 2. in [START_REF] Asplund | Gradients of convex functions[END_REF]) that f * has a weak * -strong minimum but not norm-strong minimum at 0. Since 0 is not a norm-strong minimum for f * , there exists a sequence (p n ) n ∈ X * minimizing f * such that p n 0. On the other hand, since f * has a weak * -strong minimum at 0, and (p n ) n minimize f * , we have that (p n ) n weak * converges to 0. Thus, (p n ) n weak * converges to 0 but p n 0. Hence, there exists a JN-sequence.

Canonical construction of PGNF-function.

There exist different way to build a PGNFfunction in infinite dimentional Banach spaces. We can find examples of such constructions in [START_REF] Borwein | On convex functions having points of Gâteaux differentiability which are not points of Féchet-differentiability[END_REF]. We present below a different method for constructing a PGNF-function on a Banach space X canonically from a JN-sequence. Given a JN-sequence (p n ) n ⊂ X * , we set K = co w * {p n : n ∈ N}. Using Lemma 2, there exists a sequence (x n ) n ∈ S X which separates the points of K, and as in the proof of Proposition 1, there exist a continuous seminorm h which is weak * lower semicontinuous and weak * sequentially continuous such that h |K has a weak * -strong minimum at 0. The function h is given explicitly as follows

h(x * ) = ( n≥0 2 -n ( x * , x n ) 2 ) 1 2 , ∀x * ∈ X * .
Since (p n ) n weak * converges to 0, it follows that (p n ) is a minimizing sequence for h |K . Since (p n ) n is a JN-sequence, it follows that 0 is not a norm-strong minimum for h |K . Define the function f by f

(x) = (h + δ K ) * (x), ∀x ∈ X,
where δ K denotes the indicator function, which is equal to 0 on K and equal to +∞ otherwise and where for each x ∈ X, we denote by x ∈ X * * the linear map x * → x * , x for all x * ∈ X * . Then f is convex Lipschitz continuous, Gâteaux differentiable at 0 (since h + δ K has a weak *strong minimum) but not Fréchet differentiable at 0 (because 0 is not a norm-strong minimum for h + δ K ).

4 Appendix.

There exists a class of Banach spaces (E, . E ) such that the canonical embedding i : E -→ E * * is a limited operator. This class contains in particular the space c 0 and any closed subspace F of c 0 (This class is also stable by product and quotient. For more information see [START_REF] Carrión | Banach spaces whose bounded sets are bounding in the bidual[END_REF]). In this setting, Theorem 4 gives immediately the following corollary.

Corollary 1. Suppose that the canonical embedding i : E -→ E * * is a limited operator. Let g : E * * -→ R ∪ {+∞} be a convex lower semicontinuous function. Suppose that x ∈ E belongs to the interior of dom(g) and that g is Gâteaux differentiable at x ∈ E (we use the identification i(x) = x), then the restriction of g to E is Fréchet differentiable at x. In particular, if f : E -→ R ∪ {+∞} is convex lower semicontinuous function, x ∈ E belongs to the interior of dom((f * ) * ) and (f * ) * is Gâteaux differentiable at x, then f is Fréchet differentiable at x.

We obtain the following corollary by combining Proposition 2 and a delicate result due to Zajicek (see [Theorem 2; [START_REF] Zajicek | On the differentiation of convex functions in finite and infinite dimensional spaces[END_REF]]), which say that in a separable Banach space, the set of the points where a convex continuous function is not Gâteaux differentiable, can be covered by countably many d.c (that is, delta-convex) hypersurf ace. Recall that in a separable Banach space Y , each set A which can be covered by countably many d.c hypersurf ace is σ-lower porous, also σ-directionally porous; in particular it is both Aronszajn (equivalent to Gauss) null and Γ-null. For details about this notions of small sets we refer to [START_REF] Zajicek | On sigma-porous sets in abstract spaces[END_REF] and references therein. Note that a limited set in a separable Banach space is relatively compact [START_REF] Bourgain | Limited operators and strict cosingularity[END_REF].

Proposition 2. Let Y and X be Banach spaces and T : Y -→ X be a limited operator with a dense range. Let f : X -→ R be a convex continuous function. Then f • T is Gâteaux differentiable at a ∈ Y if and only if, f • T is Fréchet differentiable at a ∈ Y .

Proof. Suppose that f • T is Gâteaux differentiable at a ∈ Y . It follows that f is Gâteaux differentiable at T (a) with respect to the direction T (Y ) which is dense in X. It follows (from a classical fact on locally Lipschitz continuous functions) that f is Gâteaux differentiable at T (a) on X. So by Theorem 4, f • T is Fréchet differentiable at a ∈ Y . The converse is always true.

Corollary 2. Let Y be a separable Banach space, X be a Banach spaces and T : Y -→ X be a compact operator with a dense range. Let f : X -→ R, be a convex and continuous function. Then, the set of all points at which f • T is not Fréchet differentiable can be covered by countably many d.c hypersurf ace.

( 1 )

 1 =⇒ (2) is the deeper Josefson-Nissenzweig theorem [[7], Chapter XII].


Abdelhamid Amroun

PERIODIC GEODESICS IN RANK 1 SURFACES

.

Introduction and main result

In [START_REF] Gelfert | Pressures for geodesic flows of rank one manifolds[END_REF] K. Gelfert and B. Schapira, have proved a formula for a compact connected surfaces which express the topological pressure of a class of potentials as the exponential growth rate of rank 1 geodesics with Lyapunov exponent bounded bellow by a positive constant depending on the potential. The potential they consider are Hölder continuous, hyperbolic (in the sens of [START_REF] Gelfert | Pressures for geodesic flows of rank one manifolds[END_REF] Theorem 1.2 and Lemma 3) and constant on the subset of singular unit tangent vectors to the surface. In this paper, we want to generalize this formula to all Hölder continuous potentials which are constant on the singular part, i.e which are not necessarily hyperbolic. This leads to a formula which includes periodic orbits close to the singular part (Lyapunov exponents may be close to zero ). Before we explain the details and state precisely the main result, we introduce the notations and the settings. In what follows, by curvature we mean sectional curvature.

Let M be a compact connected rank 1 manifold and G = (g t ) t∈IR its geodesic flow. According to [START_REF] Burns | Manifolds of nonpositive curvature and their buildings[END_REF], the rank 1 condition is generic in the set of nonpositively curved compact manifolds, i.e "most" of the compact manifolds of nonpositive curvature are rank 1. The rank 1 condition implies the existence of a rank 1 geodesic, i.e which does not admit a parallel perpendicular Jacobi field. The space of all parallel Jacobi fields along these geodesics is then one dimensional. We say that a rank 1 geodesic is hyperbolic or regular. A tangent vector v ∈ T M has rank 1 if it defines a rank 1 geodesic. The set reg of all the rank 1 vectors (regular vectors) is open and dense in T 1 M [2] and invariant with respect to the action of G on T 1 M . The singular part sing of T 1 M consists of all non-regular vectors in T 1 M ; the curvature vanishes along any singular geodesic (a geodesic defined by a singular vector). The existence of singular geodesics implies that the geodesic flow is not Anosov. For a complete exposition of this subject see [START_REF] Eberlein | Geometry of Nonpositively Curved Manifolds[END_REF][START_REF] Eberlein | When is a geodesic flow of Anosov-type?[END_REF][START_REF] Eberlein | Geodesic flows on negatively curved manifolds[END_REF].

Let G(M ) be the set of all primitive closed (periodic) geodesics, which are distinct and represent different free homotopy classes. Note that two distinct regular closed geodesics can not be contained in the same free homotopy class, otherwise, they will bound a flat strip which contradicts the regularity property.

Given Γ ⊂ G(M ) we set

P Γ (t) := {γ ∈ Γ : l(γ) ≤ t}.
Here l(γ) is the least period of the closed geodesic γ. The number of periodic geodesics in a free homotopy class may be infinite but the corresponding periods are equal. Set P (t) := P G(M ) (t) and let P reg (t) ⊂ P (t) be the subset of rank 1 geodesics. It follows from the result of G. Knieper [14] that, [START_REF] Amroun | Equidistribution results for geodesic flows[END_REF] lim

t→+∞ 1 t log #P (t) = lim t→+∞ 1 t log #P reg (t) = h top (G),
where h top (G) is the topological entropy of the geodesic flow G.

For compact rank one surfaces, K. Gelfert and B. Schapira ([12] Theorem 1.2) proved, among other results, a generalization of (1) to the case where geodesics are counted with weights. Namely, for Hölder continuous "hyperbolic potentials" f (see [START_REF] Gelfert | Pressures for geodesic flows of rank one manifolds[END_REF] Theorem 1.2 and Lemma 3) which are constant on the singular set sing, they proved that there exists β = β(f ) > 0 such that, [START_REF] Ballmann | Axial isometries of manifolds of nonpositive curvature[END_REF] lim sup

t→+∞ 1 t log γ∈P (t):χ(γ)>β e l(γ) 0 f (g t ( γ(0)))dt = P (G; f ),
where χ(γ) is the smallest Lyapunov exponent of the periodic geodesic γ and P (G; f ) the topological pressure of G corresponding to the potential f . Recall briefly that the P (G; f ) satisfies the variational principle,

P (G; f ) = sup h m (G) + f dm ,
where the sup is over the set of invariant probability measures for G and h m (G) = h m (g 1 ) is the measure theoretical entropy of the measure m (see [START_REF] Katok | Introduction to the modern theory of dynamical systems. Encyclopedia of Math and its App[END_REF], [START_REF] Walters | An introduction to ergodic theory[END_REF]).

Formula (2) suggests that the topological pressure of hyperbolic potentials can be recovered from the exponential growth rate of periodic geodesics with positive Lyapunov exponents (geodesics which stay far from the zero curvature part of the surface). Note that the geodesics in the left hand side of (2) dont not intersect the singular set sing (since in sing the Lyapunov exponents are 0). Thus the sum in ( 2) is over the subset of regular geodesics where the curvature is negative. So, what is the contribution of the rest of all regular geodesics? On the other hand, examples are given in [START_REF] Gelfert | Pressures for geodesic flows of rank one manifolds[END_REF] of non hyperbolic potentials which contradict [START_REF] Ballmann | Axial isometries of manifolds of nonpositive curvature[END_REF]. Thus, what can be said if there is no hyperbolicity assumption on the potential f ? Note that we always have, [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF] lim sup

t→+∞ 1 t log γ∈P (t) e l(γ) 0 f (g t ( γ(0)))dt ≤ P (G; f ),
for any continuous function f on T 1 M . Indeed, two geodesics representing two distinct free homotopy classes are ( , t)-separated for any < inj(M ) and t > 0, i.e for any pair of geodesics γ 1 , γ 2 ∈ P (t), there exists s ∈ [0, t] such that d(γ 1 (s), γ 2 (s)) > [START_REF] Knieper | The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds[END_REF].

Motivated by these questions, we prove (Theorem 1) that for Hölder continuous potentials (not necessarily hyperbolic) which are constant on the singular set sing we have, f (g t ( γ(0)))dt. This means that, in spite of the existence of periodic regular geodesics which come close to the zero curvature part of the surface, the exponential growth is still determined by P (f ) as in the formula (2) of K. Gelfert and B. Schapira. Thus, by (2), ( 3) and ( 4), if no hyperbolicity is required on f , we only have an inequality in (2) unless we add all regular geodesics with Lyapunov exponents close to zero.

The proof of our result does not focus on large deviations technics, as in [START_REF] Gelfert | Pressures for geodesic flows of rank one manifolds[END_REF]. Indeed, large deviations estimates suppose that we are able to built a "large deviation functional" on the space of invariant measures of the flow, which must be positive on the complement of the subspace of equilibrium states. But this is not the case here due to the lack of hyperbolicity of the potential f . Instead, we use a result of R. Bowen and D. Ruelle [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF] to prove a formula for the topological pressure P (G |Λ , f ) of the restriction of the geodesic flow to a basic set Λ ⊂ T 1 M . To recover formula (4) we apply a result of K. Burns and K. Gelfert [START_REF] Burns | Lyapunov spectrum for geodesic flows of rank 1 surfaces[END_REF] which asserts that there exits an increasing sequence Λ 1 ⊂ Λ 2 ⊂ • • • ⊂ reg of basic sets, all contained in the regular part reg, such that the topological pressures P (G| Λ l , f ) converge to P (G; f ) as l → +∞. In fact, they proved that

lim l→+∞ P (G| Λ l , f ) = sup l≥1 P (G| Λ l , f ) = P (G; f ).
The main result of the paper is the following.

Theorem 1. Let M be a compact connected rank 1 surface. Then, for any Hölder continuous potential f on T 1 M which is constant on sing, the exponential growth rate of rank 1 geodesics is given by the topological pressure, [START_REF] Burns | Lyapunov spectrum for geodesic flows of rank 1 surfaces[END_REF] lim sup

t→+∞ 1 t log γ∈Preg(t) e γ f = P (G; f ).
It is not clear how this result can be generalized, or not, to compact rank 1 manifolds of dim > 2. Indeed, in the case of surfaces, closed hyperbolic subsets of T 1 M are contained in the regular set [START_REF] Burns | Lyapunov spectrum for geodesic flows of rank 1 surfaces[END_REF], and this fact is not true in general (see [START_REF] Crovisier | Une remarque sur les ensembles hyperboliques localement maximaux[END_REF][START_REF] Fisher | Hyperbolic sets that are not locally maximal. Ergodic Theory Dynam[END_REF]).

Each periodic geodesic γ defines a probability measure on

T 1 M with support in { γ(t), 0 ≤ t ≤ l(γ)} by, δ γ (A) := leb(t ∈ [0, l(γ)] : g t ( γ(0)) ∈ A)/l(γ).
The following corollary is an easy and standard consequence of Theorem 1 (on can adapt for example the method in [START_REF] Amroun | Equidistribution results for geodesic flows[END_REF]). It gives a way of approximating equilibrium states.

Corollary 1. In the conditions of Theorem 1, the weak* limits of the measures

µ t = 1 γ∈Preg(t) e γ f γ∈Preg(t)
e γ f δ γ are measures of maximal pressure for G, i.e equilibrium states of G corresponding to the potential f .

As stated above, the main idea in our approach consists in proving the main theorem above for the restriction of the geodesic flow to basic sets. For this, recall that a basic set Λ is a compact locally maximal hyperbolic subset of T 1 M on which the flow is transitive. The hyperbolic set Λ is closed and invariant with no fixed point. The set P er(Λ) of the periodic orbits of G| Λ is dense in Λ and there is an open set U ⊃ Λ with Λ = t∈IR g t U . By ([5] Lemma 4.1) the compact and invariant set Λ is contained in the regular part reg of T 1 M . In particular, the orbits in P er(Λ) are regular. Each γ ∈ P er(Λ) is rank 1 and γ is uniquely determined in its free homotopy class. It is well known (see [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF] Theorem 3.3, [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF] Corollary 20.3.8) that the geodesic flow on Λ admits a unique equilibrium state µ f,Λ . The probability measure µ f,Λ is ergodic and satisfies other interesting properties that can be found in [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF]. The following proposition gives an equidistribution result for the restriction of geodesic flow to the basic set Λ. Let P er Λ (t) be the subset of P er(Λ) of (primitive) periodic geodesics γ with length l(γ) ≤ t.

Proposition 1. Let M be a compact finite dimensional smooth manifold of nonpositive curvature and Λ ⊂ T 1 M a basic set. Let f be a Hölder continuous function on T 1 M .

(1) The topological pressure of G| Λ is given by,

P (G| Λ ; f ) = lim t→+∞ 1 t log γ∈P er Λ (t) e γ f .
(2) For any continuous function K we have,

Kdµ Λ,t := γ∈P er Λ (t) e γ f δ γ (K) γ∈P er Λ (t) e γ f t→+∞ -→ Kdµ f,Λ .

Proof of the main result

Proof. Since the restriction of the potential f to the singular part sing is a constant function, by ([5] Theorem 1.4 and Theorem 6.1), we can find a family of compact locally maximal hyperbolic sets (basic sets)

Λ 1 ⊂ Λ 2 • • • ⊂ reg such that, (6)
lim l→+∞ P (G| Λ l ; f ) = sup l≥1 P (G| Λ l ; f ) = P (G; f ).
For all l ≥ 1 we have,

P (G| Λ l ; f ) = lim t→+∞ 1 t log γ∈P er Λ l (t)
e γ f (by P roposition1)

≤ lim sup t→+∞ 1 t log γ∈Preg(t)
e γ f (since Λ l ⊂ reg).

By ( 6)

P (G; f ) ≤ lim sup t→+∞ 1 t log γ∈Preg(t) e γ f .
Equality follows now from (3). This proves formula (5) and Theorem 1.

Proof of Proposition 1

We begin with the following simple lemma.

Lemma 1. Let h be a smooth diffeomorphism acting on a compact metric space X with a unique equilibrium state ν f corresponding to a continuous potential f . Let A ⊂ X, A = X, be closed and invariant with ν f (A) = 0. Then

P (h| A ; f ) < P (h; f ),
where P (h| A ; f ) and P (h; f ) are the topological pressure of the h on A and X respectively.

Remark 1. It is not clear if the conclusion of the above lemma remains true if ν f (A) = 0. However, if the potential f is constant everywhere, then under certain conditions on h, by (Lemma 6.2 [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF]) we have h top (h| A ) < h top (h).

Proof. Let ν A be an invariant probability measure for h| A , which satisfies h(ν A ) + A f dν A = P (h| A ; f ). h being a diffeomorphism and ν A (A) = 1, the measure ν A is also an invariant measure for the dynamical system (X, h). We have ν f (A) = 0 so that ν A = ν f . Thus the lemma follows since ν f is unique.

A special flow (or suspension) is associated to the restriction of the geodesic flow to the basic set Λ. It was proved in [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF] and ( [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF] Proposition 3.1) that the geodesic flow G on the basic set Λ can be conjugated to a special flow (Λ(Σ), ψ) over a topologically mixing subshift of finite type σ : Σ → Σ (the mixing property was proved in [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF] Proposition 3.1). Furthermore, the conjugacy ρ : Λ(Σ) → Λ is an isomorphism of the measurable flows (Λ(Σ), ψ) and (Λ, G). In particular, the suspension admits a unique equilibrium state µ f * ,Λ for the Hölder continuous potential on Λ(Σ) given by f * = f • ρ and µ f * ,Λ = ρ * µ f,Λ .

Denote by P er(Λ(Σ)) the set of periodic orbits of the flow ψ in Λ(Σ) and, by P er Λ(Σ) (t) the subset of orbits τ with (least) period less than t. Recall that the suspension is expansive since this is the case for the subshift. The topological pressure of a suspension over a topologically mixing subshift of finite type is given by, ( 7)

P (ψ; f * ) = lim t→+∞ 1 t log τ ∈P er Λ(Σ) (t)
e τ f * .

We have P (G| Λ ; f ) = P (ψ; f * ) (see [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF]). There are two sets Z ⊂ Λ and Z ⊂ Λ(Σ) such that ρ : Z → Z is an homeomorphism [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF]. By ([3] Proposition 4.1), τ ∈ Λ(Σ) is periodic if and only if ρ(τ ) is periodic. Thus, There are two closed subsets (see [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF] p452) E s , E u which are invariant in one direction :

g t (E s ) ⊂ E s , g -t (E u ) ⊂ E u for t ≥ 0, such that P er(Λ) = (P er(Λ) ∩ Z) ∪ (P er(Λ) ∩ (E s ∪ E u )),
and

P er(Λ(Σ) = (P er(Λ(Σ)) ∩ Z ) ∪ (P er(Λ(Σ)) ∩ (ρ -1 (E s ) ∪ ρ -1 (E u )). Set E s,u = E s ∪ E u . We have µ f * ,Λ (ρ -1 (E s,u )) = µ f,Λ (E s,u ) = 0 (see [3], [4]). Set f * 1 (x) := 1 0 f * (ψ t (x))dt. By Lemma 2 applied to (ρ -1 (E s ), ψ 1 , f * 1 ) and (ρ -1 (E u ), ψ -1 , f * 1 ), we have (9) lim t→+∞ τ ∈P er Λ(Σ) (t):τ (0)∈ρ -1 (E s,u ) e τ f * τ ∈P er Λ(Σ) (t) e τ f * = 0. Indeed, P (ψ 1 | ρ -1 (E s ) ; f * 1 ) < P (ψ 1 ; f * 1 ) = P (ψ; f * ), P (ψ -1 | ρ -1 (E u ) ; f *
1 ) < P (ψ -1 ; f * 1 ) = P (ψ; f * ). On the other hand, since the suspension is expansive, if two periodic points are not dynamically separated (i.e by ψ) then they belong to the same periodic orbit of ψ. We have then, lim sup We deduce from [START_REF] Fisher | Hyperbolic sets that are not locally maximal. Ergodic Theory Dynam[END_REF], [START_REF] Eberlein | Geometry of Nonpositively Curved Manifolds[END_REF] and the fact that P (ψ; e γ f = P (G| Λ ; f ).

t→+∞ 1 t log τ ∈P er Λ(Σ) (t):τ (0)∈ρ -1 (E s,u ) e τ f * ≤ P (ψ 1 | ρ -1 (E s ) ; f * 1 )+P (ψ -1 | ρ -1 (E u ) ; f * 1 ). Thus (
f * ) = P (G| Λ ; f ), (12)
This proves the first part of Proposition 1.

We prove now the second part of Proposition 1. The following equidistribution result for the hyperbolic suspension (Λ(Σ), ψ) can be found in ([17] p120). However, it can be easily proved using the asymptotic formula [START_REF] Crovisier | Une remarque sur les ensembles hyperboliques localement maximaux[END_REF] and uniqueness of the equilibrium state µ f * ,Λ for the suspension. Recall that δ τ (K

) := 1 l(τ ) l(τ ) 0 K(τ (t))dt. Theorem 2 ([17]). For any continuous function K : Λ(Σ) → IR we have, τ ∈P er Λ(Σ) (t) e τ f * δ τ (K) τ ∈P er Λ(Σ) (t) e τ f * t→+∞ -→ Kdµ f * ,Λ .
It follows from Theorem 2, ( 9) and ( 10), for any continuous function K, i.e µ Z Λ(Σ),t converges weakly to µ f * ,Λ , as t → +∞. Observe that we can not deduce this directly from Theorem 2 by considering the function 1 Z ; indeed, it is not clear whether or not it's continuity (or equivalently the continuity of 1 Z ) can be deduced from [START_REF] Bowen | Symbolic dynamics for hyperbolic flows[END_REF] and the construction there.

µ Z Λ(Σ),t (
Define the probability measures, µ Z Λ,t (K) := γ∈P er Λ (t): γ(0)∈Z e γ f δ γ (K)

γ∈P er Λ (t): γ(0)∈Z e γ f . Since ρ is an isomorphism, we have ρ * µ Z Λ,t = µ Z Λ(Σ),t . Now, ρ being continuous, the measure µ Z Λ,t converges weakly to ρ * µ f * ,Λ = µ f,Λ . We show now how to deduce the convergence of the measures µ Λ,t to the equilibrium state µ f,Λ , as t → +∞. For any continuous function K we have [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF] µ Λ,t (K) = µ Z Λ,t (K)x t + y t (K) where, . By part 1 of Proposition 1 and the same arguments in the above proof of this part, x t converges to 1 and y t converges to 0, as t → +∞. Finally, by [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF] we see that µ Λ,t (K) and µ Z Λ,t (K) converge to the same limit, i.e µ f,Λ (K). This proves that µ Λ,t converges to µ f,Λ .

e

  γ f = P (G; f ), where we have set γ f := l(γ) 0

τ

  ∈P er Λ(Σ) (t):τ (0)∈Z e τ f * .

τ

  10) limt→+∞ τ ∈P er Λ(Σ) (t):τ (0)∈Z e τ f * τ ∈P er Λ(Σ) (t) e τ f * ∈P er Λ(Σ) (t):τ (0)∈Z e τ f * = P (ψ; f * ).

  K) := τ ∈P er Λ(Σ) (t):τ (0)∈Z e τ f * δ τ (K) τ ∈P er Λ(Σ) (t):τ (0)∈Z e τ f * t→+∞ -→ Kdµ f * ,Λ ,

  x t = γ∈P er Λ (t): γ(0)∈Z e γ f γ∈P er Λ (t) e γ f , andy t (K) = γ∈P er Λ (t): γ(0)∈Z c e γ f δ γ (K) γ∈P er Λ (t) e γ f

  lim

	Finally, combining (12) and (3) we get,
	lim t→+∞	1 t	log	γ∈P er Λ (t)
	t→+∞	1 t	log	γ∈P er Λ (t): γ(0)∈Z	e γ f = P (G| Λ ; f ).