Abdelhamid Adouani
email: arbi.abdelhamid@gmail.com

Habib Marzougui
email: habib.marzouki@fsb.rnu.tn

COMPUTATION OF ROTATION NUMBERS FOR A CLASS OF PL-CIRCLE HOMEOMORPHISMS

Keywords: 2000 Mathematics Subject Classification. Primary: 37C15, 37E10, 37E45 piecewise linear homeomorphism, rotation number, conjugacy, break point, jump

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Computation of rotation numbers for a class of PL-circle homeomorphisms

Introduction

Denote by S 1 = R/Z the circle and p : R -→ S 1 the canonical projection. Let f be an orientation preserving homeomorphism of S 1 . The homeomorphism f admits a lift f : R -→ R that is an increasing homeomorphism of R such that p • f = f • p. Conversely, the projection of such a homeomorphism of R is an orientation preserving homeomorphism of S 1 .

Historically, the dynamic study of circle homeomorphisms was initiated by H. Poincaré ([10], 1886), he introduced the rotation number of a homeomorphism f of S 1 as

ρ(f ) = lim n-→+∞ f n (x) -x n (mod 1), x ∈ R.
This limit exists and is independent of the choice of the point x and the lift f of f . For example, if R α : x → x + α (mod 1) is the rotation by angle α then it is obviously that ρ(R α ) = α (mod 1). From the definition, ρ(h • f • h -1 ) = ρ(f ) holds for any orientation preserving homeomorphism h of S 1 . Assuming f is a C r -diffeomorphism (r ≥ 2) and ρ(f ) is irrational, A. Denjoy ([4]) proved:

Denjoy's theorem, [START_REF] Denjoy | Sur les courbes définies par les équations différentielles à la surface du tore[END_REF] Every C r -diffeomorphism f (r ≥ 2) of S 1 with irrational rotation number ρ(f ) is topologically conjugate to rotation R ρ(f ) . This means that there exists an orientation preserving homeomorphism h of S 1 such that f = h -1 • R ρ(f ) • h. Denjoy noted that this result can be extended (with the same proof) to a large class of circle homeomorphisms: the class P (see [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF], Chapter VI) and in particular for piecewise linear (PL) circle homeomorphisms. Definition 1.1. An orientation preserving homeomorphism f of S 1 is called piecewise linear (P L) if f is derivable except in finitely many points (c i ) 1≤i≤p of S 1 , called break points of f at which left and right derivatives, denoted respectively by Df -and Df + exist, and such that the derivative Df is constant on each ]c i , c i+1 [.

Denote by

-Homeo + (S 1 ) the group of orientation-preserving homeomorphisms of S 1 . -PL + (S 1 ) the set of orientation preserving PL-homeomorphisms of S 1 , it is a subgroup of Homeo + (S 1 ) which contains rotations.

Definition 1.2. A homeomorphism B ∈ PL + (S 1 ) with two break points x 0 and B(x 0 ) is called an affine Boshernitzan of S 1 of last break point B(x 0 ). The map B is identified with the bijection B (mod 1) of [0, 1[, where B is a lift to R of B (Fig. 1). When B(x 0 ) = 0, B is called a standard affine Boshernitzan, it is an affine exchange of 2 intervals on [0, 1[ (Fig. 1). It is plain that if B 0 is a standard affine Boshernitzan with B 0 (x 0 ) = 0 then

B := R x 0 • B 0 • R -1
x 0 is an affine Boshernitzan of last break point x 0 = B(R x 0 (x 0 )). Conversely, if B is an affine Boshernitzan of last break

point x 0 then B 0 := R -1 x 0 • B • R x 0 is a standard affine Boshernitzan with B 0 (B • R x 0 ) -1 (x 0 ) = 0.
Notice that Boshernitzan was the first who studied these examples in order to built examples of "rational" PL-homeomorphisms with irrational rotation numbers (cf. [START_REF] Boshernitzan | Dense orbits of rationals[END_REF]).

In this paper, we are mainly concerned with the computation of rotation numbers of PL-homeomorphisms with the (D)-property (see definition 1.4). The lecture of the preprint [START_REF] Liousse | Nombre de rotation dans les groupes de Thompson généralisés, automorphismes[END_REF] was the source of our interest in the subject.

Here the theoretical support for the formula of rotation numbers is provided by the explicite conjugacy given by Proposition 2.4. Notice that the question of the smoothness of the conjugacy of P L-homeomorphism to rotation have been studied in ( [START_REF] Liousse | PL Homeomorphisms of the circle which are piecewise C 1 conjugate to irrational rotations[END_REF], [START_REF] Adouani | Sur les Homéomorphismes du cercle de classe P C r par morceaux (r ≥ 1) qui sont conjugués C r par morceaux aux rotations irrationnelles[END_REF], [START_REF] Adouani | On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations[END_REF]). Before stating the main result, we need the following notations and definitions.

Throughout the paper, we assume that all PL-homeomorphisms have finitely many break points. Here and later, we identify a map f ∈ PL + (S 1 ) with the bijection f (mod 1), where

f is a lift to R of f . A point x ∈ S 1 is identified with its representative x in [0, 1[. For f ∈ PL + (S 1 ) and x ∈ S 1 , denote by O f (x) := {f n (x) : n ∈ Z} called the orbit of x by f . Definition 1.3. (Maximal connections). Let f ∈ PL + (S 1 ) and c ∈ C(f ). A maximal f -connection of c is a segment [f -p (c), . . . , f q (c)] := {f s (d) : -p ≤ s ≤ q} of the orbit O f (c)
which contains all the break points of f contained on O f (c) and such that f -p (c) (resp. f q (c)) is the first (resp. last) break point of f on O f (c).

We have the following properties:

-Two break points of f are on the same maximal f -connection, if and only if, they are on the same orbit.

-Two distinct maximal f -connections are disjoint.

Notations. Let f ∈ PL + (S 1 ). We let -C(f ) be the set of break points of f . -M i (f ) = [c i , . . . , f N i (c i )], N i ∈ N), the maximal f -connections of c i ∈ C(f ), (1 ≤ i ≤ p). -N := max 1≤i≤p N i . -M (f ) = p i=1 M i (f )
. So, we have the decomposition:

C(f ) = p i=1 C i (f ) where C i (f ) = C(f ) ∩ M i (f ), 1 ≤ i ≤ p. -σ f (x) : = Df -(x) Df + (x) called the f -jump in x. -π s,O f (c) (f ) := x∈O f (c) σ f (x) = x∈C(f )∩O f (c) σ f (x), for every c ∈ S 1 . Note that -π s (f ) := c∈S 1 π s,O f (c) (f ) = c∈C(f ) σ f (c) = 1.
-

d∈C i (f ) σ f (d) = d∈M i (f ) σ f (d).
Let n ∈ N * and x ∈ S 1 . We have:

-C(f n ) ⊂ {f -k (c) : k = 0, 1, . . . , n -1; c ∈ C(f )}, -Df n (x) = Df(x)Df(f (x)) × • • • × Df (f n-1 (x)), -The jump of f n in x is then: σ f n (x) = σ f (x)σ f (f (x)) × • • • × σ f (f n-1 (x)).
Hence we have

π O f (c i ) (f ) :=      N i k=1 σ f N+1 (f k (c)), if N i ≥ 1 1, if N i = 0 Write π(f ) := p i=1 π O f (c i ) (f ) Definition 1.4. Let f ∈ PL + (S 1
). We say that f has the (D)-property

([8]) if the product of f -jumps in the break points on each orbit O f (c) is trivial; that is π s,O f (c) (f ) = x∈C(f )∩O f (c) σ f (x) = 1.
In particular, if all break points belong to the same orbit then f has the (D)-property, this is the case in particular if f is an affine Boshernitzan.

Hence f satisfies the (D)-property means that: for every 1 ≤ i ≤ p, the product of f -jumps in the break points of C i (f ) is trivial; that is

d∈C i (f ) σ f (d) = 1. Assume that f (0) ∈]0, 1[. Set -M * (f ) := M (f )\{c 1 , . . . , c p }, -S := M * (f ) ∩ [0, 1[ -S ∪ {0, f (0)} = {b 0 , . . . , . . . , b n } where b 0 = 0 < b 1 < • • • < b m+1 := f (0) < • • • < b n < b n+1 = 1. -q =      2(N + 1), if 0 ∈ f -(N +1) (M (f ))

N, otherwise

We are in the position to give our main result.

Theorem 1.5. Let f ∈ PL + (S 1 ) with the (D)-property and irrational rotation number ρ(f ). Under the notations above, we have:

(i) if π(f ) = 1, ρ(f ) = log 1 Df -(0) c∈S∩[0,f (0)[ σ f N+1 (c) log(π(f )) (mod 1) (ii) if π(f ) = 1, ρ(f ) = m k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1 n k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1
(mod 1)

Corollary 1.6. Let f ∈ PL + (S 1 ) with the (D)-property and irrational rotation number ρ(f ).

Let M i = [c i , . . . , f N i (c i )], N i ∈ N * , i ∈ {1, . . . , p} be the maximal f -connections. Assume that M i ∩ C(f ) = {c i , f N i (c i )}. Then (i) π(f ) = p i=1 (σ f (c i )) -N i (ii) if π(f ) = 1, then ρ(f ) = log Df -(0) + p i=1 l i log σ f (c i ) p i=1 N i log σ f (c i ) (mod 1)
where

l i := card{1 ≤ k ≤ N i : f k (c i ) ∈ [0, f (0)[}.
Corollary 1.7. Let f ∈ PL + (S 1 ) with two break points c and f N (c), N ∈ N * and irrational rotation number ρ(f ). Then π(f ) = (σ f (c)) -N = 1 and hence

ρ(f ) = 1 N log Df -(0) log σ f (c) + l N ( mod 1)
where

l := {1 ≤ k ≤ N : f k (c) ∈ [0, f (0)[}.
Corollary

ρ(f ) = log Df -(0) log σ f (c) (mod 1)
Notice that the formula of ρ(f ) given in Corollary 1.8 coincides with that given by Coelho et al. ([5], Theorem 2) and Boshernitzan [START_REF] Boshernitzan | Dense orbits of rationals[END_REF].

On the other hand, let ∧ ⊂ R * + be a multiplicative subgroup and A ⊂ R be an additive subgroup, invariant by multiplication by elements of ∧ such that 1 ∈ A. Denote by T ∧,A the subgroup of PL + (S 1 ) consisting of elements with slopes in ∧, break points and their images in A.

Corollary 1.9. ([8], Lemma 3) Under the notation above, let f ∈ T ∧,A with the (D)-property and irrational rotation number ρ(f ). Then:

(i) if π(f ) = 1, then ρ(f ) = α β , where α, β ∈ A. (ii) if π(f ) = 1, then ρ(f ) = log α log π(f ) , where α ∈ Λ. Corollary 1.10. Let f ∈ T ∧,A with the (D)-property and ∧, A ⊂ Q. If π(f ) = 1 then the rotation number ρ(f ) is rational. Remark 3. Let f ∈ T ∧,A with the (D)-property and ∧, A ⊂ Q. If π(f ) = 1,
the rotation number ρ(f ) can be rational or not: For example, let consider the standard affine Boshernitzan B with two break points a and 0 = B(a) and slopes λ = DB + (0), λ ′ = DB + (a). We have

π(B) = σ B (0) = λ λ ′ = 1. By Corollary 1.8, ρ(B) = log λ ′ log λ-log λ ′ ( mod 1
). If we take λ = 4 3 p and λ ′ = 1 2 l , l, p ∈ N, then we see that

ρ(B) = -l log 2 (l + 2) log 2 -p log 3 / ∈ Q
since log 2 and log 3 are independent over Q.

Let m ≥ 2 be an integer and let T m denote the subgroup of PL + (S 1 ) consisting of elements f with slopes are integral powers of m, break points and their images under f are m-adic rational numbers (i.e. of the form

k m l : k, l ∈ Z). For m = 2, T 2 is the classical Thompson group.
Corollary 1.11. If f ∈ T m with the (D)-property (in particular if all break points belong to the same orbit) then the rotation number ρ(f ) is rational.

Note that Corollary 1.11 also hold whenever f does not satisfy the (D)property ([8]).

Some results

The purpose of this section is to introduce the basic properties of PLhomeomorphism that we need in the paper. We refer to [START_REF] Adouani | On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations[END_REF] for details. From now on, f is a PL-homeomorphism with irrational rotation number ρ(f ). Let first recall the following properties. Let f, g ∈ PL + (S 1 ). If f and g are conjugated in PL + (S 1 ) by h then π s,O f (c) (f ) = π s,Og(h(c)) (g), for every c ∈ S 1 . In particular, f satisfies the (D)-property if and only if so does g.

Proposition 2.2. ([2], Corollary 2.13)(PL-invariance of π). Let f, g ∈ PL + (S 1 ) with the (D)-property. If f and g are conjugated in PL + (S 1 ) then π(f ) = π(g). Lemma 2.3. Let σ 0 , . . . , σ n ∈ R * + and let b 0 < b 1 < • • • < b n < b n+1 be real numbers with b n+1 = b 0 + 1. If σ 0 × • • • × σ n = 1 then there exists L ∈ PL + (S 1 ) with break points b 0 = p( b 0 ), . . . , b n = p( b n ) and slopes σ L (b 0 ) = σ 0 , . . . , σ L (b n ) = σ n .
Proof. The PL-homeomorphism L of S 1 will be defined by its lift L restricted to [ b 0 , b 0 + 1] as follows:

-

L( b 0 + 1) = L( b 0 ) + 1, -b 0 , . . . , b n are the break points of L. -σ j := σ e L ( b j ) the jump of L in b j , j = 0, . . . , n. Write λ j = D L + ( b j ) the slope of L on [ b j , b j+1 [. Since σ 0 × • • • × σ n = 1, we have λ j = λ j λ j+1 × • • • × λ n-1 λ n λ n = σ j+1 × • • • × σ n λ n = (σ 0 × • • • × σ j ) -1 λ n . Now from the identity L( b 0 + 1) = L( b 0 ) + 1, it follows that 1 = L( b 0 + 1) -L( b 0 ) = n j=0 λ j ( b j+1 -b j ) = λ n n j=0 ( b j+1 -b j )(σ 0 × • • • × σ j ) -1 .
Hence,

λ n = 1 n j=0 ( b j+1 -b j )(σ 0 × • • • × σ j ) -1
and therefore for all j = 0, . . . , n

λ j = (σ 0 × • • • × σ j ) -1 n j=0 ( b j+1 -b j )(σ 0 × • • • × σ j ) -1 .
We conclude that L is uniquely determined by fixing L( b 0 ).

(i) if π(f ) = 1, f is conjugate to a rotation through a PL-homeomorphism. (ii) if π(f ) = 1, f is conjugate to a standard affine Boshernitzan through a PL-homeomorphism. Proof. For 0 ≤ k ≤ N i , 1 ≤ i ≤ p, denote by a k,i := σ f (f k (c i )) and σ k,i := σ f N+1 (f k (c i )) where N := max 1≤i≤p N i . Then σ k,i = N i j=k a j,i .
In particular, σ 0,i = π s,O f (c i ) (f ). Since f satisfies the (D)-property, we have σ 0,i = 1. We also have

π(f ) := p i=1 N i k=1 σ k,i .
By Lemma 2.3, there exists L ∈ PL + (S 1 ) with the following properties:

(i) L(0) = 0, (ii) C(L) = {f k (c i ) : 1 ≤ k ≤ N i , 1 ≤ i ≤ p}, (iii) σ L (f k (c i )) =        σ k,i , if (k, i) = (1, 1) σ 1,1 π(f ) , if (k, i) = (1, 1)
In particular, we have π s (L) = 1. We let

F = L • f • L -1 . A priori, the break points of F are: -The break points of L -1 : L(f k (c i )), 1 ≤ k ≤ N i , 1 ≤ i ≤ p, -The image by L of break points of f : L(f k (c i )), 0 ≤ k ≤ N i , 1 ≤ i ≤ p, -The image by L•f -1 of break points of L: L(f k (c i )), 0 ≤ k ≤ N i -1, 1 ≤ i ≤ p.
Therefore the possible break points of F are among:

L(f k (c i )), 0 ≤ k ≤ N i , 1 ≤ i ≤ p.
Compute the jumps of F in these points:

σ F L(f k (c i )) = σ L f (f k (c i )) σ f (f k (c i )) σ L (f k (c i )) = σ L f k+1 (c i ) σ f (f k (c i )) σ L (f k (c i ))
It follows that

σ F L(f k (c i )) =                          σ k+1,i σ k,i a k,i , if (k, i) = (0, 1) and (k, i) = (1, 1) σ 1,1 π(f ) a 0,1 σ 0,1 , if (k, i) = (0, 1) π(f ) σ 2,1 σ 1,1 a 1,1 , if (k, i) = (1, 1)
Therefore

σ F L(f k (c i )) =                  1 π(f ) , if (k, i) = (0, 1) π(f ), if (k, i) = (1, 1)

1, otherwise

We conclude that

F = L•f •L -1 ∈ P L + (S 1 ) with C(F ) ⊂ {L(c 1 ), L(f (c 1 ))} where σ F (L(c 1 )) = 1 π(f ) and σ F (L(f (c 1 ))) = π(f ). Proof of (i): If π(f ) = 1 the map F = L • f • L -1 ∈ PL + (S 1
) and has no break points, hence it is a rotation of angle ρ(f ).

Proof of (ii): If π(f ) = 1 then F satisfies the (D)-property with

C(F ) = {L(c 1 ), L(f (c 1 ))}. So F is an affine Boshernitzan of last break point L(f (c 1 )). If we conjugate F by a rotation R L(f (c 1 )) , then B 0 = R -1 L(f (c 1 )) • F • R L(f (c 1 )
) is a standard affine Boshernitzan with the (D)-property. This completes the proof.

Computation of the rotation numbers of PL homeomorphisms with the (D)-property

In this section, we compute the rotation number of f by using the explicite conjugacy given in the proof of Proposition 2.4. Let consider first the case where f is an affine Boshernitzan with the (D)-property. Proposition 3.1. ([3], [START_REF] Coelho | Absolutely continuous invariant measures for a class of affine interval interval exchange maps[END_REF]). Let B be a standard affine Boshernitzan with two break points a and 0 = B(a) and slopes λ = DB + (0), λ ′ = DB + (a). Then

(i) ρ(B) = log λ ′ log λ -log λ ′ (mod 1). (ii) ρ(B) = - log DB(x) log π(B) (mod 1), for every x ∈ S 1 . Proof. Let σ = 1 π(B) = λ λ ′ ∈ R * + \ {1}.
Let h σ denote the homeomorphism of S 1 with lift h σ : R -→ R restricted to [0, 1[ is given by:

h σ (x) = σ x -1 σ -1 , x ∈ [0, 1[.
We identify h σ with its lift h σ . Then h σ ∈ PL + (S 1 ) with one break point 0 and such that σ hσ (0) = σ. Moreover, h σ is analytic on S 1 \{0}. Then one can check easily (see [START_REF] Boshernitzan | Dense orbits of rationals[END_REF]) that h

-1 σ • B • h σ = R α is the rotation by angle α = log λ ′ log λ-log λ ′ (mod 1). Hence ρ(B) = log λ ′ log λ -log λ ′ (mod 1). Since π(B) = σ B (0) = λ ′ λ and log λ log λ -log λ ′ = log λ ′ log λ -log λ ′ (mod 1), it follows that for every x ∈ S 1 , - log DB(x) log π(B) = log λ log λ -log λ ′ (mod 1).
Therefore, (ii) holds.

Remark. Under the hypothesis of Proposition 3.1, Coelho et al. ([5],

Theorem 3) proved that B has a unique invariant probability measure which is absolutely continuous with respect to the Haar measure and with density ϕ explicitely calculated. Here from

h -1 σ • B • h σ = R ρ(B)
, we deduce that ϕ = Dh -1 σ .

Lemma 3.2. Let f ∈ PL + (S 1 ) and L ∈ PL + (S 1 ) be as in Proposition 2. [START_REF] Denjoy | Sur les courbes définies par les équations différentielles à la surface du tore[END_REF].

Then (i) σ L (b k ) = σ f q+1 (b k ) for every k = 0, . . . , n (ii)
n j=0 σ f q+1 (b j ) = 1. Proof. Observe that C(L) = M * (f ) = M(f)\{c 1 , . . . , c p }, S = M * (f )∩[0, 1[. Proof of (i): (i)-1: k = 0, m + 1. Then b k ∈ S and we have σ L (b k ) = σ f N+1 (b k ) = σ f q+1 (b k ) since q ≥ N + 1. (i)-2: k = 0. Then b 0 = 0. If 0 ∈ S then σ L (0) = σ f N+1 (0) = σ f q+1 (0). Now assume that 0 / ∈ S then σ L (0) = 1. Let's prove that σ f q+1 (0) = 1: -If 0 / ∈ ( p ∪ i=1 O f (c i )) ∩ [0, 1[ then σ f q+1 (0) = q k=0 σ f (f k (0)) = 1. -If 0 ∈ ( p ∪ i=1 O f (c i )) ∩ [0, 1[ then 0 = f j (c i ) for some 1 ≤ i ≤ p and j / ∈ {1, . . . , N }. Hence σ f q+1 (0) = q k=0 σ f (f k (0)) = q+j k=j σ f (f k (c i )). If j > N then obviously σ f q+1 (0) = 1. If -N -1 ≤ j ≤ -1 then q = 2(N + 1)
and

σ f q+1 (0) = q+j k=0 σ f (f k (c i )) = 2N +2+j k=0 σ f (f k (c i )) = π s,O f (c i ) (f ) = 1 since 2N +2+j ≥ N +1. If j < -N -1 then q = N and σ f q+1 (0) = q+j k=j σ f (f k (c i )) = N +j k=j σ f (f k (c i )) = 1. If = 0 then q = N and σ f q+1 (0) = q k=0 σ f (f k (c i )) = N k=0 σ f (f k (c i )) = π s,O f (c i ) (f ) = 1.
In either cases,

σ f q+1 (0) = 1 = σ L (0). (i)-3: k = m + 1. Then b m+1 = f (0). If f (0) ∈ S then σ L (f (0)) = σ f N+1 (f (0)) = σ f q+1 (f (0)). Now assume that f (0) / ∈ S then σ L (f (0)) = 1. Let's prove that σ f q+1 (f (0)) = 1: -If f (0) / ∈ ( p ∪ i=1 O f (c i )) ∩ [0, 1[ then σ f q+1 (f (0)) = q k=0 σ f (f k (f (0))) = 1. -If f (0) ∈ ( p ∪ i=1 O f (c i )) ∩ [0, 1[ then f (0) = f j (c i ) for some 1 ≤ i ≤ p and j / ∈ {1, . . . , N }. Hence σ f q+1 (f (0)) = q+1 k=1 σ f (f k (0)) = q+j+1 k=j+1 σ f (f k (c i )). If j > N then obviously σ f q+1 (f (0)) = 1. If -N ≤ j ≤ 0 then q = 2(N +1) and σ f q+1 (f (0)) = q+j+1 k=0 σ f (f k (c i )) = 2N +3+j k=0 σ f (f k (c i )) = π s,O f (c i ) (f ) = 1 since 2N + 3 + j ≥ N + 3. If j < -N -1 then q = N and σ f q+1 (f (0)) = q+j+1 k=j+1 σ f (f k (c i )) = N +j+1 k=j+1 σ f (f k (c i )) = 1. If j = -N -1 then q = N and σ f q+1 (f (0)) = 0 k=-N σ f (f k (c i )) = σ f (0) = σ f (f -N -2 (c i )) = 1.
In either cases,

σ f q+1 (f (0)) = 1 = σ L (f (0)). Proof of (ii): if 0, f (0) ∈ S then π s (L) = 1 = n j=0 σ L (b j ) = n j=0 σ f q+1 (b j ). If 0 / ∈ S (resp.f (0) / ∈ S) then σ L (0) = 1 (resp. σ L (f (0)) = 1) and n j=0 σ f q+1 (b j ) = c∈S σ L (c) = π s (L) = 1.
Proof of Theorem 1.5. Let L ∈ PL + (S 1 ) be given by Proposition 2.4.

Case (i): π(f ) = 1. Then B = L • f • L -1
is an affine Boshernitzan with break-points b := L(c 1 ) and B(b) = L(f (c 1 )) and jumps at these points:

σ B (b) = 1 π(f ) and σ L (B(b)) = π(f ) = 1. We have ρ(f ) = ρ(B). Since π(f ) = π(B) (Proposition 2.
2), it follows by Proposition 3.1, (ii), that

ρ(f ) = - log(DB -(0)) log(π(B)) = - log(DB -(0)) log(π(f )) (mod 1).
We have

DB -(0) = DL -(f (0))Df -(0) DL -(0)
and

DL -(0) DL -(f (0)) = c∈S∩[0,f (0)[ σ L (c) (since if 0 / ∈ S, σ L (0) = 1) = π(f ) ε c∈S∩[0,f (0)[ σ f N+1 (c) (since σ f q+1 (c) = σ f N+1 (c) if c ∈ S)
where ε ∈ {0, 1}. It follows that

ρ(f ) = log 1 Df -(0) c∈S∩[0,f (0)[ σ f N+1 (c) log(π(f )) (mod 1).
Case (ii): π(f ) = 1. In this case, the map

F = L • f • L -1 is a rotation of angle ρ(f ) with L(b 0 ) = 0. By Lemma 3.2, we have σ k := σ L (b k ) = σ f q+1 (b k ). Compute λ k = DL + (b k ) for k = 0, . . . , n : we have λ k = λ k λ k+1 × • • • × λ n-1 λn λ n = σ k+1 × • • • × σ n λ n with σ k+1 × • • • × σ n = 1 if k = n. By Lemma 3.2, σ 0 × • • • × σ n = 1, it follows that λ k = σ 0 × • • • × σ k -1 λ n = k j=0 σ f q+1 (b j ) -1 λ n . Now from the identity L(b 0 + 1) = L(b 0 ) + 1, it follows that 1 = L(b 0 +1)-L(b 0 ) = n k=0 λ k (b k+1 -b k ) = n k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1 λ n .
Therefore,

λ n = 1 n k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1 . Since L(b 0 ) = 0 and b m+1 = f (b 0 ), it follows that ρ(f ) = L(f (b 0 )) = L(b m+1 ) = m k=0 (L(b k+1 ) -L(b k )) = m k=0 λ k (b k+1 -b k )
We conclude that

ρ(f ) = m k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1 n k=0 (b k+1 -b k ) k j=0 σ f q+1 (b j ) -1
.

Proof of Corollary 1.6. Since f has the (D)-property, we have for every i = 1, . . . , p,

σ f N+1 (f k (c i )) =      σ f (f N (c i )) = (σ f (c i )) -1 , if k ≥ 1 1, if k = 0 It follows that π(f ) := p i=1 N i k=1 σ f N+1 (f k (c i )) = p i=1 (σ f (c i )) -N i . Write l i = card{1 ≤ k ≤ N i : f k (c i ) ∈ [0, f (0)[}. Then we have x∈S∩[0,f (0)[ σ f N+1 (x) = p i=1 (σ f (c i )) -l i .
By the formula in Theorem 1.5, we get

ρ(f ) = log 1 Df -(0) p i=1 (σ f (c i )) -l i log p i=1 (σ f (c i )) -N i = log (Df -(0)) + p i=1 l i log σ f (c i ) p i=1 N i log σ f (c i ) (mod 1).
Proof of Corollary 1.7. By hypothesis, f has one connection

M (f ) = [c, . . . , f N (c)], N ≥ 1 with M (f ) ∩ C(f ) = {c, f N (c)}.
So the formula holds by applying the formula in Corollary 1.6 for p = 1, since in this case, we have

π(f ) = (σ f (c)) -N = 1.
Proof of Corollary 1.8. This follows directly from Corollary 1.7 for N = 1.

Proof of Corollary 1.9. This follows directly from the formula in Theorem 1.5.

Proof of Corollary 1.10. This folllows from Corollary 1.9.

Proof of Corollary 1.11. By Corollary 1.10, one can suppose that π(f ) = 1. Since the slopes of f are of the form m l , l ∈ Z, and

π(f ) := p i=1 N i k=1 σ k,i where σ k,i = σ f N+1 (f k (c i )), it follows that π(f ) = m k for some k ∈ Z. Moreover Df -(0) = m s for some k ∈ Z. Therefore the formula for )
given by Theorem 1.5, implies that ρ(f

) = log m l log m k = l k for some l ∈ Z. So ρ(f ) ∈ Q.

Examples

Example 4.1. (Herman's example). Let λ > 1 and β > 0 be two real numbers. Define the P L homeomorphism f β,λ of S 1 with break points 0 and a defined by the restriction of its lift on [0, 1], noted also f β,λ by:

f β,λ =    λx, if 0 ≤ x < a; λ -β (x -1) + 1, if a ≤ x ≤ 1.
where a ∈]0, 1[ is given by λa = λ -β (a -1) + 1. Then f β,λ have jumps in 0 and a given by

σ f β,λ (0) = λ -1-β and σ f β,λ (a) = 1 σ f β,λ (0) = λ 1+β . (See Fig. 2). λ λ -β λ a 0 1 1 3

Figure 2. Herman's Example

Herman has considered the family of PL homeomorphisms R b • f β,λ of S 1 where b ∈ [0, 1] and R b is the rotation of angle b, these homeomorphisms have the break points 0 and a with jumps λ -1-β and λ 1+β respectively. He showed (see [START_REF] Herman | Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations[END_REF], chapter III), using continuity of the rotation number, that the rotation map ρ

: b ∈ [0, 1[-→ ρ(R b • f β,λ ) is a continuous increasing and surjective map; moreover, ρ is injective on (R \ Q)∩]0, 1[; that is for every α ∈ (R \ Q)∩]0, 1[, there exists a unique b = b(α) ∈ [0, 1[ such that ρ(R b • f β,λ ) = α. Denote by f α,β,λ = R b(α) • f β,λ . Then f α,β,λ ∈ P L + (S 1 )
with irrational rotation number α.

Suppose that f α,β,λ has the (D)-property (i.e. 0 and a belong to the sam orbit). Set f N (a) = 0, N ≥ 1. Then

(i) α = - β N (1 + β) + l N (mod 1) where l := card{1 ≤ k ≤ N : f k (a) ∈ [0, f (0)[}. In particular, β 1 + β ∈ Zα (mod 1).
(ii) the conjugacy H from f α,β,λ to R α is piecewise analytic but not PL. (iii) in the special case N = 1, we obtain:

α = 1 1 + β (mod 1) and H = h σ where σ > 0, σ = 1. Proof. (i): We have D(f α,β,λ ) -(0) = λ -β , σ f α,β,λ (0) = λ -1-β and σ f α,β,λ (a) = λ 1+β . Hence by Corollary 1.7, α = 1 N log λ -β log λ 1+β + l N = - β N (1 + β) + l N ( mod 1)
(ii): Now we determine the conjugacy from f α,β,λ to R α : By Proposition 2.4, there exists

L ∈ PL + (S 1 ) such that B := L • f α,β,λ • L -1 is an affine Boshernitzan with two break points L(a) and B(L(a)) = L(f α,β,λ (a)
) and jumps at these points

1 π(f α,β,λ ) = (σ f α,β,λ (a)) N = λ N (1+β) and π(f α,β,λ ) = (σ f α,β,λ (a)) -N = λ -N (1+β)
respectively. We determine the conjugating L in this case: We have

L(0) = 0, C(L) = {f k α,β,λ (a) : k = 1, . . . , N } and σ L (f k α,β,λ (a)) =            σ f N+1 α,β,λ (f k (a)), if 1 < k ≤ N σ f N+1 α,β,λ (f α,β,λ (a)) π(f α,β,λ ) , if k = 1 =      (σ f α,β,λ (a)) -1 , if 1 < k ≤ N (σ f α,β,λ (a)) -1+N , if k = 1 =      λ -1-β , if 1 < k ≤ N (λ (1+β)(-1+N ) , if k = 1 Now B 0 = R -1 L(f α,β,λ (a)) • B • R L(f α,β,λ (a)
) is a standard affine Boshernitzan with break points b : R -1 L(f α,β,λ (a)) (L(a)) and 0 = B 0 (b) and we have

π(B 0 ) = π(f α,β,λ ) = λ -N (1+β) .
We let σ = λ N (1+β) . Then by the proof of Proposition 3.1, we have

B 0 = h σ •R α •h -1
σ where h σ is the homeomorphism of S 1 defined by the restriction to [0, 1[ of its lift denoted also by h σ :

h σ (x) = σ x -1 σ-1 , x ∈ [0, 1[. We conclude that H • f α,β,λ • H -1 = R α where H := h -1 σ • R -1 L(f α,β,λ (a)) • L.
Note that H is piecewise analytic but not PL (since σ = 1).

(iii): in the special case N = 1, the map f α,β,λ has two break points a and 0 = f α,β,λ (a) with α = 1 1 + β (mod 1), σ = λ 1+β . Since L is the identity, it

follows that f α,β,λ = h σ • R α • h -1 σ .
λ 2 = λ 4 . Therefore ) =            log λ 2 log λ 1 -2 log λ 2 + log λ 3 (mod 1), if π(f ) = 1 -λ 1 b λ 1 c + (1 -c) (mod 1), if π(f ) = 1 b c d f (
π(f ) = (σ f (b)) -1 (σ f (d)) -1 = λ 2 λ 1 λ 2 λ 3 = λ 2 2 λ 1 λ 3 .
Moreover, since f (d) = c < f (0) and f (b) = 0, we get l 1 = l 2 = 1. Therefore:

-if π(f ) = 1, ρ(f ) = log λ 2 + log σ f (b) + log σ f (d) log σ f (b) + log σ f (d) = log λ 2 log λ 1 λ 3 λ 2 2 = log λ 2 log λ 1 -2 log λ 2 + log λ 3
(mod 1).

We have

S = (M (f )\{b, d}) ∩ [0, 1[= {a, c} ∩ [0, 1[= {a, c}. -S ∪ {0, f (0)} = {b 0 , b 1 , b 2 } where b 0 = a = 0 < b 1 = c < b 2 = f (0) < b 3 = 1. Hence n = 2 and m = 1. -S ∩ [0, f (0)[= {b 0 , b 1 } -if π(f ) = 1 then λ 2 2 λ 1 λ 3 = 1.
ρ(f ) = 1 k=0 (b k+1 -b k )( k j=0 σ f 2 (b j )) -1 2 k=0 (b k+1 -b k )( k j=0 σ f 2 (b j )) -1 = (b 1 -b 0 )(σ f 2 (b 0 )) -1 + (b 2 -b 1 )( 1 j=0 (σ f 2 (b j )) -1 ) (b 1 -b 0 )(σ f 2 (b 0 )) -1 + (b 2 -b 1 )( 1 j=0 (σ f 2 (b j )) -1 ) + (b 3 -b 2 )( 2 j=0 (σ f 2 (b j )) -1 ) = c(σ f (0)) -1 + (f (0) -c)(σ f (0)) -1 (σ f (c)) -1 c(σ f (0)) -1 + (f (0) -c)(σ f (0)) -1 (σ f (c)) -1 + (1 -f (0))(σ f (0)) -1 (σ f (c)) -1 = cλ 1 + (f (0) -c) cλ 1 + (1 -c) = f (0) -1 cλ 1 + (1 -c) (mod 1). As λ 1 b = 1 -f (0), we obtain ρ(f ) = -λ 1 b λ 1 c + (1 -c) (mod 1). Example 4.3. Let f ∈ PL + (S 1
) with five break points a 0 = 0 < a 1 < a 2 < a 3 < a 4 with f k (0) = a k , k = 0, 1, 2, 3, 4. (see Fig. 4). The break points of f are on the orbit of 0, then f satisfies the (D)-property. The rotation number of f is given by: Proof. The map f has one maximal connection M (f ) = [0, . . . , f 4 (0)] corresponding to N = 4. We have

ρ(f ) =          - log λ 1 log(π(f )) (mod 1), if π(f ) = 1 a 1 4a 1 -a 4 + 1 (mod 1), if π(f ) = 1 where π(f ) = λ 2 λ 3 λ 4 λ 5 λ 4
a 2 = (1 + λ 2 )a 1 , a 3 = (1 + λ 2 + λ 2 λ 3 )a 1 , a 4 = (1 + λ 2 + λ 2 λ 3 + λ 2 λ 3 λ 4 )a 1 , a 5 = f 5 (0) = (1 + λ 2 + λ 2 λ 3 + λ 2 λ 3 λ 4 + λ 2 λ 3 λ 4 λ 5 )a 1 , λ 1 = 1 -a 5 + a 1 1 -a 4 , -0 < a 1 < 1 1 + λ 2 + λ 2 λ 3 + λ 2 λ 3 λ 4 + λ 2 λ 3 λ 4 λ 5 , We have σ f 5 (0) = 1, σ f 5 (a 1 ) = λ 2 λ 1 , σ f 5 (a 2 ) = λ 3 λ 1 , σ f 5 (a 3 ) = λ 4 λ 1 , σ f 5 (a 4 ) = λ 5 λ 1 . Hence π(f ) = λ 2 λ 3 λ 4 λ 5 λ 4 1 . We have S = (M (f )\{0}) ∩ [0, 1[. -S ∪ {0, f (0)} = {b 0 , b 1 , b 2 , b 3 , b 4 } where b 0 = 0 < b 1 = f (0) = a 1 < b 2 = a 2 < b 3 = a 3 < b 4 = a 4 < b 5 = 1. Hence n = 4 and m = 0. -S ∩ [0, f (0)[= {b 0 } -Df -(0) = λ 1 . By Theorem 1.5, we get: -if π(f ) = 1 then ρ(f ) = log 1 Df -(0) log(π(f )) (mod 1) = - log λ 1 log(π(f )) (mod 1) -if π(f ) = 1 then λ 2 λ 3 λ 4 λ 5 λ 4 1 = 1. Let L ∈ PL + (S 1
) be the conjugating from f to the rotation R ρ(f ) as given in the proof of Proposition 2.4, then we have

C(L) = S and Df + (a 4 ) = DL + (a 4 ) DL + (f (a 4 )) = 1, so λ 1 = 1. Then 4 k=0   (b k+1 -b k ) k j=0 σ f N+1 (b j ) -1   = (a 1 -0) + (a 2 -a 1 ) λ 1 λ 2 + (a 3 -a 2 ) λ 2 1 λ 2 λ 3 + (a 4 -a 3 ) λ 3 1 λ 2 λ 3 λ 4 + (1 -a 4 ) λ 4 1 λ 2 λ 3 λ 4 λ 5 = a 1 + λ 1 a 1 + λ 2 1 a 1 + λ 3 1 a 1 + (1 -a 4 ) = 4a 1 -a 4 + 1. Hence ρ(f ) = a 1 4a 1 -a 4 + 1 (mod 1). Example 4.4. Let f ∈ PL + (S 1
) with five break points a 0 = 0, a 1 , a 2 , a 3 and a 4 with f k (0) = a k , k = 0, 1, 2, 3, 4. (see Fig. 5). The break points of f are on the orbit of 0, then f satisfies the (D)-property. The rotation number of f is given by:

ρ(f ) =           
= w + v + w w + v + w + v + v = v + 2w 3v + 2w = -2v 3v + 2w = 2a 2 -2 1 + 2a 3 -3a 2
(mod 1).

Figure 1 .

 1 Figure 1. A standard affine Boshernitzan

Proposition 2 . 1 .

 21 ([2], Proposition 2.1)(Invariance of the (D)-property).

Proposition 2 . 4 .

 24 ([2], Corollary 3.6) If f ∈ PL + (S 1 ) with the (D)-property and irrational rotation number ρ(f ). Then:

Example 4 . 2 .

 42 Let f ∈ PL + (S 1 ) with four break points a = 0, b, c and d with f (b) = a and f (d) = c (see Fig. 3). Then f has two maximal connections [b, f (b)] and [d, f (d)] corresponding to N = N 1 = N 2 = 1. We see that f satisfies the (D)-property if

Figure 3 .

 3 Figure 3. PL with four break points

  Moreover, let L ∈ PL + (S 1 ) be the conjugating from f to the rotation R ρ(f ) as given in the proof of Proposition 2.4, then we have C(L) = {a, c} and Df + (b) = DL + (b) DL + (f (b)) = DL + (b) DL + (0) = 1, it follows that λ 2 = 1 and λ 1 λ 3 = 1. By Theorem 1.5, we get:

Figure 4 .

 4 Figure 4. PL with five break points

log(λ 2 λ 4 ) 1 2a 2 Figure 5 .λ 1 , σ f 5 (a 2 ) = λ 2 λ 4 λ 1 λ 3 , σ f 5 (a 3 ) = λ 2 λ 4 λ 3 λ 5 ,σ f 5 (a 4 ) = λ 4 λ 5 .σ f 5 (b 0 )σ f 5 (b 1 )λ 2 λ 4 ) 1 λ 3 λ 5 λ 2 λ 4 + (b 3 -b 2 ) λ 1 λ 3 λ λ 2 2 λ 4 +(b 4 -b 3 ) λ 1 λ 3 λ 2 5 λ 2 λ 4 2 + ( 1 -b 4 ) λ 1 λ 3 λ 5 2 λ 2 λ 4 3 = (a 3 -λ 5 λ 2 λ 4 + ((a 4 - 1 ) -a 1 ) λ 1 λ 3 λ 5 λ 2 2 λ 4 + (a 2 -(a 4 - 1 )) λ 1 λ 3 λ 2 5 λ 2 λ 4 2 + ( 1 -a 2 ) λ 1 λ 3 λ 5 2 λ 2 λ 4 3 1  2 + (a 2 -(a 4 - 1 ))λ 5 + ( 1 -σ f 5 (b j )) - 1  2 . 2 (a 3

 4125153555555141432443521423344114241521224122415151223 Figure 5. PL with five break points


Andrew John Treharne

Martin Christopher Grossel

Andrew John Lotery
email: a.j.lotery@soton.ac.uk

Heather Anne

Jane Thomson

Heather A Thomson

Prof Andrew Lotery

The influence of polymer scaffold properties on retinal cell adhesion and control

Keywords: Retinal Pigment Epithelium, Transplantation, Retinal Degeneration, Biomaterials, Cell Adhesion

Background

Despite differing aetiology and pathology, degenerative retinal diseases including age related macular degeneration (AMD) and retinitis pigmentosa (RP) culminate in photoreceptor cell death. Loss or damage to the retinal pigment epithelium (RPE) is also a common feature of these somewhat disparate conditions. [START_REF] Hartong | Retinitis pigmentosa[END_REF][START_REF] Jager | Age-related macular degeneration[END_REF] Currently no treatments are known to halt or reverse this cellular loss. Recent advances in gene therapy for degenerative retinal diseases have shown promise in both animal models [START_REF] Acland | Long-term restoration of rod and cone vision by single dose rAAV-mediated gene transfer to the retina in a canine model of childhood blindness[END_REF][START_REF] Bennicelli | Reversal of blindness in animal models of leber congenital amaurosis using optimized AAV2-mediated gene transfer[END_REF][START_REF] Tschernutter | Long-term preservation of retinal function in the RCS rat model of retinitis pigmentosa following lentivirus-mediated gene therapy[END_REF] and clinical trials. [START_REF] Bainbridge | Effect of gene therapy on visual function in Leber's congenital amaurosis[END_REF][START_REF] Hauswirth | Treatment of leber congenital amaurosis due to RPE65 mutations by ocular subretinal injection of adeno-associated virus gene vector: short-term results of a phase I trial[END_REF][START_REF] Maguire | Safety and efficacy of gene transfer for Leber's congenital amaurosis[END_REF] However, longterm safety and efficacy of gene replacement therapy is yet to be established. In addition the effectiveness of such treatments will be somewhat limited in patients with advanced retinal degeneration, as a result of significant pre-existing cell loss. Thus treatments which aim to replace dystrophic cells are desirable either alone or as an adjunct to gene therapy.

Transplantation and the retina

Positive outcome of cell transplantation studies in animal models, such as the Royal College of Surgeons rat (RCS), have paved the way for human studies. [START_REF] Gias | Preservation of visual cortical function following retinal pigment epithelium transplantation in the RCS rat using optical imaging techniques[END_REF][START_REF] Pinilla | Preservation of outer retina and its synaptic connectivity following subretinal injections of human RPE cells in the Royal College of Surgeons rat[END_REF][START_REF] Schraermeyer | Transplantation of iris pigment epithelium into the choroid slows down the degeneration of photoreceptors in the RCS rat[END_REF] Some efficacy has been demonstrated with transplantation of healthy photoreceptors, [START_REF] Berger | Photoreceptor transplantation in retinitis pigmentosa: short-term follow-up[END_REF][START_REF] Kaplan | Human photoreceptor transplantation in retinitis pigmentosa. A safety study[END_REF] sheets of foetal neuroretina [START_REF] Humayun | Human neural retinal transplantation[END_REF][START_REF] Radtke | Vision change after sheet transplant of fetal retina with retinal pigment epithelium to a patient with retinitis pigmentosa[END_REF][START_REF] Radtke | Vision improvement in retinal degeneration patients by implantation of retina together with retinal pigment epithelium[END_REF] and donor iris pigment epithelium tissue. [START_REF] Abe | Auto iris pigment epithelial cell transplantation in patients with age-related macular degeneration: short-term results[END_REF][START_REF] Lappas | Translocation of iris pigment epithelium in patients with exudative age-related macular degeneration: long-term results[END_REF][START_REF] Thumann | Transplantation of autologous iris pigment epithelium after removal of choroidal neovascular membranes[END_REF] In particular transplantation or translocation of RPE cells with or without choroid has shown promise. [START_REF] Chen | Long-term visual and microperimetry outcomes following autologous retinal pigment epithelium choroid graft for neovascular age-related macular degeneration[END_REF][START_REF] Tezel | Adult retinal pigment epithelial transplantation in exudative age-related macular degeneration[END_REF][START_REF] Treumer | Autologous retinal pigment epithelium-choroid sheet transplantation in age related macular degeneration: morphological and functional results[END_REF] Autologous RPE transplantation can in principle restore vision, by supporting photoreceptor cell function. However surgical complications remain high and visual improvement so far has been limited. [START_REF] Maclaren | Autologous transplantation of the retinal pigment epithelium and choroid in the treatment of neovascular age-related macular degeneration[END_REF][START_REF] Da Cruz | RPE transplantation and its role in retinal disease[END_REF] Thus stem cells and/or progenitor cells are being assessed as alternative sources of cells for transplantation.

Stem cell transplantation

Stem cells are typically described as undifferentiated cells which are capable of self-renewal and have the ability to give rise to one or more differentiated cell types. [START_REF] Mooney | Cell delivery mechanisms for tissue repair[END_REF] Within the adult eye there are several areas which harbour progenitor cell populations including the ciliary margin of the retina, [START_REF] Coles | Facile isolation and the characterization of human retinal stem cells[END_REF][START_REF] Tropepe | Retinal Stem Cells in the Adult Mammalian Eye[END_REF][START_REF] Macneil | Comparative analysis of progenitor cells isolated from the iris, pars plana, and ciliary body of the adult porcine eye[END_REF] the iris pigment epithelium [START_REF] Macneil | Comparative analysis of progenitor cells isolated from the iris, pars plana, and ciliary body of the adult porcine eye[END_REF][START_REF] Asami | Multipotent cells from mammalian iris pigment epithelium[END_REF][START_REF] Sun | Retinal stem/progenitor properties of iris pigment epithelial cells[END_REF] and Müller cells. [START_REF] Fischer | Muller glia are a potential source of neural regeneration in the postnatal chicken retina[END_REF][START_REF] Limb | In vitro characterization of a spontaneously immortalized human Muller cell line (MIO-M1)[END_REF][START_REF] Osakada | Wnt signaling promotes regeneration in the retina of adult mammals[END_REF] These progenitor cells have been shown to differentiate into a range of neuronal, glial and retinal specific cell types following maintenance in a conducive in vitro environment. It is this obligate requirement for complex cellular interactions and often limited self renewal capacity in vitro which defines these as progenitor cells rather than stem cells. [START_REF] Ohta | Neuronal stem/progenitor cells in the vertebrate eye[END_REF] Stem cell therapy to repair and regenerate the corneal epithelium has been successfully employed clinically for a number of years. [START_REF] Limb | Ocular regeneration by stem cells: present status and future prospects[END_REF] Retinal cell replacement however remains elusive. To date, a number of studies have investigated various stem cell types as potential sources for retinal transplantation including embryonic stem cells (ESCs), adult stem/progenitor cells and more recently induced pluripotent stem cells (iPS). [START_REF] Osakada | Toward the generation of rod and cone photoreceptors from mouse, monkey and human embryonic stem cells[END_REF][START_REF] Chacko | Transplantation of ocular stem cells: the role of injury in incorporation and differentiation of grafted cells in the retina[END_REF][START_REF] Buchholz | Derivation of functional retinal pigmented epithelium from induced pluripotent stem cells[END_REF] Results have been encouraging. Following differentiation into retinal pigment epithelium cells in vitro, subretinally transplanted ESCs have led to improvements in visual acuity. [START_REF] Idelson | Directed differentiation of human embryonic stem cells into functional retinal pigment epithelium cells[END_REF][START_REF] Lu | Long-term safety and function of RPE from human embryonic stem cells in preclinical models of macular degeneration[END_REF] In addition, human iPS cells have been differentiated towards functional RPE cells. This has resulted in short-term survival and maintenance of photoreceptor cells following transplantation into the RCS rat model. [START_REF] Carr | Protective effects of human iPS-derived retinal pigment epithelium cell transplantation in the retinal dystrophic rat[END_REF] Despite these apparently positive findings, efficiency of cell delivery and degree of visual rescue often remain unsatisfactory. This lack of efficacy may be due to a number of reasons for example, RPE cells are anchorage dependent and therefore must quickly (re)attach to an accommodating matrix following transplantation. Bruch's membrane (BM), the inner most layer of the choroid, is the supportive structure upon which RPE cells are normally attached. However, in advanced retinal disease, with age or following macular surgery the basal lamina layer of BM may be damaged or absent.[42] Thus, it is difficult for newly transplanted cells to attach in such a non-permissive environment. Transplanted cells may therefore undergo anoikis, or cells may clump together rather than forming appropriately polarised cell monolayers. Lack of cell to cell contact may also lead to transition of RPE cells to inappropriate phenotypes. [START_REF] Tamiya | Epithelial-mesenchymal transition and proliferation of retinal pigment epithelial cells initiated upon loss of cell-cell contact[END_REF] Use of stem cells for retinal repair holds enormous promise for generation of sufficient appropriate cell populations for transplantation, due to their proliferative nature and ability to give rise to multiple cell types. The gap between theory and clinical exploitation however remains considerable. [START_REF] Maclaren | Stem cell therapy and the retina[END_REF] Significant challenges remain both from a basic biology and practical application standpoint. Principal among which is identification and use of cell sources which are viable, karyotypically stable and which will yield sufficient cell numbers. [START_REF] Djojosubroto | Chromosomal number aberrations and transformation in adult mouse retinal stem cells in vitro[END_REF] The ability to isolate an homogenous, mature population of cells for transplantation is also vitally important to avoid the possibility of teratoma formation, especially when using embryonic derived cell sources. Safe and efficient tissue delivery also needs to be considered, as does survival and integration of the transplanted cells within the host. [START_REF] Macneil | Comparative analysis of progenitor cells isolated from the iris, pars plana, and ciliary body of the adult porcine eye[END_REF][START_REF] West | Cell transplantation strategies for retinal repair[END_REF][START_REF] Tucker | The use of progenitor cell/biodegradable MMP2-PLGA polymer constructs to enhance cellular integration and retinal repopulation[END_REF] Any transplanted material must also be capable of maintaining an appropriate state of differentiation in vivo. In addition immune surveillance is a significant issue, thus autologous sources of cells for transplantation, to negate problems with graft rejection, would be the ideal. [START_REF] Streilein | Immunobiology and privilege of neuronal retina and pigment epithelium transplants[END_REF]

Cell delivery scaffolds

In this review we aim to discuss how substrates can be tailored to enhance retinal cell attachment and viability for transplantation. It has been documented that cells in suspension have a lower immune privilege than those delivered as an intact sheet. [START_REF] Streilein | Immunobiology and privilege of neuronal retina and pigment epithelium transplants[END_REF] Furthermore cells injected as a suspension often fail to regain a fully differentiated phenotype. [START_REF] Sheridan | Basement membranes and artificial substrates in cell transplantation[END_REF][START_REF] Binder | Transplantation of the RPE in AMD[END_REF] In addition, the viability of RPE cells delivered to the subretinal space is often dependent on the integrity of the underlying substrate: Bruch's membrane. [START_REF] Gullapalli | Impaired RPE survival on aged submacular human Bruch's membrane[END_REF][START_REF] Tezel | Reengineering of aged Bruch's membrane to enhance retinal pigment epithelium repopulation[END_REF] Thus it may be prudent to transplant sheets of cells rather than suspensions of single cells or cell aggregates. Many techniques and devices are being developed to improve the delivery and survival of transplanted cells, one such being the use of scaffolds for cell attachment and growth prior to transplantation. Natural scaffold materials have been investigated including Descemet's membranes, [START_REF] Thumann | Descemet's membrane as membranous support in RPE/IPE transplantation[END_REF] lens capsules [START_REF] Hartmann | Human and porcine anterior lens capsule as support for growing and grafting retinal pigment epithelium and iris pigment epithelium[END_REF][START_REF] Lee | Spatial cues for the enhancement of retinal pigment epithelial cell function in potential transplants[END_REF] and amniotic membranes. [START_REF] Capeans | Amniotic membrane as support for human retinal pigment epithelium (RPE) cell growth[END_REF][START_REF] Ohno-Matsui | In vitro and in vivo characterization of iris pigment epithelial cells cultured on amniotic membranes[END_REF] Synthetic polymeric biomaterials have also been utilised as cell delivery vehicles for retinal cell types as detailed in Table 1.

Biomaterials and the eye

Young and co-workers demonstrated that use of a polymer scaffold for stem cell delivery to the subretinal space presented advantages over simple injection of cell suspensions. [START_REF] Tomita | Biodegradable polymer composite grafts promote the survival and differentiation of retinal progenitor cells[END_REF].

When a suspension of cells was injected into the eye, problems such as immune response, disorganisation of cells and cell death were encountered. However, when a scaffold made of biodegradable polyester was used, cell survival and organisation were improved. There was also evidence that the scaffold promoted differentiation of the retinal progenitor cells towards mature retinal cell phenotypes. [START_REF] Tomita | Biodegradable polymer composite grafts promote the survival and differentiation of retinal progenitor cells[END_REF] Poly (L-lactic acid) (PLLA) and poly (D,L-lactic-coglycolic acid) (PLGA) were used in this investigation. These polymers were selected as they are biocompatible, easily processable and have been successfully used for tissue engineering applications such as bladder substitution [START_REF] Atala | Tissue engineering for bladder substitution[END_REF] and bone tissue regeneration. [START_REF] Shin | Effect of ionic activity products on the structure and composition of mineral self assembled on three-dimensional poly(lactide-coglycolide) scaffolds[END_REF] The degradation rate of these polymers can also be manipulated by changing properties such as molecular weight and the ratio of lactic to glycolic units. Thus polymers can be designed to degrade over the most appropriate timescale for the desired application.

Aside from PLLA and PLGA, several different polymers and preparation techniques have been investigated in the search for a polymer with suitable biocompatible and mechanical properties to be employed as a cellular scaffold in the eye. Many factors such as surface chemistry, mechanical properties and surface topology can affect the usefulness of different materials for cell attachment and survival. Poly(methyl methacrylate) (PMMA) has been used to manufacture ultra-thin, micro-machined scaffolds for retinal progenitor cells (RPCs). [START_REF] Tao | Survival, migration and differentiation of retinal progenitor cells transplanted on micro-machined poly(methyl methacrylate) scaffolds to the subretinal space[END_REF] Murine RPCs isolated from enhanced green fluorescent protein positive (GFP+) transgenic mice (C57BL/6 background) were shown to grow and differentiate well on these scaffolds both in vitro and in wild-type C57BL/6 mice. Similarly, poly(glycerol sebacate) (PGS) [START_REF] Redenti | Engineering retinal progenitor cell and scrollable poly(glycerol-sebacate) composites for expansion and subretinal transplantation[END_REF] has been used to manufacture a porous, elastic scaffold and poly(ε-caprolactone) (PCL) [START_REF] Redenti | Retinal tissue engineering using mouse retinal progenitor cells and a novel biodegradable, thin-film poly(e-caprolactone) nanowire scaffold[END_REF] to produce ultra-thin nanowire scaffolds. Again, successful growth of murine RPCs was shown both in vitro and wild-type and degenerative mouse models. Both types of scaffold were coated with laminin prior to cell seeding. Laminin is an extracellular matrix (ECM) protein which can aid the survival of transplanted cells by mimicking their natural environment. It was shown that, in culture expression of mature bipolar and photoreceptor cell markers such as rhodopsin, recoverin and PKCα were upregulated on these scaffolds, while several early progenitor markers including Pax6, Hes1, nestin and Sox2 were downregulated. Biodegradable poly(hydroxybutyrate-co-hydroxyvalerate) has been used to manufacture non-porous thin films. [START_REF] Tezcaner | Retinal pigment epithelium cell culture on surface modified poly(hydroxybutyrate-co-hydroxyvalerate) thin films[END_REF] These films were modified using an oxygen plasma to generate a more hydrophilic surface. This approach was found to provide a more suitable environment for attachment and growth, of a human RPE cell line, compared to untreated films. Use of an air plasma treatment by Williams et al. was found to increase the surface wettability of commercial poly(urethanes) [START_REF] Williams | Polyurethanes as potential substrates for subretinal retinal pigment epithelial cell transplantation[END_REF] and poly(dimethylsiloxane) (silicone rubber). [START_REF] Krishna | Polydimethylsiloxane as a substrate for retinal pigment epithelial cell growth[END_REF] These surfaces were found to be well suited to growth of an intact monolayer of RPE cells compared to an untreated surface. Besides synthetic polymers, proteins such as collagen have been manufactured into thin films and shown to be compatible with RPE cells both in vitro and in vivo. [START_REF] Thumann | The in vitro and in vivo behaviour of retinal pigment epithelial cells cultured on ultrathin collagen membranes[END_REF] However, as the collagen used was derived from equine sources, problems may be encountered at a later stage due to implantation of a non-native protein into the eye.

Surface modification and tissue engineering

The influence of surface modification on biomaterial applications has been extensively studied in other fields of tissue engineering (Table 1). The introduction of an amine surface coating, for example, may increase the stability of a biodegradable polymer in the initial stages of degradation. Attempts have been made to use wet chemical and plasma techniques for creating an amine surface layer on PLGA. [START_REF] Djordjevic | Morphological and surface compositional changes in poly(lactide-co-glycolide) tissue engineering scaffolds upon radio frequency glow discharge plasma treatment[END_REF] PLGA has also been treated with ammonia plasma to facilitate collagen attachment for cartilage tissue engineering. [START_REF] Thissen | Synthetic biodegradable microparticles for articular cartilage tissue engineering[END_REF] A separate study demonstrated that treatment of PGLA with ammonia plasma improves surface roughness thereby providing enhanced attachment of collagen. [START_REF] Yang | Enhanced cell affinity of poly (D,L-lactide) by combining plasma treatment with collagen anchorage[END_REF] Poly(urethane) has been surface modified with primary amine groups to facilitate the attachment of ECM molecules producing enhanced murine fibroblast cell attachment and signalling. [START_REF] Józwiak | Surface functionalization of polyurethane for the immobilization of bioactive moieties on tissue scaffolds[END_REF] Nelea et al. [START_REF] Nelea | Selective inhibition of type X collagen expression in human mesenchymal stem cell differentiation on polymer substrates surface-modified by glow discharge plasma[END_REF] were able to show that surface-modifying polypropylene and nylon-6 with ammonia plasma can inhibit expression of type X collagen in human mesenchymal stem cells (MSCs). This has the potential to produce MSCs of the appropriate phenotype for use in intervertebral disc tissue engineering. Carbon dioxide plasma treatments have been used to improve rodent fibroblast and nerve cell adhesion on PLLA surface. [START_REF] Khorasani | Comparison of fibroblast and nerve cells response on plasma treated poly (L-lactide) surface[END_REF] While plasma coating of a PLGA surface with titanium oxide has also been shown to improve surface hydrophilicity for human dermal fibroblast cell adhesion. [START_REF] Ryu | Plasma surface modification of poly (-lactic-co-glycolic acid) (65/35) film for tissue engineering[END_REF] Extracellular matrix (ECM) proteins have natural cell adhesion qualities. A number of researchers have investigated chemical attachment of these proteins to the surface of various synthetic polymers.The effects of adhering the ECM protein laminin to the surface of PLGA by chemical and plasma methods have been investigated. [START_REF] Huang | Surface modification and characterization of chitosan or PLGA membrane with laminin by chemical and oxygen plasma treatment for neural regeneration[END_REF] In this study using rodent Schwann cells, laminin coating was shown to significantly improve cell attachment and affinity for directing peripheral nerve regeneration. Another approach has been to attach collagen to polyester surfaces to increase human articular chondrocyte cell attachment. [START_REF] Chen | Surface modification of porous scaffolds with nanothick collagen layer by centrifugation and freeze-drying[END_REF] There is some evidence, however, that treatment of poly(ε-caprolactone) (PCL) with an air plasma is more effective at improving proliferation of Schwann cells than PCL-collagen scaffolds. [

Improved cell affinity

Improvement to surface hydrophilicity and cell adhesion

Increased attachment and affinity of cells to surface [START_REF] Yang | Enhanced cell affinity of poly (D,L-lactide) by combining plasma treatment with collagen anchorage[END_REF] [74]

[75] Polypropylene Ammonia plasma

Mesenchymal stem cells

Inhibition of type X collagen [START_REF] Nelea | Selective inhibition of type X collagen expression in human mesenchymal stem cell differentiation on polymer substrates surface-modified by glow discharge plasma[END_REF] Poly(L-lactide) CO 2 plasma B65 glial cells L929 fibroblast cells Increased cell adhesion [START_REF] Khorasani | Comparison of fibroblast and nerve cells response on plasma treated poly (L-lactide) surface[END_REF] Poly(glycolic acid) Surface hydrolysis

Vasular smooth muscle cells

Increased cell seeding density and better cell spreading [START_REF] Gao | Surface hydrolysis of poly(glycolic acid) meshes increases the seeding density of vascular smooth muscle cells[END_REF] Table 1-List of synthetic polymers which have been surface modified to improve cellular affinity in a range of cell types.

Despite these positive examples, paradoxically it is possible that surface modification of polymers with whole proteins may result in the adhered proteins being being unable to adopt the orientation required for cellular adhesion. As a solution to this, many groups have attempted to employ short cell recognition peptides instead. These short peptide sequences, found naturally in ECM proteins, bind to integrins on the surface of cells. The most commonly used peptides contain the tripeptide Arginyl-Glycyl-Aspartic acid (RGD) cell adhesion motif. Since the discovery in 1984 that the sequence could promote cell adhesion many methods have been developed for attachment of such peptides to a surface. [START_REF] Pierschbacher | Cell attachment activity of fibronectin can be duplicated by small synthetic fragments of the molecule[END_REF] These methods have been extensively reviewed by Hersel et al. [START_REF] Hersel | RGD modified polymers: biomaterials for stimulated cell adhesion and beyond[END_REF] Other peptide sequences such as Tyrosine-Isoleucine-Glycine-Serine-Arginine (YIGSR), a cell adhesion sequence in the ECM protein laminin, have also been shown to have useful endothelial cell mobility properties. [START_REF] Fittkau | The selective modulation of endothelial cell mobility on RGD peptide containing surfaces by YIGSR peptides[END_REF] These peptides appear to act synergistically with the RGD sequence.

Attachment of the YIGSR sequence to collagen scaffolds has previously been utilised in corneal tissue engineering to increase cell survival and proliferation. [START_REF] Duan | Biofunctionalization of collagen for improved biological response: Scaffolds for corneal tissue engineering[END_REF] Other studies have also investigated the response of adipose-derived stem cells to PCL with different attached laminin fragments. [START_REF] Santiago | Peptide-surface modification of poly(caprolactone) with laminin-derived sequences for adipose-derived stem cell applications[END_REF] The sequence Isoleucine-Lysine-Valanine-Alanine-Valanine (IKVAV) was found to significantly improve cell adherence when compared to other sequences. There have also been studies to compare the effect of linear and cyclic RGDcontaining peptides in osteoprogenitor cell culture. [START_REF] Verrier | Function of linear and cyclic RGD-containing peptides in osteoprogenitor cells adhesion process[END_REF] Cyclic peptides are more resistant to proteolysis and can have higher binding affinities for integrin receptors compared to their linear counterparts. This behaviour may confer advantages for cell binding.

Besides the use of plasmas, proteins and peptides, other surface modifications have been shown to modulate the response of cells to a biomaterial surface. Surface hydrolysis of poly(glycolic acid) fibres was found to increase cell-seeding density of vascular smooth muscle cells and result in better spreading of cells across the polymer surface. [START_REF] Gao | Surface hydrolysis of poly(glycolic acid) meshes increases the seeding density of vascular smooth muscle cells[END_REF] It has been found that incorporation of different functional groups on the surface of polymer scaffolds can control interactions with adsorbed proteins and subsequently cellular adhesion. [START_REF] Keselowsky | Surface chemistry modulates fibronectin conformation and directs integrin binding and specificity to control cell adhesion[END_REF] For example surface functional groups have been found to affect the adhesion, migration and differentiation of neural stem cells. [START_REF] Ren | In vitro behavior of neural stem cells in response to different chemical functional groups[END_REF] Other biomaterial properties to consider It is not only the surface chemistry of a biomaterial which controls cellular behaviour, surface topography may also have a significant influence (Figure 2). [START_REF] Dunn | Alignment of fibroblasts on grooved surfaces described by a simple geometric transformation[END_REF] Basement membranes which support cellular growth within the body have a surface topology appropriate to their function. [START_REF] Berthiaume | Effect of extracellular matrix topology on cell structure, function, and physiological responsiveness: hepatocytes cultured in a sandwich configuration[END_REF][START_REF] Abrams | Nanoscale topography of the basement membrane underlying the corneal epithelium of the rhesus macaque[END_REF] The interaction of cells with fibres has been of interest ever since the successful use of carbon fibers in inducing tendon and ligament formation. [START_REF] Jenkins | Induction of tendon and ligament formation by carbon implants[END_REF] More recently, the texture and morphology of polymer fibers produced by electrospinning have been found to affect the shape and proliferation of mesenchymal stem cells. [START_REF] Moroni | Fiber diameter and texture of electrospun PEOT/PBT scaffolds influence human mesenchymal stem cell proliferation and morphology, and the release of incorporated compounds[END_REF] Cellular responses to grooves and ridges have also been investigated. [START_REF] Clark | Topographical control of cell behaviour. I. Simple step cues[END_REF] However, it remains difficult to ascertain whether cells are reacting to topological or chemical cues. Experiments by Britland et al. [START_REF] Britland | Morphogenetic guidance cues can interact synergistically and hierarchically in steering nerve cell growth[END_REF] suggest that both types of cues may act synergistically. Thus biomaterials need to be engineered to mimic these natural membranes as closely as possible. [START_REF] Han | Electrospun bioscaffolds that mimic the topology of extracellular matrix[END_REF] The influence of surface topology on cellular growth has been reviewed

previously. [START_REF] Curtis | Topographical control of cells[END_REF] Studies have also been conducted to assess how material topography can be used to build 3D tissue structures including corneal stroma. [START_REF] Guillemette | Surface topography induces 3D self-orientation of cells and extracellular matrix resulting in improved tissue function[END_REF] This approach was found to increase the functionality of tissue-engineered cellular sheets.

The mechanical properties of substrates also have an important role to play in creating an appropriate tissue engineering scaffold for cellular attachment and proliferation. Mechanical stiffness of materials has been shown to influence the differentiation of neural stem cells. [START_REF] Saha | Substrate modulus directs neural stem cell behavior[END_REF] These properties are also important when considering the method of implanting biomaterials into the body. Materials based on poly(N-isopropylacrylamide) (PNiPAAM) exhibit a phase change at or around 37 o C which is known as the Lower Critical Solution Temperature (LCST) of the material. Below this temperature, PNiPAAM gels are transparent and pliable while, above the LCST, the gels become more rigid and opaque. This results in a material which is easily injectable at room temperature while becoming a more rigid support at body temperature. Such materials have been shown to be capable of supporting cellular growth in vitro and have significant potential as thermally responsive tissue-engineering scaffolds. [START_REF] Stile | Synthesis and Characterization of Injectable Poly( N -isopropylacrylamide)-Based Hydrogels That Support Tissue Formation in Vitro[END_REF] Biomaterials based on peptides which can respond to other environmental stimuli such as pH, temperature and light have also shown promise in the area of regenerative medicine. [START_REF] Mart | Peptide-based stimuli-responsive biomaterials[END_REF] In conclusion, routine clinical transplantation of RPE cells or indeed stem cells for retinal repair remains a distant prospect. Many basic cell biology questions need to be addressed prior to successful retinal transplantation. Not least among these is how cells can be transplanted most efficiently to sustain viability and functionality. Biomaterial scaffolds offer a novel transplantation strategy, with the ability to manipulate the extracellular environment both in vitro and following implantation. Biomaterial characteristics which afford the most appropriate degree of cell attachment, differentiation and maintenance of apposite function for individual organs including the eye are currently being elucidated.

Figure 1 -Figure 2 -

 12 Figure 1-Image depicting attachment of a mature retinal pigment epithelium (ARPE-19) cell line to a laminin coated porous Poly(L-lactide) (PLLA) with Poly(DL-lactide-co-glycolide) (PLGA) (25%:75%) scaffold following 1 week in vitro. Immunocytochemical characterization was utilised to confirm retention of phenotype using an antibody directed against the RPE marker-retinal pigment epithelium-specific protein-65kDa (RPE 65) (A) Cell nuclei labelled in blue with 4',6-diamidino-2-phenylindole DAPI (B). Merged image (C). Scale bar-13μm. [100]

  [START_REF] Prabhakaran | Surface modified electrospun nanofibrous scaffolds for nerve tissue engineering[END_REF]

	Polymer	Surface	Cells used	Cellular response	Reference
		modification
	Poly(L-lactide)(PLLA)/	Laminin	RPCs	Increased cell survival	[58]
	Poly(DL-lactide-co-	coating
	glycolide)(PLGA)
	Poly(methyl	Laminin	RPCs	Increased cell adherence	[61]
	methacrylate)(PMMA)	coating
	Poly(glycerol-sebacate)	Laminin	RPCs	Differentiation towards	[62]
	(PGS)	coating		mature retinal neurons
	Poly(caprolactone)	Laminin	RPCs	Proliferation and	[63]
	(PCL)	coating		expression of mature
				retinal proteins
		Air plasma	Schwann cells	Increased cell proliferation	[77]
				compared to PCL-collagen
				scaffolds
		Laminin	Adipose-	IKVAV sequence	[78]
		fragments	derived stem	significantly increased
			cells	number of cells bound to
				surface
	Poly(hydroxybutyrate-	Oxygen	RPE cell line	More hydrophilic surface	[64]
	co-hydroxyvalerate)	Plasma	D407	and better adhesion
	Poly(urethane)	Air Plasma	ARPE-19	Increase surface wetability	[65]
				resulting in RPE
				monolayer growth
	Poly(dimethylsiloxane)	Air Plasma	ARPE-19	Increase surface wetability	[66]
				leading to enhanced cell
				growth
	Poly(DL-lactide-co-	Ammonia	Mouse 3T3
	glycolide)	plasma/	fibroblasts
		Collagen
		TiO 2	Dermal
			fibroblasts
		Laminin	Schwann cells

Acknowledgements

-We thank the Biomedical Imaging Unit, School of Medicine and the Science and Engineering Electron Microscopy Centre, School of Chemistry, University of Southampton for assistance with SEM. Funding-Financial support was provided by National Institute Health Research, National Eye Research Centre, British Retinitis Pigmentosa Society and Foresight RP.

Licence for Publication-The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in BJO and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence (http://group.bmj.com/products/journals/instructions-forauthors/licence-forms).

Competing Interests-None


MD Hanna Åkerblom

MD Eva Larsson

PhD Urban Eriksson

MD Gerd Holmström

MD Gerd Holmstrom
email: gerd.holmstrom@neuro.uu.se

Central macular thickness is correlated with gestational age at birth in prematurely-born children

Background/aim: Previous studies have revealed various subnormal visual functions in prematurely-born children. The present study aimed to determine the retinal macular thickness in prematurely-born children and compare with children born at term. Methods: The eyes of 65 prematurely-born children aged 5-16 years, were examined with Stratus Optical coherence tomography (OCT) 3, and the results were compared with those of 55 children born at term. The retinal macular thickness in the nine EDTRS macular areas (A1-A9), the foveal minimum and the total macular volume were determined. Results: The central macular thickness (A1 and foveal minimum) was significantly thicker in the prematurely-born children than in those born at term. There was no correlation between macular thickness and visual acuity or refraction. Children with previous retinopathy of prematurity (ROP) had significantly thicker central maculae than those without it. Prematurely born children without previous ROP had significantly thicker central macula than the control group. Multiple regression analyses showed that gestational age at birth was the only risk factor for a thick central macula. Conclusion: Prematurely-born children had thicker central maculae than those born at term. Regardless of ROP, the degree of prematurity was the most important risk factor for abnormal foveal development. Major neurological, pulmonary and ophthalmological dysfunctions are common in prematurely-born children. However, minor sequelae of preterm birth and their effects on daily life of these children as they grow up are less well-known. Visual impairment may occur

secondary to retinopathy of prematurity (ROP) and / or damage of the visual pathways.

Although the prevalence of visual impairment is relatively low in high-income countries, population-based follow-up studies of prematurely-born children and adolescents have recently shown that a significant proportion of them have subnormal visual functions affecting visual acuity, contrast sensitivity, visual fields and stereopsis. [START_REF] Larsson | A population-based study of the refractive outcome in 10-year-old preterm and full-term children[END_REF][START_REF] Larsson | Peripheral and central visual fields in 11-year-old children who had been born prematurely and at term[END_REF][START_REF] Larsson | A Population-Based Study on the Visual Outcome in 10-Year-Old Preterm and Full-Term Children[END_REF][START_REF] Holmström | Development of spherical equivalent refraction in prematurely-born children during the first 10 years of life[END_REF][START_REF] Larsson | Contrast sensitivity in 10 year old preterm and full term children: a population based study[END_REF][START_REF] Holmström | Long-term follow-up visual functions in prematurely-born children -a prospective population-based study up to 10 years of age[END_REF][START_REF] O´connor | Visual function in low birth weight children[END_REF][START_REF] Stephenson | Children born weighing less than 1701 g: visual and cognitive outcomes at 11-14 years[END_REF][START_REF] O´connor | Long-term ophthalmic sequelae of prematurity[END_REF][START_REF] Hellgren | Visual and cerebral sequelae of very low birth weight in adolescents[END_REF][START_REF] Lindqvist | Visual acuity, contrast sensitivity, peripheral vision and refraction in low birth weight teenagers[END_REF] In a 10-year follow-up of children with a birth weight of 1500 grams or less in the Stockholm area of Sweden, the frequency of subnormal visual acuity was significantly higher than in a control group of children born at term. [6] It could not be determined whether the aetiology was cerebral or retinal.

Macular development is a very sophisticated process, including both a centrifugal displacement of inner retinal cells and a centripetal displacement of photoreceptors. [START_REF] Hendrickson | The Morphological Development of the Human Fovea[END_REF][START_REF] Provis | Ontogeny of the primate fovea: a central issue in retinal development[END_REF][START_REF] Provis | The foveal avascular region of developing human retina[END_REF] The development is not completed until about three to four years of life and the fovea is the last structure to become fully developed. [START_REF] Hendrickson | The Morphological Development of the Human Fovea[END_REF] It is therefore not surprising that children with a history of various degrees of ROP have foveal abnormalities. [START_REF] Hammer | Foveal fine structures in retinopathy of prematurity: an adaptive optics fourier domain optical coherence tomography study[END_REF][START_REF] Soong | Macular structures and vision of patients with macular heterotopia secondary to retinopathy of prematurity[END_REF][START_REF] Ecsedy | A comparison of macular structure imaged by optical coherence tomography in preterm children and full-term children[END_REF][START_REF] Do Lago | Stratus optical coherence tomography findings in patients with retinopathy of prematurity[END_REF][START_REF] Recchia | Foveal Dysplasia Evident by Optical Coherence Tomography in Patients with a History of Retinopathy of Prematurity[END_REF][START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] We hypothesized that preterm birth affects macular development and causes changes in the macular and foveal structures, even in infants without ROP. Recently, we reported on macular thickness with the help of optical coherence tomography (OCT) in full-term children. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] The aim of the present study was to compare the macular thickness of prematurely-born children with that of fullterm children and to determine whether the findings were related to prematurity and ROP.

Materials and Methods

The present study included 68 prematurely-born children with gestational ages (GA) of 32 weeks or less, who had been delivered at Uppsala University Hospital in Sweden between 1993 and 2000. All of them had been included in the routine ROP-screening, starting from the fifth postnatal week and performed every other week or more often, if necessary. The screening continued until the retina was fully vascularized or in the case of ROP, until it had resolved completely. The criterion for treatment at that time was ROP stage 3 in at least four contiguous clock hours, even in the absence of plus disease. Mild ROP comprised stages 1-2 and severe ROP stages 3-5, which includes the children treated with laser or cryopexy.

The hospital records were used to locate the prematurely-born children and their caregivers and they were both asked by letter to participate in the study.

The control group included 55 children born in Uppsala County, Sweden, who were randomly selected from the birth register of the Swedish National Board of Health and Social Welfare.

They were born at term (GA ≥ 37 weeks) with normal birthweights (BW ≥ 2500g). [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] One randomized eye from each child in the control group was used for comparison with the study group. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] Best-corrected visual acuity (VA) was assessed monocularly with linear LogMar charts.

Cycloplegic retinoscopy was done after dilating the pupil with a mixture of phenylephrine 1.5 % and cyclopentholate 0.85%. The spherical equivalent was calculated. An eye examination, including ophthalmoscopy, was performed. A medical history was taken, including neurological complications defined as intraventricular haemorrhage, cerebal palsy, periventricular leukomalacia and epilepsy.

The macular thickness was measured with Stratus OCT 3, version 4.0.1. The fast macular map protocol was used, consisting of six scans arranged in a radial pattern over the macula with the fovea as a central point. During the scanning procedure, the examiner could observe the fundus and continuously control fixation. The average retinal thickness is automatically calculated by an algorithm and presented as numeric values for nine ETDRS areas (Figure 1). The total area measured has a diameter of 6 mm and is divided into three concentric regions.

The central region is 1 mm in diameter (A1) and the inner and outer circles are divided into four quadrants (A2-A9). The total macular volume and the foveal minimum were also calculated.

Most OCT assessments were performed by one of the authors (UE), and a few by another experienced examiner. Measurements with a signalstrength of at least five were required.

Informed consent was obtained from all caregivers before enrollment. The study was approved by the Ethics Committee of Uppsala University and was conformed to the tenets of the Declaration of Helsinki.

Statistical methods

The Kolmogorov-Smirnov test was used to test for a normal distribution. Prematurely-born and full-term children were compared using the independent t-test and one-way ANOVA for continuous data. As regards area A1 and foveal minimum, we used analyses of covariance, ANCOVA, with the covariates age, gender, VA and spherical equivalent. Pearson's correlation was used for bivariate correlations. In the prematurely-born children, forward stepwise multiple linear regression analyses were done to evaluate how much the variation in the area A1 and foveal minimum could be explained by GA, BW, ROP (yes/no), neurological complications, age at examination, gender, VA and spherical equivalent. Right (RE) and left eyes (LE) were analyzed separately and p<0.05 was considered statistically significant.

Results

Sixty-eight prematurely-born children were included in the study. All except one cooperated well in the assessments and two children were excluded due to technical problems. Therefore 65 REs and LEs were included. Of these, 22 REs and LEs had mild ROP and 7 REs and LEs severe ROP, including treated ROP (5 RE,6 LE) . The values of macular thickness were compared with a control material of 55 eyes in healthy full-term children with normal birth weights. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] Demographic data of both prematurely-born and full-term children are shown in Table 1. The range of ages was the same in the two groups although the mean age was slightly lower in the prematurely-born children (p=0.02). There was a statistically significant difference between the groups as regards VA (p < 0.001 RE and LE), but not the spherical equivalent. The proportion of boys and girls were similar in the two groups. ROP had the thickest central macula, but the difference was not statistically significant compared to those with mild ROP. (Table 2, Figure 4). *55 randomized eyes. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] The thicknesses of A1-A9, foveal minimum and total macular volume showed no correlation with age at examination, VA or spherical equivalent. Boys had a thicker macula than girls, but the difference was not significant. The findings were similar in the full-term control group. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] In the prematurely-born children, low GA and BW (Figure 5) correlated significantly with a thicker central macula (A1) in the univariate analyses, (GA r = -0.45 p<0.001 RE, r = -0.43 p<0.001 LE; BW r = -0.35 p = 0.004 RE, r = -0.33 p = 0.008 LE). This correlation was also true of the thickness in foveal minimum (GA r = -0.53 p<0.001 RE, r = -0.51 p<0.001 LE; BW r = -0.44 p<0.001 RE, r = -0.42 p = 0.001 LE). However, in a multiple regression analysis, including GA, BW, ROP (yes/no), neurological complications, age at examination, gender, VA and spherical equivalent, low GA was the only risk factor for a thicker central macula later in childhood (A1 p<0.001 RE and p<0.001 LE; foveal minimum p<0.001 RE and p<0.001 LE).

Discussion

This study presents the findings using OCT of the maculae in 65 prematurely-born children, with and without ROP. The data are compared with those of full-term children, with normal gestational ages and birth weights. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] The prematurely-born children had thicker central foveas than those born at term. Among the prematurely-born children, the ones with previous ROP had thicker foveas than those without ROP. No correlation between macular thickness and visual acuity or refraction was detected. Gestational age at birth was the only significant risk factor for a thicker central macula in childhood/adolescence in a multiple regression analysis, even when ROP was taken into account.

Three features of the present study should be noted. First, it has a population-based control group of 55 children aged 5 -16 years who were born at term in Uppsala County, -i.e. the same geographical area as that of the preterm group. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] Secondly, all prematurely-born children were prospectively screened and, if necessary, treated for ROP in the neonatal period. Thirdly, this study includes more children than previous reports, [15, 17-19] and also a substantial number of children without any previous ROP (Table 3).

Table 3. OCT studies on macular thickness in prematurely-born and fullterm children and adults.

Author OCT type Preterms

(n)

Preterms

With ROP

(n) Preterms Without ROP (n) Fullterms (n)

Age

Hammer et al [START_REF] Provis | Ontogeny of the primate fovea: a central issue in retinal development[END_REF] FDOCT* 5 5 0 5 We found thicker foveas in the prematurely-born children than in those born at term, but no differences in the more "peripheral" parts of the macula. Our findings accord with those of Hammer et al [15] and Ecsedy et al [START_REF] Ecsedy | A comparison of macular structure imaged by optical coherence tomography in preterm children and full-term children[END_REF], who reported data in five and thirty prematurely-born children / adults respectively. The reason for the foveal abnormality in preterms is probably related to the foveal development, which seems to start around the 22nd week of gestation as a thickened region of ganglion cells. [START_REF] Hendrickson | The Morphological Development of the Human Fovea[END_REF][START_REF] Provis | Ontogeny of the primate fovea: a central issue in retinal development[END_REF] In the 24th to 26th week, the ganglion cells and the inner nuclear layer cells move laterally, giving rise to the beginning of the foveal depression. This peripheral migration of cells does not seem to be completed until 15 -45 months after birth. In contrast to peripheral migration, photoreceptors migrate centrally. [START_REF] Hendrickson | The Morphological Development of the Human Fovea[END_REF][START_REF] Provis | Ontogeny of the primate fovea: a central issue in retinal development[END_REF] The number of foveal cone nuclei increases from a single layer at about 22 weeks to four to eight layers in the completely mature macula, but the cones gradually thin and elongate for at least 45 months after birth. It is therefore not surprising that this complex foveal development is disturbed in the prematurely-born infant and results in thicker foveas than in the those born at term.

In accord with Recchia & Recchia [START_REF] Recchia | Foveal Dysplasia Evident by Optical Coherence Tomography in Patients with a History of Retinopathy of Prematurity[END_REF] and Ecsedy et al [START_REF] Ecsedy | A comparison of macular structure imaged by optical coherence tomography in preterm children and full-term children[END_REF], the inner retinal structures seemed to be preserved in the prematurely-born children. However, we could not evaluate the various retinal layers with our OCT technique. Hammer et al [START_REF] Hammer | Foveal fine structures in retinopathy of prematurity: an adaptive optics fourier domain optical coherence tomography study[END_REF], using sophisticated techniques, showed that all inner retinal layers were present and contiguous across the central fovea in four of their five subjects with previous ROP.

In our study, ROP seemed to affect the development of the fovea. Children with previous ROP had significantly thicker foveas than those without ROP. Since only few children with previous ROP were included, no conclusion could be drawn about the effects of the stage of ROP on the foveal thickness. However, although not statistically significant, children with severe ROP, including those treated for ROP, had the thickest maculae. Our findings accord with the data reported in the OCT study by Ecsedy et al [START_REF] Ecsedy | A comparison of macular structure imaged by optical coherence tomography in preterm children and full-term children[END_REF]. They are also in agreement with the delay in macular development in infants with ROP observed by Isenberg [21], who ascribed it to a direct macular insult by ROP.

We found no correlation between foveal thickness and refraction or visual acuity (VA) in the present study or in the full-term controls. [START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] This concurs with the study of 12 children by Recchia & Recchia [19], who found no relation between a decrease in VA and absence of a foveal depression, preservation of inner retinal layers or an increase in foveal thickness. They concluded that the increased foveal thickness may only reflect previous prematurity, and has no functional importance. This is in line with the view of Marmor et al [START_REF] Marmor | Visual Insignificance of the foveal pit[END_REF], who states that abnormal foveal morphology is not always associated with a reduction in VA.

The present study, which consists of 36 prematurely-born children without ROP, enabled us to study the effect of prematurity per se on the foveal thickness. We found that children without previous ROP indeed had thicker foveas than those born at term . The role of preterm birth was also evaluated by Ecsedy et al [START_REF] Ecsedy | A comparison of macular structure imaged by optical coherence tomography in preterm children and full-term children[END_REF], but they reached no definite conclusion because of the small size of their cohort.

Finally, our study showed that the degree of prematurity correlated with the thickness of the fovea. Moreover, in a multiple regression analysis which included GA and BW as continuous variables as well as ROP, neurological complications, refraction, VA and age at examination, gestational age at birth was the only risk factor for a thicker macula in prematurely-born children. Therefore, the degree of prematurity of the child seems to be of greatest importance for foveal development and more important than ROP. To our knowledge this has not been reported elsewhere.

In conclusion, we found, using OCT, that prematurely-born children have thicker central foveas than those born at term. We believe that this may be due to an arrest in the lateral migration of cells during normal development of the fovea. Infants with ROP seemed to be more prone to a delay in maturation than those without ROP. Regardless of ROP, however, the degree of prematurity was the most important risk factor for abnormal foveal

  ) 5.0 -16.0 -0.2 -0.0* (-0.5) -(+3.0)* Range RangeRange * 55 randomized eyes[START_REF] Eriksson | A population-based study of macular thickness in full-term children assessed with Stratus OCT: normative data and repeatability[END_REF] F=female, M=maleThe mean values of macular thickness (A1-9, respectively) in prematurely-born and full-term children are given in Figure2. The mean values of "foveal minimum" in the prematurely-born children were 196 (SD 28) μm in RE and 197 (SD 30) μm in LE and the mean values of "total macular volume" were 7.10 (SD 0.42) mm 3 in RE and 7.12 (SD 0.40) mm 3 in LE. The corresponding mean values in the full-term eyes were 166 (SD 15) μm and 7.11 (SD 0.35) mm 3 , respectively.[20] All macular areas, foveal minimum and total macular volume were normally distributed in the prematurely-born and full-term children.[20] The central macula was significantly thicker in prematurely-born children than in full-term ones, as regards area A1 and foveal minimum (p<0.001 RE and LE, respectively), although the age at examination, gender, VA and spherical equivalent were taken into account. For illustration, see OCT image of one of the thickest maculae in the preterm group in comparison with an image of a child born at term (Figure3). There was no difference between the groups as regards the inner (A2-A5) or outer (A6-A9) circles or the total macular volume.Statistically significant differences in the central macula (A1) and foveal minimum were found between children with and without previous ROP, the former having a thicker central macula (A1 p<0.001 RE and LE, foveal minimum p<0.001 RE and LE). Moreover, the differences were also statistically significant between the children without previous ROP and the full-term children as regards A1 and foveal minimum (A1 p=0.01 RE and p=0.01 LE, foveal minimum p=0.001 RE and p<0.001 LE). The children with severe, including treated,

FigurelegendsFigure 1 .

 1 Figurelegends

Figure 2 ,

 2 Figure 2, Retinal thickness (µm) in the 9 ETDRS areas, Mean (SD). A. Preterm group 65

Figure 3A .

 3A Figure 3A. Optical coherent tomography (OCT) image from one of the thickest maculae in

Figure 4 .

 4 Figure 4. Central macular thickness (A1) in children born at term (control eyes) and

Figure 5 .

 5 Figure 5. Central macular thickness (A1) in relation to gestational age at birth in the

Table 1 .

 1 Descriptive data of the 65 prematurely born children and the 55 children born at term.

	Gender	Age	Gestational	Birth	Visual acuity	Spherical equivalent
			age	weight
					(logMar)
	F/M	(years)	(weeks)	(grams)	Right eyes Left eyes	Right eyes	Left eyes
	Premature 31/34	8.6 (1.8)	28.6 (2.9)	1299 (497)
		Mean (SD)	Mean (SD)	Mean (SD)

Table 2 .

 2 Central macular thickness (A1 and foveal minimum) in the 65 prematurely born children in relation to stage of ROP and in the 55 children born at term.

		No ROP	Mild ROP	Severe ROP incl. treated	Controls
		n=36		n=22		n=7		n=55*
		Right eyes	Left eyes	Right eyes	Left eyes	Right eyes	Left eyes
	A1 (μm)
	Mean (SD)	218 (20)	186 (24)	228 (17)	231 (19)	250 (32)	252 (34)	204 (19)
	Range	181-268	179-271	194-265	195-269	213-310	212-320	162-234
	Foveal min. (μm)
	Mean (SD)	187 (22)	186 (24)	199 (25)	205 (26)	230 (36)	233 (39)	166 (15)
	Range	155-249	149-256	149-252	149-258	185-297	188-306	130-194

Acknowledgements

The authors would like to thank Elisabeth Berg, for statistical advice, and Eva Nuija and Bengt Schepke for helpful participation in the study.

The study was supported by the Crown Princess Margareta Foundation for Visually Impaired, The Mayflower Charity Foundation for Children and the Swedish Society for Medicine.

development.

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