Mar Ía

Isabel Cortez

Samuel Petite

SET OF INVARIANT MEASURES OF GENERALIZED TOEPLITZ SUBSHIFTS

Keywords: 2000 Mathematics Subject Classification. Primary: 37B10; Secondary: 37B05, discrete group actions, invariant measures, orbit equivalence

We show that for every metrizable Choquet simplex K and for every group G, which is amenable, finitely generated and residually finite, there exists a Toeplitz G-subshift whose set of shift-invariant probability measures is affine homeomorphic to K. Furthermore, we get that for every integer d ≥ 1 and every minimal Cantor system (X, T ) whose dimension group is divisible, there exists a minimal Toeplitz Z d -subshift which is topologically orbit equivalent to (X, T ).

Introduction

The Toeplitz subshifts are a rich class of symbolic systems introduced by Jacobs and Keane in [START_REF] Jacobs | 0 -1-sequences of Toeplitz type[END_REF], in the context of Z-actions. Since then, they have been extensively studied and used to provide series of examples with interesting dynamical properties (see for example [START_REF] Downarowicz | The Choquet simplex of invariant measures for minimal flows[END_REF][START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF][START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF][START_REF] Williams | Toeplitz minimal flows which are not uniquely ergodic[END_REF]). Generalizations of Toeplitz subshifts and some of their properties to more general group actions can be found in [START_REF] Cortez | Z d -Toeplitz arrays[END_REF][START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF][START_REF] Downarowicz | The royal couple conceals their mutual relationship: a noncoalescent Toeplitz flow[END_REF][START_REF] Krieger | Sous-décalages de Toeplitz sur les groupes moyennables résiduellement finis[END_REF]. For instance, in [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF] Toeplitz subshifts are characterized as the minimal symbolic almost 1-1 extensions of odometers (see [START_REF] Downarowicz | Almost 1 -1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows[END_REF] for this result in the context of Z-actions). In this paper, we give an explicit construction that generalizes the result of Downarowicz in [START_REF] Downarowicz | The Choquet simplex of invariant measures for minimal flows[END_REF], to Toeplitz subshifts given by actions of groups which are amenable, residually finite and finitely generated. The following is our main result.

Theorem A. Let G be an amenable, residually finite and finitely generated group. For every metrizable Choquet simplex K and any G-odometer O, there exists a Toeplitz Gsubshift whose set of invariant probability measures is affine homeomorphic to K and such that it is an almost 1-1 extension of O.

Typical examples of the groups G involved in this theorem are the finitely generated subgroups of upper triangular matrices in GL(n, C). The proof of Theorem A deals with combinatorics on Følner sequences and is independent on the known results for G = Z (see [START_REF] Downarowicz | The Choquet simplex of invariant measures for minimal flows[END_REF] or [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF]). Furethermore, we obtain some consequences in the orbit equivalence problem. Two minimal Cantor systems are (topologically) orbit equivalent, if there exists an orbitpreserving homeomorphism between their phase spaces. Giordano, Matui, Putnam and Skau show in [START_REF] Giordano | Orbit equivalence for Cantor minimal Z dsystems[END_REF] that every minimal Z d -action on the Cantor set is orbit equivalent to a minimal Z-action. Such a result can not be extended to any countable group. For instance, by using the notion of cost, Gaboriau [START_REF] Gaboriau | Coût des relations d'équivalence et des groupes[END_REF] proves that if two free actions of free groups Fn and Fp are (even measurably) orbit equivalent then their rank are the same i.e. n = p. It is still unknown for which groups the result in [START_REF] Giordano | Orbit equivalence for Cantor minimal Z dsystems[END_REF] is true and which are the Z-orbit equivalence classes that the Z d -actions (or more general group actions) realize. We give a partial answer for the second question. As a consequence of the proof of Theorem A we obtain the following result.

Theorem B. Let (X, T ) be a minimal Cantor system having an associated Bratteli diagram B which satisfies the equal path number property. For each n ≥ 1, let Vn and rn be the set of vertices of the level n of B and the number of edges arriving to a vertex v in Vn respectively. If there exists a constant C > 0 such that Crn ≥ |Vn| for each n ≥ 1, then for every d ≥ 1 there exists a Toeplitz Z d -subshift which is orbit equivalent to (X, T ). In particular, every minimal Cantor system (X, T ) whose dimension group is divisible, is orbit equivalent to a Toeplitz Z d -subshift.

It remains open the problem if the previous result can be generalized to more general group actions and to any Toeplitz subshift given by a Z-action.

This paper is organized as follows. Section 2 is devoted to introduce the basic definitions. In Section 3 we give a characterization of any Choquet simplex as an inverse limit defined by suitable sequences of matrices that we call "manageable". For an amenable discrete group G and a decreasing sequence of finite index subgroups of G with trivial intersection, we construct in Section 4 an associated sequence (Fn) n≥0 of fundamental domains, so that it is Følner and each Fn+1 is tilable by translated copies of Fn. Next we use the representation established in Section 3 and the previous Følner sequence to construct a Toeplitz G-subshift with a prescribed set of invariant probability measures. This construction improves and generalizes that one given in [START_REF] Cortez | Realization of a Choquet simplex as the set of invariant probability measures of a tiling system[END_REF] for Z d -actions, and moreover, allows to characterize the associated dimension group. We use this result to prove Theorem B in the last section.

Basic definitions and background

In this article, by a topological dynamical system we mean a triple (X, T, G), where T is a continuous left action of a finitely generated group G on the compact metric space (X, d). For every g ∈ G, we denote T g the homeomorphism that induces the action of g on X. The unit element of G will be called e. The system (X, T, G) or the action T is minimal if for every x ∈ X the orbit oT (x) = {T g (x) : g ∈ G} is dense in X. We say that (X, T, G) is a minimal Cantor system or a minimal Cantor G-system if (X, T, G) is a minimal topological dynamical system with X a Cantor set. An invariant probability measure of the topological dynamical system (X, T, G) is a probability Borel measure µ such that µ(T g (A)) = µ(A), for every Borel set A. We denote by M(X, T, G) the space of invariant probability measures of (X, T, G).

2.1. Subshifts. For every g ∈ G, denote Lg : G → G the left multiplication by g ∈ G.

That is, Lg(h) = gh for every h ∈ G. Let Σ be a finite alphabet. Σ G denotes the set of all the functions x : G → Σ. The (left) shift action σ of G on Σ G is given by σ g (x) = x•L g -1 , for every g ∈ G. Thus σ g (x)(h) = x(g -1 h). We consider Σ endowed with the discrete topology and Σ G with the product topology. Thus every σ g is a homeomorphism of the Cantor set Σ G . The topological dynamical system (Σ G , σ, G) is called the full G-shift on Σ. For every finite subset D of G and x ∈ Σ G , we denote x|D ∈ Σ D the restriction of x to D. For F ∈ Σ D (F is a function from D to Σ) we denote by [F ] the set of all x ∈ Σ D such that x|D = F . The set [F ] is called the cylinder defined by F , and it is a clopen set (both open and closed). The collection of all the sets [F ] is a base of the topology of Σ G . A subshift or G-subshift of Σ G is a closed subset X of Σ G which is invariant by the shift action. The topological dynamical system (X, σ|X , G) is also called subshift or G-subshift. See [START_REF] Ceccherini-Silberstein | Coornaert Cellular automata and groups[END_REF] for details.

2.1.1. Toeplitz G-subshifts. An element x ∈ Σ G is a Toeplitz sequence, if for every g ∈ G there exists a finite index subgroup Γ of G such that σ γ (x)(g) = x(γ -1 g) = x(g), for every γ ∈ Γ. A subshift X ⊆ Σ G is a Toeplitz subshift or Toeplitz G-subshift if there exists a Toeplitz sequence x ∈ Σ G such that X = oσ(x). Such subshifts are minimal because the Toeplitz sequences are regularly recurrent. See [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF] and [START_REF] Krieger | Sous-décalages de Toeplitz sur les groupes moyennables résiduellement finis[END_REF] for details about Toeplitz sequence and subshifts for G-actions.

2.2. Inverse and direct limit. Given a sequence of continuous maps fn : Xn+1 → Xn, n ≥ 0 on topological spaces Xn, we denote the associated inverse limit by

lim ←n (Xn, fn) = X0 o o f 0 X1 o o f 1 X2 o o f 2 • • • := {(xn)n; xn ∈ Xn, xn = fn(xn+1) ∀n ≥ 0}.
Let us recall that this space is compact when all the spaces Xn are compact and the inverse limit spaces associated to any increasing subsequences (ni)i of indices are homeomorphic. In a similar way, we denote for a sequence of maps gn : Xn → Xn+1, n ≥ 0 the associated direct limit by lim →n (Xn, gn) = X0

g 0 / / X1 g 1 / / X2 g 2 / / • • • := {(x, n), x ∈ Xn, n ≥ 0}/ ∼,
where two elements are equivalent (x, n) ∼ (y, m) if and only if there exists k ≥ m, n such that

g k • . . . • gn(x) = g k • . . . • gm(x)
. We denote by [x, n] the equivalence class of (x, n). When the maps gn are homomorphisms on groups Xn, then the direct limit inherits a group structure.

Odometers.

A group G is said to be residually finite if there exists a nested sequence (Γn) n≥0 of finite index normal subgroups such that n≥0 Γn is trivial. For every n ≥ 0, there exists then a canonical projection πn : G/Γn+1 → G/Γn. The G-odometer or adding machine O associated to the sequence (Γn)n is the inverse limit

O := lim ←n (G/Γn, πn) = G/Γ0 o o π 0 G/Γ1 o o π 1 G/Γ2 o o π 2 • • • .
We refer to [START_REF] Cortez | G-odometers and their almost one-to-one extensions[END_REF] for the basic properties of such a space. Let us recall that it inherits a group structure through the quotient groups G/Γn and it contains G as a subgroup thanks the injection

G ∋ g → ([g]n) ∈ O,
(Z kn , An) = Z k 0 A 0 / / Z k 1 A 1 / / Z k 2 A 2 / / • • • , and J + = {[v, n] : a ∈ (Z + ) kn , n ∈ Z + }.
) + = {[a] : a ∈ H + }.
For more details about ordered groups and dimension groups we refer to [START_REF] Effros | Dimensions and C * -algebras[END_REF] and [START_REF] Goodearl | Partially ordered abelian groups with interpolation[END_REF].

Lemma 1. Let (H, H + ) be a simple dimension group equals to the direct limit

lim →n (Z kn , Mn) = Z k 0 M 0 / / Z k 1 M 1 / / Z k 2 M 2 / / • • • .
Then for every z = (zn) n≥0 in the inverse limit

lim ←n ((R + ) kn , M T n ) = (R + ) k 0 o o M T 0 (R + ) k 1 o o M T 1 (R + ) k 2 o o M T 2 • • • , the function φz : H → R given by φ([n, v]) =< v, zn >, for every [n, v] ∈ H,
is well defined and is a homomorphism of groups such that φz(H + ) ⊆ R + . Conversely, for every group homomorphism φ : H → R such that φ(H + ) ⊆ R + , there exists a unique z ∈ lim←n((R + ) kn , M T n ) such that φ = φz.

Proof. Let z ∈ lim ← -n ((R + ) kn , M T n ). If (m, w) ∈ [n, v] then there exists k ≥ m, n such that M k • • • Mmw = M k • • • Mnv. Then v, zn = v T M T n • • • M T k z k+1 = w T M T m • • • M T k z k+1 = w T zm = w, zm .
This implies that φz is well defined. If [n, v] is in H + , we can assume that v ≥ 0, then φz([n, v]) = v, zn ≥ 0. It is straightforward to show that φz is a group homomorphism so that φz(H + ) ⊆ R + . Convesly, let φ : H → R be a group homomorphism such that φ(H + ) ⊆ R + . For every n ≥ 0, the function φn :

Z kn → R such that φn(v) = φ([n, v]
), is a group homomorphism verifying φn((Z + ) kn ) ⊆ R + . Then zn = (φn(e1), • • • , φn(e kn )) ≥ 0, where e1, • • • , e kn are the unitary vectors in Z kn . We have φ([n, v]) = φn(v) = v, zn , for every v ∈ Z kn and n ≥ 0. Thus if we show that z = (zn) n≥0 is in lim ← -n ((R + ) kn , M T n ), we get φ = φz for z in the inverse limit. But this is direct because φn(v) = φn+1(Mnv), for every v ∈ Z kn . Applying this to the unitary vectors, we get zn = M T n zn+1, for every n ≥ 0. Different elements in lim ← -n ((R + ) kn , M T n ) define different homomorphisms, so z is unique.

2.5. Associated ordered group and orbit equivalence. Let (X, T, G) be a topological dynamical system such that X is a Cantor set and T is minimal. The ordered group associated to (X, T, G) is the unital ordered group

G(X, T, G) = (Dm(X, T, G), Dm(X, T, G) + , [1]),
where

Dm(X, T, G) = C(X, Z)/{f ∈ C(X, Z) : f dµ = 0, ∀µ ∈ M(X, T, G)}, Dm(X, T, G) + = {[f ] : f ≥ 0},
and [START_REF] Ceccherini-Silberstein | Coornaert Cellular automata and groups[END_REF] ∈ Dm(X, T, G) is the class of the constant function 1.

Two topological dynamical systems (X1, T1, G1) and (X2, T2, G2) are (topologically) orbit equivalent if there exists a homeomorphism F : X1 → X2 such that F (oT 1 (x)) = oT 2 (F (x)) for every x ∈ X1. Theorem 2.5 in [START_REF] Giordano | Orbit equivalence for Cantor minimal Z dsystems[END_REF] implies that for every integer d ≥ 1, the isomorphic class of G(X, T, Z d ) is a total invariant of the topological orbit equivalence class of (X, T, Z d ) for minimal Z d action on a Cantor set.

Characterization of Choquet simplices

A compact, convex, and metrizable subset K of a locally convex real vector space is said to be a (metrizable) Choquet simplex, if for each v ∈ K there is a unique probability measure µ supported on the set of extreme points of K such that xdµ(x) = v.

In this section we show that any metrizable Choquet simplex is affine homeomorphic to an inverse limit defined by sequences of matrices that we call manageable.

We say that a sequence of non-negative integer matrices (Mn) n≥0 is manageable with respect to the increasing sequence of positive integers (pn) n≥0 , if for every n ≥ 0 the integer pn divides pn+1, and if the matrix Mn verifies the following properties:

(1) Mn has kn ≥ 3 rows and kn+1 ≥ 3 columns.

(2)

kn i=1 Mn(i, k) = p n+1 pn , for every 1 ≤ k ≤ kn+1, (3) kn+1 ≤ min{Mn(i, k) : 1 ≤ i ≤ kn, 1 ≤ k ≤ kn+1}.
Let p be a positive integer. For every n ≥ 1 we denote by △(n, p) the closed convex hull generated by the vectors 1 p e

(n) 1 , • • • , 1 p e (n)
n , where e

(n) 1 , • • • , e (n) n
is the canonical base in R n . Thus △(n, 1) is the unitary simplex in R n . Observe that if (Mn) n≥0 is a manageable sequence of matrices with respect to (pn) n≥0 , then for each n ≥ 0, the map Mn : △(kn+1, pn+1) → △(kn, pn) is well defined, where kn is the number of rows of Mn.

3.1.

Finite dimensional Choquet simplices. For technical reasons, we have to separate the finite and the infinite dimensional cases. Lemma 2. Let K be a finite dimensional metrizable Choquet simplex with exactly d ≥ 1 extreme points. Let (pn) n≥0 be an increasing sequence of positive integers such that for every n ≥ 0 the integer pn divides pn+1, and let k ≥ max{3, d}. Then there exist an increasing subsequence (ni) i≥0 of indices and a sequence (Mi) i≥0 of square k-dimensional matrices which is manageable with respect to (pn i ) i≥0 and such that K is affine homeomorphic to lim ← -n (△(k, pn i ), Mi).

Proof. Let k ≥ max{3, d}, we will define the subsequence (ni) i≥0 by induction on i through a condition explained later. For every i ≥ 0, we define Mi the k-dimensional matrix by

Mi(l, j) = pn i+1 pn i -k(k -1) if 1 ≤ l = j ≤ d k if l = j, 1 ≤ l ≤ k and 1 ≤ j ≤ d and Mi(•, j) = Mi(•, d) for d < j ≤ k.
By the very definition, Mi is a positive matrix having k ≥ 3 rows and columns ;

k l=1 Mi(l, j) = pn i+1 pn i
for every 1 ≤ j ≤ k and the range of Mi is at most d. Thus the convex set lim ← -n (△(k, pn i ), Mi) has at most d extreme points. If it has exactly d extreme points, it is affine homeomorphic to K. Thus by the Hadamard's Lemma, it is enough to ensure that the submatrices (M0 . . . Mi(l, j)) 1≤l,j≤d are diagonally strictly dominant matrices. Let n0 = 0 and n1 be an integer so that

pn 1 p0 -k(k -1) ≥ k(d -1) + 1 ≥ k.
This ensures that the matrix (M0(l, j)) 1≤l,j≤d is diagonally strictly dominant. Now let us assume that the sequence n0, . . . , ni-1 is defined so that (M0 . . . Mi-1(l, j)) 1≤l,j≤d is a diagonally strictly dominant matrix. Then standard calculus show that for any l, j ∈ {1, . . . , d},

M0 . . . Mi(l, j) = ( pn i+1 pn i -k(k -1))M0 . . . Mi-1(l, j) + b l,j ,
where the coefficient b l,j does not depend on the diagonal coefficient

pn i+1 pn i -k(k -1
). The diagonal is dominant means that for any l ∈ {1, . . . , d}

( pn i+1 pn i -k(k -1))M0 . . . Mi-1(l, l) + b l,l > d j=1,j =l ( pn i+1 pn i -k(k -1))M0 . . . Mi-1(l, j) + b l,j .
Now, let ni+1 be an integer so that

pn i+1 pn i -k(k -1) > max   sup 1≤l≤d d j=1,j =l b l,j -b l,l , k   ; since M0 . . . Mi-1(l, l)-d j=1,j =l M0 . . . Mi-1(l, j) ≥ 1
, the former inequality is true, and the matrix (M0 . . . Mi(l, j)) 1≤l,j≤d is diagonally strictly dominant. It is straightforward to check that (Mi)i is manageable with respect to (pn i )i.

3.2.

Infinite dimensional Choquet simplices. Firt, we use the following characterization of infinite dimensional metrizable Choquet simplex.

Lemma 3 ([21], Corollary p.186).

For every infinite dimensional metrizable Choquet simplex K, there exists a sequence of matrices (An) n≥1 such that for every n ≥ 1

(1) An : △(n + 1, 1) → △(n, 1) is well defined and surjective, (2) K is affine homeomorphic to lim ← -n (△(n, 1), An).

Our strategy is to approximate the sequence of matrices (An)n by a manageable sequence. Then we show that the associated inverse limits are affine homeomorphic. For this, we need the following classical density result.

Lemma 4. Let r = (rn) n≥0 be a sequence of integers such that rn ≥ 2 for every n ≥ 0.

Let Cr be the subgroup of (R, +) generated by {(r0

• • • rn) -1 : n ≥ 0}. Then (Cr) p ∩ △(p, 1) ∩ {v ∈ R p : v > 0}
is dense in △(p, 1), for every p ≥ 2.

Proof. Suppose that p = 2. Let m ≥ 1 and Γm = Z(r0

• • • rm) -1 . For α = (α1, α2) ∈ △(2, 1) \ (Γm) 2 , let k1, k2 ≥ 0 be the integers such that k1 r0 • • • rm ≤ α1 < k1 r0 • • • rm + 1 r0 • • • rm and k2 r0 • • • rm ≤ α2 < k2 r0 • • • rm + 1 r0 • • • rm .
Since there is i ∈ {1, 2} such that αi / ∈ Γm, we have 2 and one of its coordinates is not positive, then α = e

1 - 1 r0 • • • rm < k1 r0 • • • rm + k2 r0 • • • rm + 1 r0 • • • rm < 1 + 1 r0 • • • rm , which implies that k1 r0 • • • rm + k2 r0 • • • rm + 1 r0 • • • rm = 1. Thus v = (r0 • • • rm) -1 (k1, k2 + 1) and w = (r0 • • • rm) -1 (k1 + 1, k2) are elements in (Γm) 2 ∩ △(2, 1) such that v -α 1 , w -α1 1 ≤ 2(r0 • • • rm) -1 . Furthermore, v or w is strictly positive. If α is in (Γm)
(2)

1 or α = e (2) 2 . Observe that (1-(r0 • • • rm) -1 , (r0 • • • rm) -1
) and ((r0

• • • rm) -1 , 1- (r0 • • • rm) -1 ) are positive elements in (Γm) 2 ∩ △(2, 1) at distance at most 2(r0 • • • rm) -1 from α.
Since m can be arbitrarily large, we conclude that the lemma is true for p = 2 For p > 2 we conclude by induction on p.

Indeed, if α = (αi) p i=1 ∈ △(p, 1) then β = (α1, • • • , αp-2, αp-1 + αp) is in △(p -1, 1). Thus for ε > 0 there exists a strictly positive element z = (z1, • • • , zp-1) in (Cr) p-1 ∩ △(p -1, 1) such that z -β 1 < ε.
Since Cr is dense in R, there exists a ∈ Cr such that 0 < a < zp-1. If αp < αp + αp-1, by taking ε small enough, we can assume that αp < zp-1. In this case we can choose a in order that |a -αp| < ε. If αp = αp + αp-1, then we can choose a in order that |αp -a| ≤ 2ε. In any case, we get that w = (z1, • • • , zp-2, zp-1 -a, a) is a strictly positive element in (Cr) p ∩ △(p, 1) such that w -α 1 < 5ε. Lemma 5. Let K be an infinite dimensional metrizable Choquet simplex, and let (pn) n≥0 be an increasing sequence of positive integers such that for every n ≥ 0 the integer pn divides pn+1. Then there exist an increasing subsequence (ni) i≥1 of indices and a manageable sequence (Mi) i≥1 of matrices with respect to (pn i ) i≥0 such that K is affine homeomorphic to the inverse limit lim ← -n (△(ki, pn i ), Mi), where ki is the number of rows of Mi, for every i ≥ 0.

Proof. For every n ≥ 0, let rn ≥ 2 be the integer such that pn+1 = pnrn. Let (An) n≥1 be the sequence of matrices given in Lemma 3. We can assume that An+2 : △(n + 3, 1) -→ △(n + 2, 1), for every n ≥ 1. Now we define the subsequence (ni)i by induction.

We set n1 = 0.

Let i ≥ 1 and suppose that we have defined ni ≥ 0. We set r (i) = (rn) n≥n i . For every

1 ≤ j ≤ i + 3, Lemma 4 ensures the existence of v (i,j) ∈ (C r (i) ) i+2 ∩ △(i + 2, 1) ∩ {v ∈ R i+2 : v > 0} such that (1) v (i,j) -Ai+2(•, j) 1 < 1 2 i .
Let Bi be the matrix given by Bi(•, j) = v (i,j) , for every 1 ≤ j ≤ i + 3.

Observe that (1) implies that

n≥1 sup{ Anv -Bnv 1 : v ∈ △n+3} < ∞. It follows from [5, Lemma 9] that K is affine homeomorphic lim ← -n (△(i + 2, 1), Bi). Let ni+1 > ni be such that rn i • • • rn i+1 -1v (i,j
) is an integer vector and such that

rn i • • • rn i+1 -1v (i,j) > i + 3, for every 1 ≤ j ≤ i + 3. We define Mi = pn i+1 pn i Bi.
Thus Mi = P -1 i BiPi+1, where Pi is the diagonal matrix given by Pi(j, j) = pn i for every 1 ≤ j ≤ i + 2 and i ≥ 1. This shows that lim ← -n (△(i + 2, 1), Bi) is affine homeomorphic to lim ← -n (△(i + 2, pn i ), Mi). The proof conclude verifying that (Mi) i≥0 is manageable with respect to (pn i ) i≥0 .

Suitable Følner sequences and connected components.

Let G be a residually finite group, and let (Γn) n≥0 be a nested sequence of finite index normal subgroup of G such that n≥0 Γn = {e}. For technical reasons it is important to notice that since the groups Γn are normal, we have gΓn = Γng, for every g ∈ G.

To construct a Toeplitz G-subshift that is an almost 1-1 extension of the odometer defined by the sequence (Γn)n, we need a "suitable" sequence (Fn)n of fundamental domains of G/Γn. More precisely, each Fn+1 has to be tileable by translated copies of Fn. To control the simplex of invariant measure of the subshift, we need in addition the sequence (Fn)n to be Følner. We did not find in the specialized litterature a result ensuring these conditions.

4.1. Suitable sequence of fundamental domains. Lemma 6. Let (Dn) n≥0 be an increasing sequence of finite subsets of G such that for every n ≥ 0, e ∈ Dn and Dn is a fundamental domain of G/Γn. Let (ni) i≥0 ⊆ Z + be an increasing sequence. Consider (Fi) i≥0 defined by F0 = Dn 0 and Fi = v∈Dn i ∩Γn i-1 vFi-1 for every i ≥ 1.

Then for every i ≥ 0 we have the following:

(1) Fi ⊆ Fi+1 and Fi is a fundamental domain of G/Γn i .

(2) Fi+1 = v∈F i+1 ∩Γn i vFi.

Proof. Since e ∈ Dn i , the sequence (Fi) i≥0 is increasing. F0 = Dn 0 is a fundamental domain of G/Γn 0 . We will prove by induction on i that Fi is a fundamental domain of G/Γn i . Let i > 0 and suppose that Fi-1 is a fundamental domain of G/Γn i-1 .

Let v ∈ Dn i . There exist then u ∈ Fi-1 and w ∈ Γn i-1 such that v = wu. Let z ∈ Dn i and γ ∈ Γn i be such that w = γz. Since z ∈ Γn i-1 ∩ Dn i and v = γzu, we conclude that Fi contains one representing element of each class in G/Γn i . Let w1, w2 ∈ Fi be such that there exists γ ∈ Γn i verifying w1 = γw2. By definition, w1 = v1u1 and w2 = v2u2, for some u1, u2 ∈ Fi-1 and v1, v2 ∈ Dn i ∩ Γn i-1 . This implies that u1 and u2 are in the same class of G/Γn i-1 . Since Fi-1 is a fundamental domain, we have u1 = u2. From this we get v1 = γv2, which implies that v1 = v2. Thus we deduce that Fi contains at most one representing element of each class in G/Γn i . This shows that Fi is a fundamental domain of G/Γn i .

The neutral element e is contained in D0 = F0 ⊆ Fi-1. Then by definition of Fi we have ve ∈ Fi for every v ∈ Dn i ∩ Γn i-1 . This shows that

Dn i ∩ Γn i-1 ⊆ Fi ∩ Γn i-1 . For v ∈ Fi ∩ Γn i-1 , let u ∈ Fi-1 and γ ∈ Dn i ∩ Γn i-1 be such that v = γu. Since v and γ are in Γn i-1 , we have u ∈ Γn i-1 , which implies that u = e and v = γ ∈ Dn i ∩ Γn i-1 .
In this paper, by Følner sequences we mean right Følner sequences. That is, a sequence (Fn) n≥0 of nonempty finite sets of G is a Følner sequence if for every g ∈ G

lim n→∞ |Fng△Fn| |Fn| = 0.
Observe that (Fn) n≥0 is a right Følner sequence if and only if (F -1 n ) n≥0 is a left Følner sequence.

Lemma 7. Suppose that G is amenable. There exists an increasing sequence (ni) i≥0 ⊆ Z + and a Følner sequence (Fi) i∈Z + , such that i) Fi ⊆ Fi+1 and Fi is a fundamental domain of G/Γn i , for every i ≥ 0.

ii) G = i≥0 Fi.

iii) Fi+1 = v∈F i+1 ∩Γn i vFi, for every i ≥ 0.

Proof. From [20, Proposition 4.1], there exists an increasing sequence (mi) i≥0 ⊆ Z + and a Følner sequence (Di) i∈Z + such that for every i ≥ 0, Di ⊆ Di+1, Di is a fundamental domain of G/Γm i , and G = i≥0 Di. Up to take subsequences, we can assume that Di is a fundamental domain of G/Γi, for every i ≥ 0, and that e ∈ D0.

We will construct the sequences (ni) i≥0 and (Fn) n≥0 as follows:

Step 0: We set n0 = 0 and F0 = D0.

Step i: Let i > 0. We assume that we have chosen nj and Fj for every 0 ≤ j < i. We take ni > ni-1 in order that the following two conditions are verified:

(2)

|Dn i g △ Dn i | |Dn i | < 1 i|Fi-1|
, for every g ∈ Fi-1.

(

) Dn i-1 ⊆ v∈Dn i ∩Γn i-1 vFi-1. 3
Such integer ni exists because (Dn) n≥0 is a Følner sequence and Fi-1 is a fundamental domain of G/Γn i-1 (then G = v∈Γn i-1 vFi-1). We define Fi =

v∈Dn i ∩Γn i-1 vFi-1.
Lemma 6 ensures that (Fi) i≥0 verifies i) and iii) of the lemma. The equation (3) implies that (Fi) i≥0 verifies ii) of the lemma. It remains to show that (Fi) i≥0 is a Følner sequence.

By definition of Fi we have

Fi \ Dn i ⊆ g∈F i-1 Dn i g \ Dn i .
Then by equation ( 2) we get

|Fi \ Dn i | |Dn i | ≤ g∈F i-1 |Dn i g \ Dn i | |Dn i | ≤ |Fi-1| 1 i|Fi-1| = 1 i . Since |Fi ∩ Dn i | + |Dn i \ Fi| = |Dn i | = |Fi| = |Fi ∩ Dn i | + |Fi \ Dn i |, we obtain |Dn i \ Fi| |Dn i | ≤ 1 i . Let g ∈ G. Since Fig \ Fi = (Fi ∩ Dn i )g \ Fi ∪ (Fi \ Dn i )g \ Fi ⊆ (Fi ∩ Dn i )g \ Fi ∪ (Fi \ Dn i )g ⊆ Dn i g \ (Fi ∩ Dn i ) ∪ (Fi \ Dn i )g, we have (4) |Fig \ Fi| |Fi| ≤ |Dn i g \ (Fi ∩ Dn i )| |Dn i | + |(Fi \ Dn i )g| |Dn i | ≤ |Dn i g \ (Fi ∩ Dn i )| |Dn i | + 1 i .
On the other hand, the relation

Dn i g \ Dn i = Dn i g \ ((Dn i ∩ Fi) ∪ Dn i \ Fi) = (Dn i g \ (Dn i ∩ Fi)) \ (Dn i \ Fi),
implies that

Dn i g \ (Fi ∩ Dn i ) = [(Dn i g \ (Fi ∩ Dn i )) ∩ (Dn i \ Fi)] ∪ (Dn i g \ (Fi ∩ Dn i )) \ (Dn i \ Fi) = [(Dn i g \ (Fi ∩ Dn i )) ∩ (Dn i \ Fi)] ∪ Dn i g \ Dn i ⊆ Dn i \ Fi ∪ Dn i g \ Dn i , which ensures that (5) |Dn i g \ (Fi ∩ Dn i )| |Dn i | ≤ |Dn i \ Fi| |Dn i | + |Dn i g \ Dn i | |Dn i | .
From equations ( 4) and ( 5), we obtain

|Fig \ Fi| |Fi| ≤ 2 i + |Dn i g \ Dn i | |Dn i | , which implies (6) lim i→∞ |Fig \ Fi| |Fi| = 0.
In a similar way we deduce that

Fi \ Fig ⊆ (Dn i \ (Fi ∩ Dn i )g) ∪ Fi \ Dn i , Dn i \ Dn i g = (Dn i \ (Dn i ∩ Fi)g) \ (Dn i \ Fi), and
Dn i \ (Fi ∩ Dn i )g ⊆ Dn i \ Fi ∪ Dn i \ Dn i g.
Combining the last three equations we get

|Fi \ Fig| |Fi| ≤ 2 i + |Dn i \ Dn i g| |Dn i | , which implies (7) lim i→∞ |Fi \ Fig| |Fi| = 0.
Equations ( 6) and [START_REF] Downarowicz | Survey of odometers and Toeplitz flows[END_REF] imply that (Fi) i≥0 is Følner.

The following result is a direct consequence of Lemma 7.

Lemma 8. Let G be an amenable residually finite group and let (Γn) n≥0 be a decreasing sequence of finite index normal subgroups of G such that n≥0 Γn = {e}. There exists an increasing sequence (ni) i≥0 ⊆ Z + and a Følner sequence (Fi) i≥0 of G such that

(1) {e} ⊆ Fi ⊆ Fi+1 and Fi is a fundamental domain of G/Γn i , for every n ≥ 0.

(2) G = i≥0 Fi.

(3) Fj = v∈F j ∩Γn i vFi, for every j > i ≥ 0.

Proof. The existence of the sequence of subgroups of G and the Følner sequence verifying (1), ( 2) and (3) for j = i+1 is direct from Lemma 7. Using induction, it is straightforward to show (3) for every j > i ≥ 0.

Connected components.

We recall here the notion of connected component of a discrete group G. This notion will be usefull to define a Toeplitz sequence. Let Γ be a finitely generated subgroup of G. Let S be a symmetric generating set of Γ and let C ⊆ Γ be a non empty set. The connected components of C with respect to S are the equivalence classes of the following equivalence relation defined on C:

g ∼ h ⇐⇒ there exist s1, • • • , s k ∈ S ∪ {e}, such that for every 1 ≤ i ≤ k, gs1 • • • si ∈ C and gs1 • • • s k = h.
The set C is said to be connected if C has only one connected component. Suppose there exists a connected subset

C ′′ of C such that |C ′′ | = r-1. Since |C| ≥ r, we can choose h ∈ C \ C ′′ . Due to C is connected, for g ∈ C ′′ there exist s1, • • • , s k ∈ S such that for every 1 ≤ i ≤ k, gs1 • • • si ∈ C and gs1 • • • s k = h. Let g ∈ C ′′ and h ′ = gs1 • • • si be such that i = min{1 ≤ j ≤ k : gs1 • • • sj / ∈ C ′′ }. The set C ′ = C ′′ ∪ {h ′ }
is connected and has exactly r elements.

Let us recall that for a group G finitely generated and Γn a subgroup of finite index, then Γn is finitely generated (see for example [1, Proposition 6.6.2]). Thus for (Γn) n≥0 and (Fn) n≥0 be as in Lemma 8. Definition 1. For every n ≥ 0, let Sn be a finite symmetric generating subset of Γn.

Since the sequence (Fn) n≥0 is increasing and the union of these sets covers G, there exists a subsequence (Fn i ) i≥0 such that e and the elements of Sn i are in the same connected component of Fn i+1 ∩ Γn i with respect to Sn i .

Proof of Theorem A.

Let G be an amenable finitely generated residually finite group. Let (Γn) n≥0 and (Fn) n≥0 be as in Lemma 8. For every n ≥ 0, we call Rn the set [(Fn ∪ F

-1 n )F -1 n ∪ Sn] ∪ [(Fn ∪ F -1 n )F -1 n
∪ Sn] -1 . This will enable us to define a "border" of each domain Fn+1. Let Σ be a finite alphabet. For every n ≥ 0, let kn ≥ 3 be an integer and let {Bn,1, • • • , B n,kn } ⊆ Σ Fn be a collection of different functions. We say that ({Bn,1, • • • , B n,kn }) n≥0 verifies conditions (C1)-(C4) if it verifies the following four conditions for any n ≥ 0:

(C1) σ γ -1 (B n+1,k )|F n ∈ {Bn,i : 1 ≤ i ≤ kn}, for every γ ∈ Fn+1 ∩ Γn, 1 ≤ k ≤ kn+1. (C2) B n+1,k |F n = Bn,1, for every 1 ≤ k ≤ kn+1. (C3) For any g ∈ Fn such that for some 1 ≤ k, k ′ ≤ kn, B n,k (gv) = B n,k ′ (v) for all v ∈ Fn ∩ Fng -1 , then g = e. (C4) σ γ -1 (B n+1,k )|F n = B n,kn for every γ ∈ (Fn+1 ∩ Γn) ∩ Fn+1 \ Fn+1g -1 , for some g ∈ Rn.
In this case, for every n ≥ 0 we define the matrix Mn ∈ M kn×k n+1 (Z + ) as

Mn(i, k) = |{v ∈ Fn+1 ∩ Γn : B n+1,k |vF n = Bn,i}|.
In the next lemma, we show that conditions (C1) and (C2) are sufficient to construct a Toeplitz sequence. The technical conditions (C3) (aperiodicity) and (C4) (also known as "forcing the border") will ensure the existence of a good sequence of partitions (Pn)n spanning the topology. This will allow to give a characterization of the set of invariant probability measures and the ordered group of the associated Toeplitz subshift.

Lemma 10. Let ({Bn,1, • • • , B n,kn }) n≥0 be a sequence that verifies conditions (C1)-(C4). Then:

(1) The set n≥0 [Bn,1] contains only one element x0 which is a Toeplitz sequence.

(2) Let X be the orbit closure of x0 with respect to the shift action. For every n ≥ 0,

Pn = {σ u -1 ([B n,k ] ∩ X) : 1 ≤ k ≤ kn, u ∈ Fn}
is a clopen partition of X. Moreover, Pn+1 is finer than Pn and (Pn) n≥0 spans the topology of X.

(3) The Toeplitz subshift (X, σ|X , G) is an almost 1-1 extension of the odometer O = lim ← -n (G/Γn, πn). (4) There is an affine homeomorphism between the set of invariant probability measures of (X, σ|X , G) and the inverse limit lim ← -n (△(kn, |Fn|), Mn).

(5) The ordered group G(X, σ|X , G) is isomorphic to (H/inf (H), (H/inf (H)) + , u + inf (H)), where H = lim -→n (Z kn , M T n ), H + = {[v, n] : v ≥ 0, n ≥ 0} and u = [|F0|(1, • • • , 1), 0].
Proof. 1. Condition (C2) implies that n≥0 [Bn,1] is non empty, and since G = n≥0 Fn, there is only one element x0 in this intersection. Let X be the orbit closure of x0. For every n ≥ 0 and 1

≤ k ≤ kn, we denote C n,k = [B n,k ] ∩ X. Let n ≥ 0. Condition (C1) and (3) of Lemma 8 imply that σ γ -1 (B m,k )|F n ∈ {Bn,i : 1 ≤ i ≤ kn}, for every m > n, 1 ≤ k ≤ km and γ ∈ Γn ∩ Fm. From this we deduce that σ γ -1 (x0)|F n ∈ {Bn,i : 1 ≤ i ≤ kn}, for every γ ∈ Γn.
The condition (C2) implies then that σ γ -1 (x0)|F n-1 = Bn-1,1 for any γ ∈ Γn. Thus, for g ∈ G and n such that g ∈ Fn-1, we get x0(γg) = Bn-1,1(g) for any γ ∈ Γn. This shows that x0 is Toeplitz.

2. If g is any element in G, then there exist u ∈ Fn and γ ∈ Γn such that g = γu. Thus

σ g -1 (x0) = σ u -1 (σ γ -1 (x0)) ∈ σ u -1 (C n,k ), for some 1 ≤ k ≤ kn. It follows that Pn = {σ u -1 (C n,k ) : 1 ≤ k ≤ kn, u ∈ Fn}
is a clopen covering of X. Condition (C3) ensures that the set of return times of x0 to kn k=1 C n,k , i.e. the set {g ∈ G : σ g -1 (x0) ∈ kn k=1 C n,k }, is Γn. This implies that Pn is a partition. From (C1) we have that Pn+1 is finer than Pn. Since every Pn is a partition, for every n ≥ 0 and every x ∈ X there are unique vn(x) ∈ Fn and 1 ≤ kn(x) ≤ kn such that

x ∈ σ vn(x) -1 (C n,kn(x) ).
The collection (Pn) n≥0 spans the topology of X if and only if (vn(x)) n≥0 = (vn(y)) n≥0 and (kn(x)) n≥0 = (kn(y)) n≥0 imply x = y. Let x, y ∈ X be two sequences such that vn(x) = vn(y) = vn and kn(x) = kn(y) for every n ≥ 0. Let g ∈ G such that x(g) = y(g).

We have then for any n ≥ 0

σ vn (x)|F n = σ vn (y)|F n ∈ {Bn,i : 1 ≤ i ≤ kn}.
And then

x| v -1 n Fn = y| v -1 n Fn . Thus by definition, we get g ∈ v -1
n Fn for any n. We can take n sufficiently large in order that g ∈ Fn-1. Let γ ∈ Γn and u ∈ Fn such that vn(x)g = γu. Observe that ug -1 / ∈ Fn. Indeed, if ug -1 ∈ Fn, then the relation vn(x) = γug -1 implies γ = e, but in that case we get vn(x)g = u ∈ Fn which is not possible by hypothesis. By the condition (C1), there exists an index 1 ≤ i ≤ kn such that σ γ -1 (σ vn (x))|F n = Bn,i and then

x(g) = σ γ -1 σ vn (x)(γ -1 vng) = Bn,i(u). Let γ ′ ∈ Γn-1 ∩ Fn and u ′ ∈ Fn-1 such that u = γ ′ u ′ . Since γ ′ u ′ g -1 = ug -1 /
∈ Fn, we get γ ′ ∈ Fn \ Fngu ′-1 . This implies that γ ′ ∈ Fn \ Fnw, for w = gu ′-1 ∈ Rn-1 and Bn,i(u) = B n-1,k n-1 (u ′ ) by the condition (C4). Thus x(g) = B n-1,k n-1 (u ′ ). The same argument implies that y(g) = B n-1,k n-1 (u ′ ) = x(g) and we obtain a contradiction. This shows that (Pn) n≥0 spans the topology of X.

3. The map π : X → O given by π(x) = ([vn(x)]n) n≥0 is well defined, is a factor map and verifies π -1 (π(x0)) = {x0}. This shows that (X, σ|X , G) is an almost 1-1 extension of O.

4. Since (Pn) n≥0 spans the topology of X, for every µ ∈ M(X, σ|X , G), the sequence (µn = (µ(Cn,1), • • • , µ(C n,kn )) n≥0 determines completely µ. Moreover, since

Mn(i, k) = |{v ∈ Fn+1 : σ v -1 (C n+1,k ) ⊆ Cn,i}|, for every 1 ≤ i ≤ kn, 1 ≤ k ≤ kn+1 and n ≥ 0,
the function µ → (µn) n≥0 , from M(X, σ|X , G) to lim ← -n (△(kn, |Fn|), Mn), is well defined and is an affine homeomorphism (with respect to the weak-star topology and the product topology). Indeed it is standard through the Følner sequence to show that a sequence in lim ← -n (△(kn, |Fn|), Mn) define an unique G-invariant probability measure.

5.

Let φ : H → Dm(X, σ|X , G) be the function given by φ(

[v, n]) = kn k=1 vi[1C n,k ], for every v = (v1, • • • , v kn ) ∈ Z kn and n ≥ 0.
It is easy to check that φ is a well defined homomorphism of groups that verifies and φ(H + ) ⊆ Dm(X, σ|X , G) + . Since (Pn) n≥0 spans the topology of X, every function f ∈ C(X, Z) is constant on every atom of Pm, for some m ≥ 0. This implies that φ is surjective. Lemma 1 and the previous point 4, imply that Ker(φ) = inf (H). Finally, φ induces a isomorphism φ :

H/inf (H) → Dm(X, σ|X, G) such that φ((H/inf (H)) + ) = Dm(X, σ|X , G) + . It remains to show that φ([|F0|(1, • • • , 1), 0]) = k |F0|[1C 0,k ] = [1X ]
. This is true because for any G-invariant measure µ, since P0 is a partition, we have

1 = 1≤k≤k 0 u∈F 0 µ(σ u -1 (C 0,k )) = 1≤k≤k 0 |F0|µ(C 0,k ).
The next result shows that, up to telescope a manageable sequence of matrices, it is possible to obtain a manageable sequence of matrices with sufficiently large coefficient to satisfy the conditions of Lemma 10.

Lemma 11. Let (Mn) n≥0 be a sequence of matrices manageable with respect to (|Fn|) n≥0 .

Let kn be the number of rows of Mn, for every n ≥ 0.

Then there exists an increasing sequence (ni) i≥0 ⊆ Z + such that for every i ≥ 0 and every

1 ≤ k ≤ kn i+1 , (i) Rn i ⊆ Fn i+1 , (ii) For every 1 ≤ l ≤ kn i , Mn i Mn i +1 • • • Mn i+1 -1(l, k) > |Sn i | + | g∈Rn i Fn i+1 \ Fn i+1 g -1 | (iii) kn i+1 < Mn i • • • Mn i+1 -1(i, k), for every 1 ≤ i ≤ kn i .
Proof. We define n0 = 0. Let i ≥ 0 and suppose that we have defined nj for every 0 ≤ j ≤ i. Let m0 > ni be such that for every m ≥ m0, Rn i ⊆ Fm.

Let 0 < ε < 1 be such that ε|Rn i | < 1. Since (Fn) n≥0 is a Følner sequence, there exists m1 > m0 such that for every m ≥ m1,

\ Fmg -1 | |Fm| < ε |Fn i +1| (8) |Fn
, for every g ∈ Rn i .

Since ε|Rn i | < 1, there exists m2 > m1 such that for every m ≥ m2,

1 -|Sn i | |Fn i +1| |Fm| > ε|Rn i |. Then |Fm| |Fn i +1| -|Sn i | > ε|Rn i | |Fm| |Fn i +1|
.

Conditions (1) and (3) for manageable sequences imply that

Mn i • • • Mm-1(l, j) ≥ |Fm| |Fn i +1|
, for every 1 ≤ l ≤ kn i , 1 ≤ j ≤ km.

Combining the last two equations we get

Mn i • • • Mm-1(l, j) -|Sn i | > ε|Rn i | |Fm| |Fn i +1| ,
and from equation (8), we obtain

Mn i • • • Mm-1(l, j) -|Sn i | > |Fm \ Fmg -1 ||Rn i |, for every g ∈ Rn i ,
which finally implies that

Mn i • • • Mm-1(l, j) > | g∈Rn i Fm \ Fmg -1 | + |Sn i |, for every 1 ≤ l ≤ kn i , 1 ≤ j ≤ km.
Condition (4) for manageable sequences implies the existence of m3 > m2 such that for every m ≥ m3 km+1 ≤ Mn i • • • Mm(i, j) for every 1 ≤ i ≤ kn and 1 ≤ j ≤ kn+1.

By taking ni+1 ≥ m3 we get the desired subsequence (ni) i≥0 ⊆ Z + .

The following proposition shows that given a manageable sequence, there exists a sequence of decorations verifying conditions (C1)-(C4). The aperiodicity condition (C3) is obtained by decorating the center of Fn in a unique way with respect to other places in Fn. The restriction on the number of columns of the matrices in a manageable sequence, gives enough choices of coloring to ensure conditions (C3) and (C4).

Proposition 1. Let (Mn) n≥0 be a sequence of matrices which is manageable with respect to (|Fn|) n≥0 . For every n ≥ 0, we denote by kn the number of rows of Mn. Then there exists a Toeplitz subshift (X, σ|X, G) verifying the following three conditions:

(1) The set of invariant probability measures of (X, σ|X, G) is affine homeomorphic to lim ← -n (△(kn, |Fn|), Mn).

(2) The ordered group G(X, σ|X , G) is isomorphic to (H/inf (H), (H/inf (H)) + , u + inf (H)), where H = lim -→n

(Z kn , M T n ), H + = {[v, n] : v ≥ 0, n ≥ 0} and u = [|F0|(1, • • • , 1), 0]. (3) (X, σ|X , G) is an almost 1-1 extension of the odometer O = lim ← -n (G/Γn, πn).
Proof. Let (ni) i≥0 ⊆ Z + be a sequence as in Lemma 11. Since (Mn) n≥0 and the sequence (Mn i • • • Mn i+1 -1)i≥0 define the same inverse and direct limits, without loss of generality we can assume that for every n ≥ 0 we have:

(9) Rn ⊆ Fn+1, (10)
Mn(i, k) > |Sn| + | g∈Rn Fn+1 \ Fn+1g -1 | for every 1 ≤ i ≤ kn, 1 ≤ k ≤ kn+1, and (11)
kn+1 < min{Mn(i, j) : 1 ≤ i ≤ kn, 1 ≤ j ≤ kn+1}.

Claim 1. Let Cn,1, • • • , Cn,r n be the connected components of (Fn+1 ∩ Γn) \ (Sn ∪ {e} ∪ g∈Rn Fn+1 \ Fn+1g -1 ) with respect to Sn. The number Mn(1, k) -1 is smaller than the number of elements which are in all the components Cn,i having more than one element.

To show the claim, notice that we have [START_REF] Downarowicz | Almost 1 -1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows[END_REF] |Cn,1|

+ • • • + |Cn,r n | ≥ Mn(1, k) -1 + Mn(2, k), because Mn(1, k) + Mn(2, k) -1 ≤ Mn(1, k) + • • • + Mn(kn -1, k) -1 = |Fn+1 ∩ Γn| -Mn(kn, 1) -1 ≤ |Fn+1 ∩ Γn| -|Sn| -| g∈Rn Fn+1 \ Fn+1g -1 | -|{e}| ≤ |Fn+1 ∩ Γn \ (Sn ∪ g∈Rn Fn+1 \ Fn+1g -1 ∪ {e})| = |Cn,1| + • • • + |Cn,r n |.
We fill the rest of the coordinates v ∈ Fn+1 ∩ Γn in order that σ v -1 (B In this section we briefly recall the concept of ordered Bratteli diagram and its relation with minimal Z-actions on the Cantor set. We refer to [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] and [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF] for the details. A Bratteli diagram is an infinite directed graph B = (V, E), such that the vertex set V and the edge set E can be partitioned into finite sets

V = V0 ∪ V1 ∪ • • • and E = E1 ∪ E2 ∪ • • •
with the following properties:

• V0 = {v0} is a singleton.

• For every j ≥ 1, each edge in Ej starts from a vertex in Vj-1 and arrives to a vertex in Vj. • All vertices in V have at least one edge starting from it, and all vertices except v0 have at least one edge arriving to it. The sequence of transition matrices or incidences matrices of the Bratteli diagram B is the sequence (Mn) n≥0 such that for every n ≥ 0, the matrix Mn ∈ MV n ×V n+1 (Z + ) is defined as Mn(v, v ′ ) = number of edges from v ∈ Vn to v ′ ∈ Vn+1. For a vertex e ∈ E we denote by s(e) the vertex where e starts and by r(e) the vertex to which e arrives. For m > n, a path from v ∈ Vn to w ∈ Vm in B, is a sequence of edges en+1en+2 • • • em such that s(en+1) = v, r(em) = w and for each n

+ 1 ≤ i ≤ m -1, r(ei) = s(ei+1). The set of infinite paths of B is XB = {(ei) i≥1 ∈ i≥1 Ei : r(ei) = s(ei+1), for each i ≥ 1}.
This set becomes a totally disconnected space when we endow every Ei with the discrete topology, i ≥ 1 Ei with the product topology and XB with the induced topology. Let (mn) n≥0 be an increasing sequence of integers such that m0 = 0. The telescoping of B to (mn) n≥0 is the Bratteli

diagram B ′ = (V ′ , E ′ ) defined by V ′ = {Vm n : n ≥ 0} and E ′ = {E ′ n : n ≥ 1}, where E ′ n contains an edge from v ∈ Vm n-1 to w ∈ Vm n in B ′ for each path from v to w in B.
The Bratteli diagram B is simple, if there exists a telescoping B ′ of B so that the incidences matrices of B ′ are strictly positive. In this case, XB is a Cantor set (when XB is infinite). An ordered Bratteli diagram (V, E, ≥) is a Bratteli diagram B = (V, E) together with a partial order ≥ on E, so that two edges are comparable if and only if they arrive at the same vertex. This order induces a transformation TB : XB → XB. If B is simple, the order ≥ can be chosen so that TB is a minimal homeomorphism on XB.

Conversely, if (X, T, Z) is a minimal Cantor system, then there exist a simple Bratteli diagram B = (V, E) and an order ≥, such that (V, E, ≥) is an ordered Bratteli diagram and (X, T, Z) is conjugate to (XB, TB, Z) (see [START_REF] Durand | Substitutional dynamical systems, Bratteli diagrams and dimension groups[END_REF] and [START_REF] Herman | Ordered Bratteli diagrams, dimension groups and topological dynamics[END_REF] for the details). In this case we say that B = (V, E) is an associated Bratteli diagram to (X, T, Z). In The Bratteli diagram B ′′ defined by the new sequence of matrices (Mn) n≥0 determines the same strong orbit equivalence class than B ′ , verifies the e.p.n-property and its incidence matrices have exactly 3 columns.

The following proposition is the essential part of Theorem B.

Proposition 2. Let (X, σ|X , Z) be a minimal Cantor system having an associated Bratteli diagram B = (V, E) which satisfies the equal path number property. If in addition, there exists a constant C > 0 such that for each n ≥ 1, C|r -1 (v)| ≥ |Vn| for every v ∈ Vn, then for every d ≥ 1 there exists a Toeplitz subshift (Y, σ|Y , Z d ) which is orbit equivalent to (X, σ|X , Z).

In [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], Gjerde and Johansen show that every Toeplitz subshift (X, σ|X, Z) has an associated Bratteli diagram verifying the e.p.n-property. In [START_REF] Sugisaki | Toeplitz flows, ordered Bratteli diagrams and strong orbit equivalence[END_REF], Sugisaki shows that every minimal Cantor system (Y, T, Z) having an associated Bratteli diagram which satisfies the e.p.n-property, is strong orbit equivalent to a Toeplitz subshift (X, σ|X , Z). At the opposite of residually finite group, let us mention [START_REF] Dahl | AF equivalence relations associated to locally finite groups[END_REF], where Dahl shows that any Brattteli diagram verifying the e.p.n-property is orbit equivalent to a free action of a locally finite group.

Proof of Proposition 2. From Lemma 12, we can assume that |Vn| ≥ 3, for every n ≥ 1. Let (Mn) n≥0 be the sequence of incidence matrices of B. u ld+j for every 2 ≤ j ≤ d.

We have p id = qi = qi,1 • • • q i,d , for every i ≥ 1.

Let Γi = d j=1 qi,jZ. The sequence of groups (Γi) i≥1 satisfies Γi ⊆ Γi+1, |Z d /Γi| = qi and i≥1 Γi = {0}. Thus from Proposition 1, there exists a Toeplitz subshift (Y, σ|Y , Z d ) such that G(Y, σ|Y , Z d ) is isomorphic to G(X, σ|X , Z). Theorem 2.5 in [START_REF] Giordano | Orbit equivalence for Cantor minimal Z dsystems[END_REF] implies that (Y, σ|Y , Z d ) is orbit equivalent to (X, σ|X , Z). Corollary 1. Let (X, T, Z) be a minimal Cantor system whose dimension group is divisible. Then for every d ≥ 1 there exists a Toeplitz subshift (Y, σ|Y , Z d ) which is orbit equivalent to (X, T, Z).

Proof. From the proof of Theorem 12 in [START_REF] Gjerde | Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows[END_REF], there exists a Toeplitz subshift (X ′ , σ| X ′ , Z) which is orbit equivalent to (X, T, Z), and such that (X ′ , σ| X ′ , Z) has an associated Bratteli diagram B = (V, E) having the e.p.n-property and such that for each n ≥ 0, 6|Vn+1| < r -1 (v) for every v ∈ Vn+1. We conclude applying Proposition 2.

We combine Proposition 2 and Corollary 1 to obtain Theorem B.

Lemma 9 .

 9 Let C ⊆ Γ be a connected set. For every finite number 1 ≤ r ≤ |C| there exists a connected subset C ′ of C such that |C ′ | = r. Proof. Let 1 ≤ r ≤ |C| be a finite number. If r = 1 then C ′ = {g}, for g ∈ C, is connected.

Lemma 12 .

 12 [START_REF] Giordano | Topological orbit equivalence and C*-crossed products[END_REF], Giordano, Putnam and Skau show that the associated Bratteli diagram (without order) of a minimal Z-action on the Cantor set is a total invariant of its strong orbit class. We say that the Bratteli diagram B = (V, E) has the equal path number property (e.p.nproperty) if for each n ≥ 1, |r -1 (v)| = |r -1 (u)| for every u, v ∈ Vn. It is straightforward to show that the equal path number property is invariant under telescoping. Every strong orbit equivalence class having an associated Bratteli diagram verifying the equal path number property, has an associated Bratteli diagram verifying the equal path number property and whose incidence matrices have at least three columns.Proof. Let B = (V, E) be a Bratteli diagram verifying the e.p.n-property. If there exist infinitely many n's such that |Vn| ≥ 3, we may telescope the diagram to those levels in order to get a new diagram B ′ whose incidence matrices have at least 3 columns. This new Bratteli diagram verifies the e.q.n-property and defines the same strong orbit equivalence class than B. Otherwise, we can telescope the diagram B in order to get a new diagram B ′ whose incidence matrices have 1 or either 2 columns. In both cases, B ′ defines the same strong orbit equivalence class than a Bratteli diagram verifying the e.p.n-property and whose incidence matrices have exactly three columns. Indeed, let (An) n≥0 be the sequence of incidence matrices of B ′ and suppose first that An = [mn] , for every n ≥ 0. We can assume that mn > 1 (otherwise X B ′ contains only one point). Since [mn] = (1, 1)(mn -1, 1) T , the diagram B ′ determines the same strong orbit equivalence class than the Bratteli diagram B ′′ defined by the sequence of incidence matrices (Bn) n≥1 , where B0 = (1, 1) and Bn = (mn-1 -1, 1) T (1, 1) for every n ≥ 1. The Bratteli diagram B ′′ determines the same orbit equivalence class than B ′ , verifies the e.p.n-property and its incidences matrices have exactly two columns. Now we are in the case where every An has exactly two columns. Then we set M0 = A0A and Mn = BnA for n ≥ 1, where

  For each n ≥ 1, let rn-1 = |r -1 (v)| for every v ∈ Vn. The e.p.n-property ensures v∈Vn Mn(v, v ′ ) = rn for every v ′ ∈ Vn+1 and n ≥ 0.Let (ni) i≥0 be a sequence of non-negative integers such that n0 = 0, ni+1 > ni + 3 and verifiesrn i +1 • • • rn i+1 -2 > C, for every i ≥ 0. Observe that Mn i • • • Mn i+1 -1(l, j) ≥ rn i +1 • • • rn i+1 -1,for every l, j and i ≥ 0.Defining Ai = Mn i • • • Mn i+1 -1, we have Ai(l, j) > Crn i+1 -1 ≥ |Vn i+1 | for every l, j and i ≥ 0.The sequence (Ai) i≥0 is the sequence of incidence matrices of the telescoping of B to the levels (ni) i≥0 . This sequence is manageable with respect to (pi) i≥0 , where p0 = 1,p1 = r0 • • • rn 1 -1 and pi+1 = pirn i • • • rn i+1 -1 for every i ≥ 1. Let ui+1 = rn i • • • rn i+1 -1,for every i ≥ 0.Let d ≥ 1 be an integer. For every i ≥ 1 we define

  Ordered groups. An ordered group is a pair (H, H + ), such that H is a countable abelian group and H + is a subset of H verifying (H + )+(H + ) ⊆ H + , (H + )+(-H + ) = H and (H + ) ∩ (-H + ) = {0} (we use 0 as the unit of H when H is abelian). An ordered group (H, H + ) is a dimension group if for every n ∈ Z + there exist kn ≥ 1 and a positive homomorphism An : Z kn → Z k n+1 , such that (H, H + ) is isomorphic to (J, J + ), where J

	unique invariant probability measure. Notice that this action is free: the stabilizer of
	any point is trivial. The Toeplitz G-subshifts are characterized as the subshifts that are
	minimal almost 1-1 extensions of G-odometers [4].
	2.4. is the direct limit
	lim -→n

where [g]n denotes the class of g in G/Γn. Thus the group G acts by left multiplication on O. When there is no confusion, we call this action also odometer. It is equicontiuous, minimal and the left Haar measure is the

  The dimension group is simple if the matrices An can be chosen strictly positive. An order unit in the ordered group (H, H + ) is an element u ∈ H + such that for every g ∈ H there exists n ∈ Z + such that g ≤ nu. If (H, H + ) is a simple dimension group then each element in H + \ {0} is an order unit. A unital ordered group is a triple (H, H + , u)

	such that (H, H + ) is an ordered group and u is an order unit. An isomorphism between
	two unital ordered groups (H, H + , u) and (J, J + , v) is an isomorphism φ : H → J such
	that φ(H + ) = J + and φ(u) = v. A state of the unital ordered group (H, H + , u) is
	a homomorphism φ : H → R so that φ(u) = 1 and φ(H + ) ⊆ R + . The infinitesimal
	subgroup of (H, H + , u) is
	inf(H) = {a ∈ H : φ(a) = 0 for all state φ}.
	It is not difficult to show that inf(H) does not depend on the order unit.
	The quotient group H/inf(H) of a simple dimension group (H, H + ) is also a simple
	dimension group with positive cone
	(H/inf(H)

  n+1,k )|F n ∈ {Bn,1, • • • , B n,kn } and such that |{v ∈ Fn+1 ∩ Γn : σ v -1 (B n+1,k )|F n = Bn,i}| = Mn(i, k), for every 2 ≤ i ≤ kn. Notice that the number of such v is at least Mn(2, k), because there are at least Mn(2, k) -1 coordinates to be filled with Bn,2 and at least 1 coordinate to be filled with B n,kn . Thus we have at least Mn(2, k) > kn+1 different ways to fill the coordinates such that the functions Bn+1,1, • • • , B n+1,k n+1 are pairwise different. By construction, every function B n+1,k verifies (C1), (C2) and (C4). Let us assume there are g ∈ Fn+1 and 1 ≤ k, k ′ ≤ kn+1 such that B n+1,k (gv) = B n+1,k ′ (v) for any v where it is defined, then by the induction hypothesis, g ∈ Γn. This implies that σ g -1 (B n+1,k )|F n = B n+1,k ′ |F n = Bn,1. By definition, if g = e then there exists s ∈ Sn such that gs ∈ Fn+1 and σ (gs) -1 (B n+1,k )|F n = Bn,1 or Bn,2. On the other hand, σ s -1 (B n+1,k ′ )|F n = B n,kn . Since σ (gs) -1 (B n+1,k )|F n and σ s -1 (B n+1,k ′ )|F n have to coincide, we deduce that g = e. This shows that the collection Bn+1,1, • • • , B n+1,k n+1 verifies (C3). We conclude applying Lemma 10. Proof of Theorem A. Let ext(K) be the set of extreme points of K. If ext(K) is finite, then the proof is direct from Lemma 2 and Proposition 1. If ext(K) is infinite, the proof follows from Lemma 5 and Proposition 1.

6. Proof of Theorem B.

The claim is then trivial if for every 1 ≤ i ≤ rn we have |Cn,i| ≥ 2. Suppose now, there is at least one component having only one element. Up to change the indexation, we may assume these components are Cn,1, • • • , C n,ln with 1 ≤ ln ≤ rn. If g ∈ Cn,i for 1 ≤ i ≤ ln, then for every s ∈ Sn we have gs ∈ Fn+1 (because g / ∈ h∈Rn Fn+1\Fn+1h -1 and Sn ⊆ Rn), which implies that gs ∈ Sn∪ h∈Rn Fn+1\Fn+1h -1 (gs = e because g / ∈ Sn). Thus if s ∈ Sn and gi ∈ Cn,i for every 1 ≤ i ≤ ln, then {g1s, • • • , g ln s} ⊆ Sn ∪ g∈Rn Fn+1 \ Fn+1g -1 . Since this set has exactly ln elements, we deduce

which contradicts [START_REF] Effros | Dimensions and C * -algebras[END_REF] and shows the claim.

For every n ≥ 0, we will construct a collection of functions Bn,1,

Let F0 be the set of B ∈ Σ F 0 verifying B(e) = 0 and B(g) ∈ Σ\{0} for every g ∈ F0 \{e}.

Let n ≥ 0. Suppose that we have defined Bn,1,

Now we will determine which are the rest of v ∈ Fn+1 ∩ Γn for which

Observe that equation [START_REF] Effros | Dimensions and C * -algebras[END_REF] implies that Mn(1, k) > 1.

By Claim 1, we can write


Yuan Lei

Nigel Garrahan

Boris Hermann

Michael P Fautsch

Michael Boulton

James Edwards Morgan
email: morganje3@cardiff.ac.uk

D H Johnson

M R Hernandez

Trans-retinal degeneration in ageing human retina: a multiphoton microscopy analysis

Keywords: retina, degeneration, glaucoma

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Most elderly people experience some loss of vision, even in the absence of any identifiable eye disease. Advancing age has brought a spectrum of degenerative or regressive events which manifest visually as a reduction in visual acuity and in peripheral and scotopic sensitivities [START_REF] Bonnel | The aging of the retina[END_REF][START_REF] Owsley | Delays in rod-mediated dark adaptation in early age-related maculopathy[END_REF] . Dark adaptation is also delayed in the elderly due to changes in the kinetics of rod function [START_REF] Owsley | Vision and aging[END_REF] . These age-related changes have functional effects that span the retina. Standard full-field electrophysiology studies indicate a significant increase in implicit times and a decrease in the amplitudes of a waves (photoreceptor responses) and b waves (bipolar and Müller cell responses) in the elderly [START_REF] Birch | Standardized full-field electroretinography. Normal values and their variation with age[END_REF][START_REF] Weleber | The effect of age on human cone and rod ganzfeld electroretinograms[END_REF] . The amplitude of multifocal oscillatory potentials, which are thought to reflect inner retinal function and rod-cone interactions, decreases linearly with age whereas their latency increases with age [START_REF] Kurtenbach | Effect of aging on multifocal oscillatory potentials[END_REF] . Histological studies have provided a neural basis for this loss in function, reporting reductions in retinal ganglion cell (RGC) [START_REF] Curcio | Retinal ganglion cells in Alzheimer's disease and aging[END_REF] , rod bipolar cells, photoreceptors (especially rods) [START_REF] Curcio | Photoreceptor topography in ageing and age-related maculopathy[END_REF][START_REF] Curcio | Aging of the human photoreceptor mosaic: evidence for selective vulnerability of rods in central retina[END_REF][START_REF] Gao | Aging of the human retina. Differential loss of neurons and retinal pigment epithelial cells[END_REF] and pigmented epithelium (RPE) [START_REF] Gao | Aging of the human retina. Differential loss of neurons and retinal pigment epithelial cells[END_REF] .

What is not clear from these observations is whether neural losses within the retinal layers occur independently or transneuronal degeneration contributes to the effects of ageing. Transneuronal degeneration refers to neuron loss in one area affecting linked neurons in neighbouring regions. Knowledge of these changes is important in the development of therapies for the restoration of vision in the elderly. Recently, neuronal plasticity was noted in senescent retinas in mouse [START_REF] Liets | Dendrites of rod bipolar cells sprout in normal aging retina[END_REF] and human [START_REF] Eliasieh | Cellular reorganization in the human retina during normal aging[END_REF] where the neurons in the inner nuclear layer (INL) extended their dendrites to the outer nuclear layer (ONL). Our group reported a pattern of neuronal loss that was consistent with retinal transneuronal degeneration in glaucoma in which neuronal loss in the inner retina influences the viability of outer retina [START_REF] Lei | Quantification of retinal transneuronal degeneration in human glaucoma: a novel multiphoton-DAPI approach[END_REF] . Transneuronal degeneration from the retina and optic nerve to the central visual systems including the lateral geniculate nucleus (LGN) and visual cortex [START_REF] Gupta | Glaucoma and the brain[END_REF] .

Multiphoton confocal microscopy has greater sensitivity compared with conventional imaging techniques, e.g. phase contrast Nomarski and single photon confocal microscopy for the quantification of neuronal population in intact tissues. It has the advantage of deeper tissue penetration as a result of a near-simultaneous absorption of multiple photons. With this imaging technique we have been able to preserve the topographic relationship between retinal layers while quantifying different populations of neurones in the RGC layer (RGCL), INL and ONL. These data provide valuable information on the pattern and mechanism of neuronal loss in the ageing human retina.

Materials and methods

Storage and research using human tissue samples was carried out according to Human Tissue Authority regulations under the UK Human Tissue Act 2004. Six young (18-33 years old) and six ageing retinas (74-90 years old) were analysed. In all cases, age, gender, ethnic origin and the cause of death were known. The retinas were fixed within 24 hours of death in 4% paraformaldehyde in 0. good agreement for the ONL ((r 2 = 0.66 ,P<0.01). Any truncated nuclei that were smaller than the threshold size were excluded. Endothelial cells, astrocytes [START_REF] Ogden | Nerve fiber layer astrocytes of the primate retina: morphology, distribution, and density[END_REF] and glial cells [START_REF] Ramirez | Structural specializations of human retinal glial cells[END_REF] were distinguished from neurons by their laminar location, small size, and nuclear morphology. Endothelial cells were identified by their small and elongated nuclei, astrocyte nuclei were mainly in the nerve fibre layer (NFL) which were not counted. Müller cell nuclei were angular and were located in the internal nuclear layer (INL). In addition, 'forbidden lines' were used in which cell nuclei that touched the top and right edges of the square region of interest were excluded from the analysis [START_REF] Gundersen | Notes on the estimation of the numerical density of arbitrary profiles: the edge effect[END_REF] to avoid overestimating cell counts. Our counting method could not distinguish between RGC and amacrine cells, therefore we used literature values to offset the count for easy comparison, i.e. 0.3% to 24% amacrine cells between 0.5 to 4 mm eccentricity in the human retina [START_REF] Curcio | Retinal ganglion cells in Alzheimer's disease and aging[END_REF][START_REF] Curcio | Topography of ganglion cells in human retina[END_REF] .

Statistics

Statistical analysis was performed using SPSS 14.0 for Windows. A general linear model was used to compare neuron densities and ratios at different eccentricities of the retina in young and aged groups. In each retinal layer, given that the same measurements were carried out on the same individuals at six different eccentricities, a repeated measures analysis was performed with a Bonferroni correction.

Differences were considered significant at the p<0.05 or lower. .

Results

At each sampling point, neurons at the RGCL, INL and ONL were imaged sequentially to preserve the topographical correlation. Typical image stacks for young and old eyes are reconstructed as 3D models in Figure 1. We found close agreement between our RGC densities and those reported by Curcio 7,[START_REF] Curcio | Photoreceptor topography in ageing and age-related maculopathy[END_REF][START_REF] Curcio | Topography of ganglion cells in human retina[END_REF][START_REF] Curcio | Human photoreceptor topography[END_REF] for similar age groups (Figure 2).

In both groups of the retinas, 56% of the total area within central 4 mm eccentricity, and 27% between 4 mm and 7 mm eccentricity were imaged. In total, 262,532 neurons were counted in young (n=6) and 228,140 in aged human retinae (n=6). The mean neuron densities in were compared for young and aged eyes in Figure 3.

Neuron density In the RGCL, in the aged eyes significantly decreased between 0.5 and 3 mm eccentricity (Figure 3A); within the central 4 mm eccentricity, the mean density decreased by approximately 15.4 % (from 14,405 to 12,186 per mm 2 ); the greatest RGC loss was at 1mm eccentricity of 24.7%. In the INL and ONL, significant reductions of neuron densities were seen at 1-2 mm eccentricity of 8.7% (Figure 3B) and 0.5-4 mm eccentricity of 15.6% (Figure 3C), respectively (P<0.05). The greatest reductions in cell densities were seen in the RGC and ONL layers with comparative preservation of the INL.

Discussion

Our results are consistent with previous reports of aged related neurodegeneration in both human and animal models [START_REF] Curcio | Retinal ganglion cells in Alzheimer's disease and aging[END_REF][START_REF] Curcio | Aging of the human photoreceptor mosaic: evidence for selective vulnerability of rods in central retina[END_REF][START_REF] Gao | Aging of the human retina. Differential loss of neurons and retinal pigment epithelial cells[END_REF][START_REF] Harman | Neuronal density in the human retinal ganglion cell layer from 16-77 years[END_REF][START_REF] Kim | Effects of aging on the densities, numbers, and sizes of retinal ganglion cells in rhesus monkey[END_REF][START_REF] Danias | Quantitative analysis of retinal ganglion cell (RGC) loss in aging DBA/2NNia glaucomatous mice: comparison with RGC loss in aging C57/BL6 mice[END_REF][START_REF] Neufeld | The inherent, age-dependent loss of retinal ganglion cells is related to the lifespan of the species[END_REF] . The RGC density in four quadrants of the retina is similar to literature values 18 (Figure 3), but the degree of RGC degeneration (19%) in the central 11 degree of vision is less than Curcio's report ('one-fourth') in ageing eye compared with retinas from young adults [START_REF] Curcio | Retinal ganglion cells in Alzheimer's disease and aging[END_REF] , which might be due to the inter-individual variability. An important limitation of the method used in the present study is that we could not reliably discriminate RGC and amacrine cells and therefore used literature values to offset the retinal ganglion cell layer count. In the same region, the densities of photoreceptors in the ageing eyes were 16% less compared with young eyes, This is comparable to previous report of a 15% reduction in photoreceptors [START_REF] Curcio | Aging of the human photoreceptor mosaic: evidence for selective vulnerability of rods in central retina[END_REF] . The neuronal loss in the central 11 degree of visual field may be responsible for the visual function deficits in aged individuals [START_REF] Owsley | Vision and aging[END_REF] . We could not distinguish between rods and cones using the muliphoton DAPI method.

The contribution of the present study is that we have matched changes in the inner and outer retina for any given retinal region of interest. Among the three layers of the retina examined, the greatest neuronal loss was seen in the ONL between 0.5 to 4 mm eccentricity in aged retinas (Figure 3). Although INL degeneration occurs in a smaller region (between 1-2 mm eccentricity), the reduction in neuron densities in the INL corresponds to the region with the greatest reduction in RGCL and ONL cell counts (Figure 3). The resilience of the INL suggests that loss in the retinal ganglion cell and ONL layers are relatively independent. This contrasts with our findings in glaucomatous eyes in which we found a gradient of cell loss from inner to outer retina which is consistent with a transneuronal degeneration initiated in the retinal ganglion cell layer [START_REF] Lei | Quantification of retinal transneuronal degeneration in human glaucoma: a novel multiphoton-DAPI approach[END_REF] .

It Ageing retinas show significant reduction in neuron densities of 30% between 0.5 and 2 mm eccentricities. In the INL and ONL, there are significant reductions of 8.7%% (1-2 mm eccentricity) for the INL 15.6% for the ONL (0.5-4 mm eccentricity).

The column graphs compare the mean neuron densities at all the eccentricities for each retinal layer, which are not significantly different. * p<0.05, **p<0.01.

13 .

 13 1 M phosphate buffer (pH 7.4). Clinical examination of the optic nerve and retina did not indicate the presence of glaucoma or macular degeneration in any of the eyes. Following the removal of anterior segment and vitreous humour, the retina was dissected as a wholemount preparation and freed from the retinal pigment epithelium. Four linear cuts were made in the retina to provide 4 approximately equal segments which were then stained by DAPI (4,6 diamidino-2-phenylindole, Sigma, UK) solution (3 μg/mL in PBS) overnight at 4 o C. The segments were then wholemounted RGC side uppermost in a hard set mounting medium for fluorescence imaging (VECTASHIELD ® Mounting Medium). Details of the multiphoton imaging strategy have been described in detail elsewhere Briefly, in order to relate retinal pathology to clinical data, a sampling grid, was used which retinotopically corresponded to the spacing stimuli in a Humphrey 30-2 visual field. DAPI molecules were excited by the 843 nm line of the Ti-Sapphire Tsunami multiphoton laser (SpectraPhysics) and images acquired using a Leica TCS SP2 confocal system with a Leica DMRE microscope (HCX PL Apo 40/1.25 NA oil CS objective lens; Leica, Milton Keynes, UK). Images were analysed using ImageJ (version 1.34, US National Institutes of Health) to derive counts for neuronal nuclei in the RGCL, INL and ONL. These DAPI labelled nucleus layers were separated by regions of unlabelled neurite. The whole depth of the RGCL was imaged, but since INL and ONL were several cells deep, the automated identification all cells in these layers within a region of interest was not possible. For the INL and ONL we therefore restricted counting to a 10um thick segment within these layers and then multiplied the count by a depth correction factor that corrected for the thickness of the respective nuclear layer. The scanning interval for all layers was 1 um, and the image stacks were analysed as a projected image (i.e. collapsed in the Z dimension). Comparison of manual and automated counting methods in two eyes (8 sample areas feach or the INL ONL and 25 sample areas for the RGC layer) indicated excellent agreement for the RGCL(r 2 = 0.84, P<0.01) and INL (r 2 = 0.96,P<0.01) and

Figure 1

 1 Figure 1 Multi-photon images of DAPI stained cell nuclei in the retinal ganglion cell

Figure 2

 2 Figure 2 Comparisons of RGC quantities measured by multiphoton-DAPI method in

Figure 3

 3 Figure 3 Comparison of neuron densities in the retinal ganglion cell layer (RGCL),

  is not clear why in a closely linked visual network both inner and outer retina undergo significant cell loss whereas the middle layer is relatively preserved. Most electrophysiological studies also point to outer retina degeneration in the elderly. Our data do not rule out the possibility of INL cell loss reflecting damages of the neighbouring layers as it is in the region where RGCL and ONL damages were greatest. The comparable loss of neurons in the RGCL and INL has implications for strategies directed at the repair of the ageing retina. While the present study did not include patients with age related macular degeneration, given the age of patients affected by this condition, it is likely that treatments directed at the replacement of neurons lost in the ONL[START_REF] Maclaren | Stem cell therapy and the retina[END_REF] may be limited by inner retina degeneration. Our findings suggest that strategies should be devised to address both inner and outer retinal degeneration in age related retinal diseases. Acknowledgement This research is supported by Research Into Ageing in UK. The technical support from Dr C.Thrasivoulou and Mr D Ciantar and statistical support from Mr P McGeoghan is very much appreciated. We are also very grateful for the comments of Dr Jennifer Singgers, Imperial College London, on the manuscript.

Research Into Ageing (Grant No. 271, YL, JEM), NIH EY 15736 (MPF), NIH EY 07065 (MPF, DHJ).

Licence for Publication

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Competing Interest

Competing Interest: None to declare.


Aman Chandra

Fiona Jayzyeri

Tom H Williamson

Mr Tom
email: tom@retinasurgery.co.uk

H Williamson

Warfarin in Vitreoretinal surgery -a case controlled series

Keywords: Warfarin, rhegmatogenous retinal detachment, vitrectomy, vitreous haemorrhage, choroidal haemorrhage

come

INTRODUCTION

Warfarin is an anticoagulant commonly used in patients with mechanical prosthetic heart valves, chronic atrial fibrillation and venous thrombo-embolism. The vitreoretinal surgeon is likely to encounter more patients on this drug because an aging population is associated with an increased number of patients taking warfarin. In general, older patients (over 65 years old) are more than twice as likely to have any surgical operation therefore the effects of warfarin on increased complications during surgery is a risk [START_REF] Politser | American College of Surgeons: socioeconomic factbook for surgery[END_REF] . With ophthalmic surgery, there is a perceived risk of ocular haemorrhage [START_REF] Tuli | Delayed suprachoroidal hemorrhage after glaucoma filtration procedures[END_REF] , and some authorities advocate modifying warfarin therapy in certain cases of ophthalmic surgery 3 .

The absolute risk of a thrombo-embolic event in patients undergoing any surgical procedure who interrupt warfarin for between 6-8 days preoperatively ranges from 0.02% in low risk patients with atrial fibrillation to 0.42% in patients with prosthetic heart valves [START_REF] Douketis | Perioperative anticoagulation management in patients who are receiving oral anticoagulant therapy: a practical guide for clinicians[END_REF] . For ophthalmic surgery, in particular vitreoretinal surgery, there is no consensus with regard to the risks or benefits of discontinuation of warfarin in the perioperative period. This study was therefore designed to compare presenting features and outcomes of pars plana vitrectomy surgery in patients anticoagulated with warfarin compared to case controls. Additionally, we surveyed members of the British & Eire Association of Vitreoretinal Surgeons (BEAVRS) to gather a consensus of current trends amongst its members.

METHODS

All vitreoretinal procedures performed or supervised by one author (THW) are entered onto an electronic patient record database (available with Vitreoretinal Surgery published by Springer) [START_REF] Williamson | Vitreoretinal Surgery[END_REF] . In this database all peri-operative complications are prospectively recorded. Patients who were taking warfarin were asked to continue their anticoagulation during the surgical period. Patients receiving pars plana vitrectomy (PPV) between January 2001 and December 2008 were examined with only the first operation for each patient included for analysis. These patients were case controlled to patients on the database for age, sex and presenting complaint. Patients taking aspirin or other antiplatelet agents were excluded from this study. Their demographics, ocular history and examination, presentation, severity of vitreous haemorrhage at presentation, number of vitreoretinal procedures, visual outcome and complication rates were collected. Approval was obtained from the institutional research ethics committee of Guy's & St. Thomas' Hospitals NHS Foundation Trust and the study conformed to the declaration of Helsinki. Informed consent was obtained from all patients. Direct comparative analyses were performed using paired T test, MacNemar test and Mann Whitney matched pair test as required using Analyse It software.

An online survey (www.freeonlinesurvey.com) was sent to all registered members of BEAVRS (see appendix). Reminder emails were sent two weeks after the initial requests were sent. The online survey was open for 10 weeks over September till December 2009.

RESULTS

The study period was 2001 to 2008 60 consecutive patients were taking warfarin on the day of pars plana vitrectomy (2% of the patients receiving vitrectomy in this period). There were 33 males and 27 females with a median age of 72.5 years (37.7 years -89.5 years). Follow up was for a mean of 0.88 years. INR ranged between 0.94 to 4.6 (median 2.3). 6.5% were subtherapeutic (INR <1.49), 19.6% were borderline therapeutic (INR 1.5 -1.99), 32.6% were therapeutic (INR 2 -2.49) and 41.3% were high therapeutic (INR > 2.5).Case matched controls, not taking warfarin, had a median age of 71.9 years (13.04-88.3 years), not significantly different from the case group (P=0.277) and were followed up for a mean of 1.06 years. 46 (76.7%) of the patients in the warfarin arm were operated under a general anaesthetic, compared to 34 (56.7%) in the control arm. All local anaesthetics were administered via subtenons. As cases and controls were collected over the same period, surgical techniques were similar between both groups. Indications for vitrectomy included retinal detachment (n=30), vitreous haemorrhage (n=15), macula pucker (n=6), macula hole (n=5), dropped nucleus (n=3) and "other" (n=1) in both groups.

Mean visual acuity improved from logMar 1.3 to logMar 0.69 at final follow up in the control group.

Mean visual acuity improved from 1.1 to 0.8 in the warfarin group. There was no significant difference between either initial visual acuity (P=0.172) and final visual acuity (P=0.568). Mean difference in visual acuity, pre and post operation, was 0.469 (warfarin group) and 0.605 (control group), which was not significantly different (P=0.282).

Two cases of suprachoroidal haemorrhages occurred in the control group, whilst none occurred in the warfarin group (p=0.479). The first case of suprachoroidal haemorrhage occurred in an 87 year old arteriopathic male who had had a vitreous haemorrhage secondary to neovascularisation in response to an ischaemic hemiretinal vein occlusion. He developed a suprachoroidal haemorrhage during vitrectomy for the vitreous haemorrhage. This gentleman subsequently underwent cataract surgery and developed the same complication during this operation. The second case occurred in an 83 year old patient who had had a dropped nucleus during cataract surgery. A previtrectomy ultrasound examination showed evidence of suprachoroidal haemorrhage, suggesting that this was therefore not necessarily a complication of the vitrectomy.

Five patients in the warfarin group required further PPVs in comparison to 11 in the control group which was not significantly different. There were 30 cases of retinal detachments in either group: 28 Rhegmatogenous (RRD) and 2 tractional. All of the patients with tractional detachment presented with vitreous haemorrhage in addition. Twelve cases out of the 28 RRD in the warfarin group presented with vitreous haemorrhage compared to only four in the control group. Vitreous haemorrhage as a presenting feature of RRD was therefore significantly higher in the warfarin group (P= 0.04). Four of the patients in the warfarin group with RRD ultimately had persistent retinal detachments, compared to one in the control group at the last follow up (not statistically significant).

The online survey yielded responses from 59 out of 149 members contacted. 51 of these (86.4%) were at consultant level. With regard to warfarin use, 48 respondents (81%) would ask patients to withhold warfarin prior to vitreoretinal surgery based on the INR. 12 (25%) of these 48 would do so for patients with INR >3.5 with the rest stopping for lower INR values. 71% of respondents who withheld warfarin stratified the thrombotic risk of the patient by assessing risk factors such as a Cerebrovascular event within the last month, left ventricular dysfunction, age >75 years, hypertension, diabetes mellitus, history of venous thromboembolism, active cancer, antiphospholipid syndrome or any mitral valve caged ball [START_REF] Douketis | Perioperative anticoagulation management in patients who are receiving oral anticoagulant therapy: a practical guide for clinicians[END_REF] .

DISCUSSION

Warfarin therapy is commonly used in patients at risk of thrombo-embolism. Surgeons considering modifying warfarin treatment need to risk stratify their patients based on the indications for anticoagulation [START_REF] Douketis | Perioperative anticoagulation management in patients who are receiving oral anticoagulant therapy: a practical guide for clinicians[END_REF] . Stopping anticoagulation should be undertaken with care and due consideration must be given to the benefits of such a course of action. It has been suggested that the effects of platelets, drugs affecting platelet function and other factors influencing primary haemostasis have a greater impact on minute surgical trauma, as in ophthalmic surgery, than indirect anticoagulants which influence clotting factors, such as warfarin [START_REF] Pandolfi | Hemorrhages in Ophthalmology[END_REF] . Our previous experience has indeed suggested that antiplatelet medication clopidogrel, can be associated with to ophthalmic haemorrhagic complications [START_REF] Herbert | Haemorrhagic vitreoretinal complications associated with combined antiplatelet agents[END_REF] . Flaxel 8 reported 3 cases with haemorrhagic complications during vitrectomy which they attributed to aspirin but we have previously found no association of aspirin with increased complications in a cohort study [START_REF] Narendran | The effects of aspirin and warfarin therapy on haemorrhage in vitreoretinal surgery 10[END_REF] .

Tabendeh et al 10 reviewed 32 patients who suffered suprachoroidal haemorrhage during pars plana vitrectomy (PPV). None of these patients were taking warfarin, and only one patient was taking aspirin. Other reviews of 13 cases 11 and 12 cases 12 of suprachoroidal haemorrhage during PPV similarly did not show warfarin to being a significant risk factor in the aetiology. We had no difference in intraoperative suprachoroidal haemorrhage rates between patients taking warfarin and controls.

The rate of suprachoroidal haemorrhage in pars plana vitrectomy has not been clarified. Ghoraba 13 reviewed 3342 pars plana vitrectomies revealing 2 intraoperative suprachoroidal haemorrhages -a rate of 0.06%. Such a retrospective analysis is however very likely to under estimate the complication rate as small bleeds may not be recorded and it is likely that a prospective recording of the complication will uncover a higher rate. Obviously with such a presumed low rate for this complication, one would need a much larger powered study to give definitive conclusions with regard to the effect of warfarin. Subgroup analysis of rhegmatogenous retinal detachments in patients on warfarin, revealed a significantly higher proportion presenting with vitreous haemorrhage. The rate of vitreous haemorrhage as a presenting feature with retinal detachment is unclear. An increased rate associated with warfarin anticoagulation has not been previously demonstrated.

Our post operative follow up data did not include information specifically about recurrent vitreous haemorrhage. However only one patient who presented with vitreous haemorrhage required repeat PPV and this patient was not on warfarin.

Guidelines with regard to anticoagulation in ophthalmic surgery are vague. Within the UK, the Royal College of Ophthalmologists recommend that "for certain glaucoma, vitreoretinal, corneal graft, orbital and oculoplastic procedures the decision to modify anticoagulant therapy will be predicated upon the risk-benefit ratio" 3 . This is due to mixed reports of associated complications. In ophthalmic anaesthesia, the largest study [START_REF] Kallio | Haemorrhage and risk factors associated with retrobulbar/peribulbar block: a prospective study in 1383 patients[END_REF] found no increase in risk of haemorrhage in 76 patients on warfarin undergoing peribulbar or retrobulbar anaesthesia. With ocular adnexal procedures, surgeons tend to stop anticoagulation, especially prior to dacryocystorhinostomy procedures [START_REF] Parkin | Aspirin and warfarin therapy in oculoplastic surgery[END_REF] . Tuli et al [START_REF] Tuli | Delayed suprachoroidal hemorrhage after glaucoma filtration procedures[END_REF] has suggested that anticoagulation is a significant risk factor in the occurrence of delayed suprachoroidal haemorrhage after glaucoma filtration procedures. With cataract surgery, most studies have shown no increased risk of haemorrhagic complications in patients taking warfarin [START_REF] Katz | Study of Medical Testing for Cataract Surgery Team. Risks and benefits of anticoagulant and antiplatelet medication use before cataract surgery[END_REF][START_REF] Mccormack | Management of the anticoagulated patient for ophthalmic surgery[END_REF] .

For vitreoretinal procedures, attitudes towards anticoagulation have been cautious. 81% of the respondents to our survey of members of the British and Eire Association of Vitreo Retinal Surgeons (BEAVRS) suggested that they would withhold warfarin before vitreoretinal procedures. Interestingly 29% of these surgeons did not risk stratify their patients before withholding warfarin.

Although our early experience suggested that warfarin may have been associated with haemorrhagic complications [START_REF] Narendran | The effects of aspirin and warfarin therapy on haemorrhage in vitreoretinal surgery 10[END_REF] , our impression since has been that the risk of stopping anticoagulation outweighs any risks intra and post operatively.

Recent studies have also suggested this. Fu et al 19 In conclusion, this is the largest case controlled study of pars plana vitrectomy in patients taking warfarin. A significantly higher proportion of patients taking warfarin presented with vitreous haemorrhage in association with retinal detachment than controls. There was no increase in perioperative or post operative complications, in particular haemorrhage, in patients taking warfarin. It is our belief that the risks of stopping warfarin anticoagulation are greater than the perceived benefits of reduced complications. We do not recommend that surgeons withhold warfarin treatment before vitrectomy.

  had good visual outcomes in 25 patients undergoing Vitreoretinal surgery whilst taking warfarin. Only one highly myopic patient suffered a submacular haemorrhage, which resolved with a final visual acuity of 20/40. 44% of their cohort had an International Normalized Ration (INR) level of <2.0, leading them to suggest that patients INR should be reduced towards 2.0 to reduce haemorrhagic complications. In this study the INR was mean 2.3 and ranged up to 4.6 without a detectable increase in peri-operative complications. Additionally, 73.9% of the patients had therapeutic (INR 2-2.49) or high therapeutic (INR >2.5) anticoagulation. Dayani et al 20 analysed 54 patients who underwent pars plana vitrectomies or silicone oil removal whilst taking warfarin. They subdivided their patients into sub therapeutic (INR range 1.2-1.49), borderline (INR 1.5-1.99), therapeutic (INR 2.0-2.49) or high therapeutic (INR 2.5 or higher). They experienced no intraoperative haemorrhagic complications. Four patients with INRs of 1.28, 1.41 (sub therapeutic), 2.68 and 2.69 (high therapeutic) suffered post operative vitreous haemorrhage, all these resolved spontaneously requiring no further surgery. They advised that patients may safely undergo vitreoretinal surgery while maintaining therapeutic levels of warfarin anticoagulation.