Peter Haïssinsky

Kevin M Pilgrim

An algebraic characterization of expanding Thurston maps

Let f : S 2 → S 2 be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for f to be homotopic, relative to its postcritical set, to an expanding map g.

Introduction

Let T 2 = R 2 /Z 2 denote the usual torus and, using the standard basis, identify T 2 with the quotient H 1 (T 2 , R)/H 1 (T 2 , Z). Equip T 2 with the Euclidean metric. The following result is well-known:

Theorem 1.1 Suppose f : T 2 → T 2 is a self-covering map of the torus. Then f is homotopic through covering maps to an expanding map g : T 2 → T 2 if and only if the spectrum of f * : H 1 (T 2 , R) → H 1 (T 2 , R) lies outside the closed unit disk.

In this setting, one may take g to be the linear map on T 2 = H 1 (T 2 , R)/H 1 (T 2 , Z) induced by f * . One may further arrange so that the map g is a factor of f via a map π :

T 2 → T 2 (that is, π • f = g • π)
where π is homotopic to the identity, is monotone, has fibers not disconnecting T 2 , and, if f is itself expanding, is a homeomorphism. The proof is now standard: one lifts the identity map h 0 : T 2 → T 2 under f and g to obtain h 1 : T 2 → T 2 and a homotopy h t , t ∈ [0, 1] from h 0 to h 1 . By induction, lifting and concatenating the homotopies, one obtains a family h t , t ∈ [0, ∞), such that g • h t+1 = h t • f . Expansion of g implies the family h t converges uniformly to the desired semiconjugacy. Similar and much more general results hold, cf. [START_REF] Shub | Endomorphisms of compact differentiable manifolds[END_REF]Theorem 3].

In this work, we formulate and prove an analog of this theorem for a certain class of continuous maps f : S 2 → S 2 of the sphere to itself, called Thurston maps.

A Thurston map f : S 2 → S 2 is an orientation-preserving branched covering of the two-sphere of degree d ≥ 2 for which the necessarily nonempty postcritical set P f = P = ∪ n>0 f •n ({branch points}) is finite. These were originally introduced by Thurston [DH] as combinatorial objects with which to classify rational functions as dynamical systems on the Riemann sphere C. Two Thurston maps f, g are combinatorially equivalent if there exist orientationpreserving homeomorphisms h 0 , h 1 : (S 2 , P f ) → (S 2 , P g ) such that h 0 • f = g • h 1 and h 0 , h 1 are homotopic relative to P f . The relation of combinatorial equivalence can be thought of as conjugacy, up to isotopy relative to postcritical sets. Recently, Thurston maps have been investigated in connection with the study of finite subdivision rules, the classification of quasisymmetry classes of metrics on S 2 , and Cannon's conjecture; see [CFP], [BM], [START_REF] Haïssinsky | Pilgrim. Coarse expanding conformal dynamics[END_REF].

A formulation of a variant of Theorem 1.1 for Thurston maps requires the development of a notion of expansion for non-locally injective maps, algebraic/ homotopy-theoretic invariants of a self-map of a simply-connected surface, and the synthetic construction of an expanding model map from the algebraic data.

Expansion. Let f be a Thurston map. Since f has branch points, it can never be expanding with respect to a Riemannian metric on S 2 , nor can it be positively expansive. We therefore reformulate the notion of expansion in terms of contraction of inverse branches. Let U 0 be a finite cover of S 2 by connected open sets, and let U n denote the covering whose elements are connected components of f -n (U ), U ∈ U 0 . We say that f is expanding if the following condition holds.

Definition 1.2 The map f is expanding if the mesh of the coverings U n tends to zero as n → ∞. That is, for any finite open cover Y of S 2 by open sets, there exists N such that for all n ≥ N and all U ∈ U n , there exists Y ∈ Y with U ⊂ Y . Equivalently: the diameters of the elements of U n , with respect to some (any) metric, tend to zero as n → ∞.

Note that this is a purely topological condition. It is at present unknown whether every expanding Thurston map is topologically conjugate to a map which is smooth away from a finite set of points. Perhaps more surprisingly, it is unknown whether every expanding Thurston map is topologically conjugate to a map which preserves some complete length structure on S 2 . Algebraic invariants. Nekrashevych [START_REF] Nekrashevych | Self-similar groups[END_REF] introduced a collection of algebraic invariants associated to a very wide class of dynamical settings, including that of a Thurston map f . These may be unfamiliar to most readers; see §3 for details. As in [DH], associated to f is an orbifold structure O on S 2 . In order to have needed finiteness properties, we assume f has no periodic branch points, so that O is compact. Once a basepoint is fixed, there is a corresponding orbifold fundamental group π 1 (O), and a so-called virtual endomorphism φ : π 1 (O) π 1 (O), well-defined up to postcomposition by inner automorphisms. Roughly speaking, φ is the homomorphism induced by the inverse of f . The property of φ being contracting plays the role of the spectral condition on the induced map f * on homology in Theorem 1.1. Synthetic model map. When the virtual endomorphism φ is contracting, by choosing suitable defining data (basis; generators) one may construct a negatively curved self-similarity complex Σ. The construction is similar to that of the boundary of a hyperbolic group from a Cayley graph. The boundary at infinity J of Σ inherits a self-map f Σ : J → J . The dynamical system f Σ : J → J plays the role of the induced linear map g in Theorem 1.1.

Our main result is

Theorem 1.3 (Characterization of expanding Thurston maps) Suppose f is a Thurston map without periodic critical points. Then f is homotopic (through Thurston maps relative to its postcritical set) to an expanding map g if and only if the virtual endomorphism φ :

π 1 (O) π 1 (O) is contracting.
By applying an observation of Rees [START_REF] Rees | A partial description of parameter space of rational maps of degree two[END_REF]§ 1.4], we conclude that a Thurston map without periodic branch points is combinatorially equivalent to an expanding map if and only if its virtual endomorphism on the orbifold fundamental group is contracting.

The hypothesis of no periodic critical points cannot be dropped: the rational function f (z) = z 2 acting on the Riemann sphere has contracting virtual endomorphism, but is not homotopic to any expanding map: the nonnegative real axis is an invariant arc joining periodic points in its postcritical set, and by Proposition 2.1 this is an obstruction to expansion.

If the defining data for Σ and an open covering U 0 of S 2 is chosen in a suitable way, one obtains a semiconjugacy π : S 2 → J from f : S 2 → S 2 to f Σ : J → J playing the role of the map π introduced after Theorem 1.1. In the setting of Theorem 1.3, J is a sphere, and g is a limit of conjugates of f via a path of conjugacies h t (with now t ∈ [0, 1]) which gradually collapse the fibers of π to points. Formally, h t is a pseudo-isotopy.

A finite subdivision rule R on the sphere (in the sense of [CFP]) is, roughly, a Thurston map f : S 2 → S 2 together with a cell structure S R on S 2 and a refinement R(S R ) of S R such that f : R(S R ) → S R is cellular and a homeomorphism on each cell. By taking preimages, one obtains a sequence of refinements R n (S R ), n ∈ N. One says that R has mesh going to zero if the diameters of the cells at level n tend to zero as n tends to infinity. It has mesh going to zero combinatorially if for some n, (i) every edge of S R is properly subdivided in R n (S R ), and (ii) given a pair of disjoint edges e, e ′ of S R in the boundary of a tile, no tile t of R n (S R ) contains edges in both e and in e ′ . Thus mesh going to zero is clearly an invariant of topological conjugacy, while mesh going to zero combinatorially is a combinatorial property.

By appealing to recent work of Bonk and Meyer [BM], we obtain Theorem 1.4 (Equivalence of combinatorial expansion) Suppose f is a Thurston map without periodic branch points. The following conditions are equivalent.

1. The map f is equivalent to an expanding map, g.

2.

There exists m such that f m is equivalent to a map g which is the subdivision map of a subdivision rule R with mesh going to zero.

3. There exists m such that f m is equivalent to a map g which is the subdivision map of a subdivision rule R with mesh going to zero combinatorially.

4. The virtual endomorphism φ f is contracting.

Proof: That (1) implies (2) follows from [START_REF] Bonk | Expanding Thurston maps[END_REF]Theorem 1.2]; we remark that they do not require the absence of periodic branch points. That (2) implies ( 1) is a straightforward consequence of the definitions. The equivalence of ( 2) and ( 3) is [START_REF] Cannon | Finite subdivision rules[END_REF]Theorem 3.2]. Now suppose (3) holds. By [START_REF] Cannon | Subdivision rules and virtual endomorphisms[END_REF]Theorem 1.1], the virtual endomorphism φ g is contracting. This implies (4). That (4) implies ( 1) is Theorem 1.3.

In Theorem 1.4, the topological conjugacy class of the expanding map g is necessarily unique, by e.g. [START_REF] Bonk | Expanding Thurston maps[END_REF]Theorem 10.4]. A corollary of Theorem 1.4 is thus that the dynamics of an expanding Thurston map is determined by either of two pieces of combinatorial data: a subdivision rule with mesh going to zero combinatorially, or by an expanding virtual endomorphism. If a Thurston map f is already presented in terms of a subdivision rule R, there is a finite, easily checkable, necessary and sufficient condition for R to have mesh going to zero combinatorially [START_REF] Cannon | Subdivision rules and virtual endomorphisms[END_REF]Theorem 6.1]. If f is not so presented, one can try to verify the contraction of φ f instead. Bartholdi has pointed out that this can be done algorithmically, e.g. by combining a normal forms algorithm for writing elements of the orbifold fundamental group with the FR package [Bar] in the computational algebra program GAP.

In the paragraphs below, we outline in detail the reduction of the proof of Theorem 1.3 to a few key steps.

Necessity. The implications (1) =⇒ (2) =⇒ (3) =⇒ (4) have already been established in the above proof of Theorem 1.4. In §7, however, we provide a new, direct proof. The idea is that the group π 1 (O) acts on the universal cover p : O → O and that we may find a lift f -of "f -1 " i.e, a map satisfying f • p • f -= p. The action of the virtual endomorphism on π 1 (O) can be interpreted in terms of the action of f -on O. Assuming that f is expanding yields a metric on O which can be lifted to O in such a way that (a) π 1 (O) acts geometrically on O implying that π 1 (O) is quasi-isometric to O and (b) the action of f -is contracting. Bringing together these two facts leads to the contraction of the virtual endomorphism.

Sufficiency. Suppose the virtual endomorphism φ is contracting and f has no periodic branch points. Necessarily #P ≥ 3. For technical reasons, it is convenient later to assume #P > 3. When #P = 3, f is equivalent to a rational function which is chaotic on the whole sphere, by Thurston's characterization [DH], and such a map is expanding.

We first construct the semiconjugacy π : S 2 → J from f : S 2 → S 2 to the model map f Σ : J → J ; recall that J arises as the boundary of the hyperbolic selfsimilarity complex, Σ. Let U 0 be an open cover of S 2 by Jordan domains. By taking inverse images, there is another associated infinite onecomplex Γ; see [START_REF] Haïssinsky | Pilgrim. Coarse expanding conformal dynamics[END_REF] and §5. Using the absence of periodic branch points, we show (Theorem 6.1) that for suitable choices of the covering U 0 defining Γ and the defining data for Σ, there is a natural quasi-isometry Φ : Σ → Γ. Thus Γ is also hyperbolic. Extending Φ to the boundary by a map of the same name, the composition π := Φ -1 • π Γ is the desired semiconjugacy.

Next, we show that J is homeomorphic to S 2 , and that the model map f Σ is a Thurston map. A key role is played by the fact, established in [START_REF] Haïssinsky | Finite type coarse expanding conformal dynamics[END_REF], that f Σ : J → J is a branched covering that, as a topological dynamical system, is expanding and is of so-called finite type. In particular, the local degrees of iterates are uniformly bounded. We first develop ( §4) some dynamical consequences of the hypothesis that φ is contracting, e.g. that there are no so-called Levy cycles. Next, after showing π is monotone (Lemma 5.1), it suffices, by a classical theorem of Moore [START_REF] Daverman | Decompositions of manifolds[END_REF]Theorem 25.1], to show that the fibers of π do not disconnect S 2 . To see this, we use a case-by-case analysis of the possibilities for how such fibers separate and intersect P and the absence of Levy cycles and other related obstructions to expansion. We conclude that J is a sphere, that f Σ : J → J is an expanding Thurston map, and that π| P is injective.

It remains to construct an isotopy from f to an expanding map g. By [START_REF] Daverman | Decompositions of manifolds[END_REF]Theorems 13.4,25.1], the decomposition G of S 2 by the fibers of the semiconjugacy π : S 2 → J has the property of being strongly shrinkable: there is a one-parameter family of continuous maps h t : S 2 → S 2 , t ∈ [0, 1], called a pseudoisotopy, such that h 0 = id S 2 , h t is a homeomorphism for t ∈ [0, 1), h t | P f = id P f for all t, and the fibers of h 1 and those of π coincide. The induced homeomorphism S 2 → J conjugates f Σ to an expanding Thurston map g : S 2 → S 2 , and the family of maps f t , t ∈ [0, 1] defined as the unique continuous solution of h t • f = f t • h t , with f 1 = g, gives an homotopy through Thurston maps. The desired isotopy follows.

In the remainder of this work, we assume we are given a degree d ≥ 2 Thurston map f : S 2 → S 2 with postcritical set P .

Outline. In §2, we collect generalities on Thurston maps and related combinatorial objects, e.g. the pullback relations on curves and on arcs. In §3, we discuss algebraic invariants and define the selfsimilarity complex, Σ. In §4, we derive dynamical consequences from the contraction of the virtual endomorphism. §5 introduces the complex Γ associated to an open cover U 0 , while §6 gives the proof that Σ and Γ can be chosen so that the natural map between them is a quasi-isometry. § §7 and 8 conclude the proof of Theorem 1.3.

Remark. Nekrashevych [Nek2,Cor. 5.13] has shown that in great generality, a pair of maps ι, f : M 1 → M 0 satisfying analogous algebraic contraction conditions is combinatorially equivalent to a pair ι ′ , f ′ : M ′ 1 → M ′ 0 where the M ′ i are simplicial complexes, the maps ι ′ , f ′ are piecewise affine, and the pair is expanding. While the result applies to our setting, the dimensions of M ′ i can be larger than two, and it seems difficult to arrange for the spaces M ′ i to coincide.

Thurston maps 2.1 Curves and arcs

Here, we introduce the pullback relations on curves and arcs defined by a Thurston map, and formulate obstructions to expansion in terms of these relations.

Denote by C the set of free homotopy classes of essential, oriented, simple, closed, nonperipheral (that is, not homotopic into arbitrarily small neighborhoods of elements of P ) curves in S 2 -P ; we use the term curve for an element of C.

Let o denote the union of the homotopy classes in S 2 -P of curves which are either inessential or peripheral; we call such curves trivial. The pullback relation ←on C ∪ {o} is defined by setting o ←o and γ 1 ←γ 2 if and only if γ 2 is homotopic in S 2 -P to a component of f -1 (γ 1 ). Thus γ ←o if and only if some preimage of γ is inessential or peripheral in S 2 -P . If γ 1 ←γ 2 we write γ 1 ←֓ γ 2 if there is a representative of γ 2 mapping injectively to a representative of γ 1 . The relation defined by ←֓ we refer to as the univalent pullback relation.

We next define a similar pullback relation on certain arcs with endpoints in P . Let A denote the set of isotopy classes (fixing P ) of embedded arcs α ⊂ S 2 whose interiors are contained in S 2 -P and whose endpoints lie in P . A preimage α of α is, by definition, the closure of a connected component of the inverse image of the interior of α under f . If the endpoints of α lie in P , then α represents an element of A. Adjoining a symbol o to the set A to stand for the collection of arcs with one or both endpoints outside P , we obtain similarly a pullback relation on the set A ∪ {o}.

We recall that a multicurve is a finite subset of C represented by disjoint curves. A Levy cycle is a sequence (γ 0 , γ 1 , . . . , γ p-1 ) with γ i ←֓ γ i+1 mod p whose elements comprise a multicurve.

The result below is not needed for the proofs of the theorems, since it could be derived from Theorems 1.3 and Theorem 4.1 below. We include the statement and proof since they are simple and help build intuition. It applies to maps which may have periodic branch points.

Proposition 2.1 Suppose f is expanding. Given any curve C 0 or arc α 0 , there exists an integer N such that if C 0 ←֓ . . . ←֓ C n or α 0 ←-. . . ←α n is an orbit with C n , α n nontrivial, then n ≤ N . In particular, the univalent pullback relation on curves has no (Levy) cycles or wandering curves, and the pullback relation on arcs has no cycles and no wandering arcs.

Abusing terminology somewhat, we summarize the conclusion by saying that the univalent pullback relation on curves and pullback relation on arcs have no elements with arbitrarily long nontrivial iterates.

Proof: If f were expanding with respect to a length structure, the conclusion would follow immediately, since lengths would decrease exponentially as one pulls back. Without this additional structure, we use instead coverings by small open balls. Let U 0 , U i , i ∈ N be as in the definition of expanding. Choose m sufficiently large so that C 0 , α 0 are covered by some number, say L, of elements of U m and the union of these elements are contained in a regular neighborhood of C 0 , α 0 which meets P in the same manner as does C 0 , α 0 (that is, either not at all, or only at the endpoints of α 0 ) and which is homotopic, relative to P , to C 0 , α 0 . It follows that for each n, C n , α n are covered by the same number L of elements of U m+n . Hence if the diameters of the elements of the U i tend to zero with i, the integer n cannot be arbitrarily large.

Since the pullback relations are natural with respect to combinatorial equivalence, Proposition 2.1 gives a necessary condition for a Thurston map f to be equivalent to an expanding map. It is not, however, a sufficient condition; see §4 for examples.

Orbifolds

We review here the definition of the orbifolds associated to a Thurston map and their fundamental groups.

For

x, y ∈ S 2 let ν 0 (y) = lcm{deg(f n , x) : f n (x) = y} and ν 1 (x) = ν 0 (f (x))/ deg(f, x). For i = 0, 1 let Σ i = {x ∈ S 2 : ν i (x) > 1},

and let

O i be the orbifold whose underlying topological space is {x ∈ S 2 : ν i (x) < ∞} and whose weight function is ν i . The sets Σ i are called the singular sets of O i . Note that Σ 0 = P . There are no singular points of infinite weight if and only if there are no periodic branch points of f . In this case, the orbifolds O i are compact. The singular sets satisfy f

-1 (Σ 0 ) ⊃ Σ 1 . Set U 0 = S 2 -Σ 0 and U 1 = S 2 -f -1 (Σ 0 ). Then U 1 ⊂ U 0 and f : U 1 → U 0 is a covering map.
Let b 0 ∈ U 0 be a basepoint and b 1 ∈ f -1 (b 0 ) be one of its preimages. For i = 0, 1 let N i denote the normal subgroup of π 1 (U i , b i ) generated by the set of elements of the form g k , where g is represented by a simple closed peripheral loop γ surrounding a puncture x of U i , and the exponent k is the weight ν i (x) < ∞; if the weight is infinite, we do not add such a loop as a generator. The orbifold fundamental groups π

1 (O i , b i ) are by definition the quotient groups π 1 (U i , b i )/N i .
In what follows, we put

O = O 0 and G := π 1 (O, b 0 ).
As the (orbifold) universal covering deck group, G acts properly discontinuously on the universal covering of O.

Note that each element of C corresponds to a conjugacy class in G whose elements are of infinite order.

Algebraic constructions

In this section, we briefly summarize constructions and results of Nekrashevych [START_REF] Nekrashevych | Self-similar groups[END_REF], specializing to the case of Thurston maps.

The virtual endomorphism

In the setup of §2.2, let f * : π 1 (U 1 , b 1 ) → π 1 (U 0 , b 0 ) be the injective homomorphism induced by the covering f : U 1 → U 0 . Since f sends peripheral loops to peripheral loops, it follows from the definitions of the weight functions ν i that f * : N 1 → N 0 is an isomorphism. This observation and the "Five Lemma" of homological algebra imply that the homomorphism f * descends to a welldefined and injective map f * :

π 1 (O 1 , b 1 ) → π 1 (O 0 , b 0 ) = G. We denote the image group f * (π 1 (O 1 , b 1 )) by H; it has finite index in G. Let α : [0, 1] → U 0 be a path joining b 1 to b 0 and α * : π 1 (U 0 , b 1 ) → π 1 (U 0 , b 0 ) the induced isomorphism. Let N ′ 0 = α -1 * (N 0 ). Since N 0 is normal, the subgroup N ′ 0 is normal and is independent of the choice of path α. Set π 1 (O 0 , b 1 ) = π 1 (U 0 , b 1 )/N ′ 0 . Again, the map α * descends to a well-defined isomorphism α * : π 1 (O 0 , b 1 ) → π 1 (O 0 , b 0 ).
Since the inclusion ι : U 1 ֒→ U 0 sends peripheral loops to loops which are either peripheral or trivial, and since ν 0 (x) divides ν 1 (x) for all x, the induced map ι * :

π 1 (U 1 , b 1 ) → π 1 (U 0 , b 1 ) is surjective and sends N 1 to N ′ 0 .
It easily follows that the map ι * also descends to a surjective map ι * :

π 1 (O 1 , b 1 ) → π 1 (O 0 , b 1 ). Definition 3.1 The virtual endomorphism induced by f is the homomor- phism φ : H → G defined by φ = α * • ι * • (f * ) -1 .
By construction, the virtual endomorphism φ associated to f is surjective.

The virtual endomorphism depends on the choices of the basepoint b 0 , the preimage b 1 , and the homotopy class of the path α. Different choices yield virtual endomorphisms which differ by pre-and/or post-composition by inner automorphisms. Up to this ambiguity, the virtual endomorphism is an invariant of the homotopy class of f relative to P . A combinatorial equivalence between Thurston maps conjugates, up to inner automorphisms, their virtual endomorphisms; the property of being contracting is preserved.

For n ≥ 2, the nth iterate φ n is the homomorphism whose domain is defined inductively by domφ = H; domφ n = {g ∈ H : φ(g) ∈ domφ n-1 } and whose rule is given by iterating φ a total of n times.

Suppose S is a finite generating set for G. We denote by ||g|| the word length of g in the generators S.

Definition 3.2 The virtual endomorphism φ : H → G is called contracting if the contraction ratio ρ = lim sup n→∞ lim sup ||g||→∞ ||φ n (g)|| ||g|| 1/n < 1.
The contraction ratio of the virtual endomorphism φ is independent of the choices used in its construction.

As an algebraic object, the virtual endomorphism is rather straightforward to describe. The next subsection gives a less familiar, but more natural, construction.

The biset M

Suppose ℓ 0 , ℓ 1 : [0, 1] → S 2 -P are two arcs joining the basepoint b 0 to a common point in S 2 -P . We say ℓ 0 , ℓ 1 are homotopic in O if the loop formed by traversing ℓ 0 first and then the reverse of ℓ 1 represents the trivial element of G. We denote by M the set of all homotopy classes [ℓ] of arcs ℓ in O joining b 0 to one of the d elements of f -1 (b 0 ). For [ℓ] ∈ M we denote by z [ℓ] ∈ f -1 (b 0 ) the corresponding common endpoint. The group G acts on M via two commuting actions as follows. On the right, it acts by pre-concatenation:

ℓ • g := ℓ * g
where first g, then ℓ is traversed. Note that the right action is free and has d orbits. The left action is obtained by taking preimages of the loops under f :

g • ℓ := f -1 (g)[z [ℓ] ] * ℓ where first ℓ, then the lift f -1 (g)[z [ℓ] ] of g based at z [ℓ]
, is traversed. The set M equipped with these two commuting actions of G defines the G-biset associated to f .

In the next subsection, we show that choosing certain representatives of elements of M and representatives of generators of G gives rise to an abstract 1-complex Σ and a projection map π Σ : Σ → S 2 . Along the way, we reformulate the notion of contraction in terms of an action of G on the set of words in an alphabet X of size d.

The selfsimilarity complex Σ

Choose a bijection Λ :

X → f -1 (b 0 ), where X is a finite set of cardinality d. For each x ∈ X, let λ x be an arc in S 2 -P f joining b 0 to Λ(x). The collection {[λ x ] : x ∈ X} of elements of M is called a basis of the biset M = M f .
For n ∈ N, consider now the nth iterate f n of f . The orbifolds O = O f and O f n coincide, so G = G f = G f n . By lifting the arcs λ x under f n , we obtain an identification Λ : X n → f -n (b 0 ) × {n} of the set X n of words of length n in the alphabet X with the fiber f -n (b 0 ) × {n} and a corresponding basis for the G-biset M f n associated to the nth iterate of f . We denote by X * = ∪ n X n , with X 0 = {∅} consisting of the empty word. We write

|w| = n if w ∈ X n .
Suppose S is a generating set for G. For each s ∈ S, let γ s be a loop in S 2 -P based at b 0 representing s. By lifting loops under iterates of f , we obtain an action of G on X * preserving the lengths of words and acting transitively on each X n .

The basepoint b 0 , arcs λ x , x ∈ X, and loops γ s , s ∈ S, we refer to as defining data.

Since the right action is free, for each word u ∈ X * , and each g ∈ G, there are a unique word v ∈ X n and a unique h ∈ G with g.u = v.h; the element h is called the restriction 1 of g to u, and is denoted g| u . The interpretation in terms of defining data is as follows. Let λ u , λ v denote the concatenations of lifts of arcs corresponding to u and v, so that the endpoints of λ u , λ v correspond to u and v. Then v is the endpoint of f -n (g)[u] and h is represented by the loop which first traverses λ u , then f -n (g)[u], then the reverse of λ v .

By [START_REF] Nekrashevych | Self-similar groups[END_REF]Prop. 2.11.11], contraction of the virtual endomorphism is equivalent to the following: there is a finite set N such that for each g ∈ G, there is an integer n such that for all v ∈ X * satisfying |v| ≥ n, the restriction g| v ∈ N . In this case, one says the biset M is hyperbolic. This is an asymptotic condition: M f is hyperbolic if and only if M f m is hyperbolic for some m ≥ 1, since the contraction coefficients for φ and φ m are necessarily simultaneously less than 1.

Suppose now that defining data has been chosen. The selfsimilarity complex Σ is the infinite 1-complex whose vertex set is X * and whose edges, which come in two types, horizontal and vertical, are defined as follows. Given w ∈ X * , a horizontal edge, labelled s, joins w to s.w; a vertical edge, labelled x, joins w to xw. The selfsimilarity complex, with edges of length 1, is a proper geodesic metric space, and the local valence at each nonempty vertex is the same if S is symmetric.

The right shift σ : X * → X * determines a cellular self-map f Σ : Σ ′ → Σ where Σ ′ is the subcomplex determined by nonempty words. When M is hyperbolic, Σ is hyperbolic in the sense of Gromov, and the boundary J is compact. Since the virtual endomorphism φ is surjective, the action is recurrent, so J is also connected and locally connected [START_REF] Nekrashevych | Self-similar groups[END_REF]Theorem 3.5.1]. The map f Σ extends to a map of the boundary f Σ : J → J , yielding a dynamical system.

Although Σ is an abstract 1-complex, path-lifting of the defining data determines a map π Σ : Σ → S 2 ; see [Pil] and [START_REF] Nekrashevych | Self-similar groups[END_REF]Theorem 5.5.3]. If f is not expanding with respect to a length structure on S 2 , however, π Σ need not extend to a map J → S 2 .

4 Dynamical consequences of the hyperbolicity of M Suppose f has no periodic branch points. Proposition 2.1 gives necessary conditions for f to be expanding. The main result of this section is to derive the same conclusion from the algebraic assumption that φ is contracting; equivalently, that M is hyperbolic. Proof: The central ingredient in the proof is the following Lemma, which will be proved later.

Lemma 4.2 Suppose C 0 ←֓ C 1 . Given any g 0 ∈ G whose conjugacy class corresponds to C 0 , there exists g 1 ∈ G whose conjugacy class corresponds to C 1 and x ∈ X with g 0

• x = x • g 1 ∈ M.
To prove the Theorem, suppose C 0 ←֓ C 1 ←֓ . . . ←֓ C n . Induction and Lemma 4.2 implies that there exist g i ∈ G, x i ∈ X such that

g 0 • x 0 x 1 . . . x i = x 0 x 1 . . . x i • g i+1 , i = 0, . . . , n -1.
In particular, upon setting w = x 0 x 1 . . . x n ∈ X n+1 , we have

g 0 • w = w • g n ;
recall that by definition, g n = g 0 | w . Since M is hyperbolic, there is a finite set N ⊂ G such that g n = g 0 | w ∈ N whenever n = |w| is sufficiently large. This immediately implies that the number of distinct elements g n of C arising in this way is bounded and hence that the collection of curves {C n } arising in this way is also bounded, independent of n.

To rule out cycles, we use topology to simplify the algebra. Consider first the case of a Levy cycle of period one, i.e. a curve C for which C ←֓ C. By varying f within its homotopy class, and using the fact that C is oriented, we may assume f = id on a neighborhood of C. Take the basepoint b 0 for M to lie on C; it becomes a fixed point of f . Let g be represented by C, regarded as a loop based at b 0 . By choosing a basis for M with one arc equal to the constant path at b 0 and corresponding to an element x in the alphabet X, we find g • x = x • g. Induction shows then that for all n ∈ N and m ∈ Z we have that if w = xx . . . x ∈ X n then

g m • w = w • g m .
But this is impossible if M is hyperbolic: g has infinite order, and the identity above shows that for all n, g m | w = g m which cannot lie in N if m is sufficiently large.

The case of a Levy cycle of period larger than one can be ruled out by passing to an iterate.

Analyzing the pullback relation on arcs is reduced to that of the pullback relation on curves. An arc which does not eventually become trivial must have both endpoints in cycles of P . For such arcs α 0 , α 1 , since f has no periodic branch points, whenever α 0 ←α 1 , the boundaries of regular neighborhoods of α 0 and α 1 yield curves C 0 and C 1 for which C 0 ←֓ C 1 .

If #P > 3 then C 0 , C 1 must be nontrivial, and the eventual triviality of curve ←֓-orbits, already established, implies eventual triviality of arc ←-orbits. So assume #P = 3. By Thurston's characterization [DH], f is equivalent to a rational function without periodic critical points. Such a map is uniformly expanding with respect to a complete length structure, so such arc orbits are always eventually trivial.

We now turn to the proof of the Lemma, which is just an exercise in definitions.

Proof: Suppose defining data has been chosen. Choose a representative γ 0 for g, and let γ 0 be a preimage of γ 0 corresponding to C 1 ; it exists, by homotopy lifting and the fact that C 1 maps to C 0 by degree one. Let b be the unique preimage of b lying on γ 0 , and suppose x corresponds to b, i.e. Λ(x) = b. The loop given by γ 1 := λ x * γ 0 * λ x (where λ x is traversed first) represents an element g 1 ∈ G. Moreover, by the definition of the biset,

x • g 1 = [λ x * λ x * γ 0 * λ] = g 0 • x. Lemma 4.2
Examples showing the conditions in Propositions 2.1 and Theorem 4.1 are necessary but not sufficient can be found among Thurston maps induced by torus endomorphisms that are hyperbolic but not expanding.

Let T 2 = R 2 /Z 2 , let A be an integral matrix with determinant larger than one, and let F : T 2 → T 2 be the noninvertible endomorphism on the torus induced by A. There is a corresponding pullback relation on curves, defined analogously as in §2. In this setting, all preimages of nontrivial curves are nontrivial, are homotopic to each other, and map by the same degree. Proof: The sufficiency in the second statement is obvious. The proof will show both the first statement, and the necessity; it is a straightforward reinterpretation of the arguments of [Nek1, Prop. 2.9.2]. Use the standard basis to identify

Q 2 with H 1 (T 2 , Q) and π 1 (T 2 ) with H 1 (T 2 , Z) = Z 2 ,
so that A is the matrix for the induced map on rational and integral homology. Let B = A -1 . Observe that if C = C 0 ←֓ C 1 then in fact all preimages of C 0 map by degree one and the monodromy corresponding to C 0 is trivial. Thus if C 0 ←֓ C 1 ←֓ C 2 ←֓ . . . is an infinite sequence of related curves (either wandering, or cycling) then as an element of π 1 (T 2 ), the curve C 0 represents a nontrivial element of the kernel U < Z 2 < Q 2 of the corresponding iterated monodromy group action. The group U is B-invariant, so U ⊗ Q is a B-invariant Q-linear subspace of Q 2 ; let C = B| U ⊗Q . Then the characteristic polynomial q of C is monic, has integer coefficients, and is a factor of the characteristic polynomial of B. If 1/λ is a root of q then λ is an eigenvalue of A. Both λ and 1/λ are algebraic integers, hence the norm of λ is equal to 1. The condition det(A) > 1 implies the algebraic degree of λ must be equal to 1, so λ = ±1.

The endomorphism F : T 2 → T 2 is hyperbolic if A has no eigenvalues of unit modulus. Under the assumption det(A) > 1, the following is easily shown via straightforward computations.

Proposition 4.4 The endomorphism F is hyperbolic but not expanding if and only if tr

(A) -det(A) > 1.
The endomorphism F : T 2 → T 2 commutes with ±id T 2 and descends to a Thurston map f : S 2 → S 2 ; the corresponding orbifold O has signature (2, 2, 2, 2). The singular points are the images of the points of order at most 2.

Theorem 4.5 Suppose det(A) > 1 and tr(A)det(A) > 1, and let f : S 2 → S 2 be the induced Thurston map. Then 1. f is not equivalent to an expanding map, but

the univalent pullback relation on curves and the pullback relation on arcs have no elements with arbitrarily long nontrivial iterates.

Proof: Proposition, 4.4, Proposition 4.3, and Theorem 1.1 imply both conclusion (1) and the first part of (2). Suppose α ⊂ S 2 is an arc. The two preimages of α under the branched covering T 2 → S 2 , when concatenated, yield a nontrivial curve, C. If α 0 ←α 1 then for the corresponding curves in T 2 we have C 0 ←֓ C 1 . Thus nontrivial arc orbits lift to nontrivial univalent curve orbits.

The covering complex Γ

In this section, we begin by summarizing constructions and results of [HP1, Chapter 3], specializing to the case of Thurston maps. We associate to f and a suitable open covering U 0 another infinite one-complex, Γ, this time a graph. When it is hyperbolic, it has a natural boundary, and there is a natural map π Γ : S 2 → ∂Γ.

For convenience, equip S 2 with the usual Euclidean length metric. Let U 0 be a finite open cover of S 2 by small balls B for which #B ∩ P ≤ 1 with equality only when the corresponding point of P is in the center of B. For n ∈ N let U n be the open cover whose elements are connected components of sets of the form f

-n (U ), U ∈ U 0 . It is convenient to set U -1 := {S 2 }. If U ∈ U n we write |U | = n.
A key observation here is that if f has no periodic branch points, then there is an upper bound on the degrees of restrictions

f n : U → U , U ∈ U n , n ∈ N.
The vertex set of Γ is defined as ∪ n≥-1 U n , i.e. one vertex for each component. We denote the vertex corresponding to the unique element of U -1 by o. Horizontal edges join

U 1 , U 2 if |U 1 | = |U 2 | and U 1 ∩ U 2 = ∅. Vertical edges join U, V if ||U | -|V || = 1 and U ∩ V = ∅.
Γ is a graph with root o. We equip Γ with the graph path metric, so that each edge has length 1. Then Γ is a proper, geodesic metric space; let d denote the corresponding distance function. If f has no periodic branch points, then the valence of a vertex of Γ is uniformly bounded, independent of the vertex.

Let ε > 0 be small, and let ̺ ε : Γ → (0, ∞) be given by ̺ ε (x) = exp(-εd(x, o)). Upon redefining the lengths of curves via

ℓ ε (γ) := γ ̺ ε ds
we obtain an incomplete length metric space. Its completion compactifies Γ by adding a boundary at infinity ∂ ε Γ. Points in its boundary ∂ ε Γ are represented by vertical geodesic rays from the root. Given a ∈ S 2 , let U n (a), n = -1, 0, 1, 2, . . . be a sequence of open sets for which a ∈ U n ∈ U n . Then {U n } defines a geodesic ray and, therefore, a point of ∂ ε Γ. We obtain a well-defined surjective continuous map π Γ : S 2 → ∂ ε Γ. In particular, ∂ ε Γ is connected and locally connected. When Γ is hyperbolic and ε > 0 is chosen small enough, then ∂ ε Γ coincides with its visual boundary ∂Γ. We will always choose ε as above when Γ is hyperbolic.

Lemma 5.1 If Γ is hyperbolic, the projection π Γ : S 2 → ∂Γ is monotone, i.e. its fibers are connected.

Proof: For z ∈ S 2 and n ≥ -1, pick U n (z) ∈ U n such that z ∈ U n (z). It follows that the sequence (U n (z)) n defines a ray in Γ. Suppose π Γ (x) = π Γ (y) = ξ ∈ ∂Γ. From the definition of π and the hyperbolicity of Γ, there is a positive integer D such that for every n ∈ N, there exist a "chain"

U 0 n , U 1 n , . . . , U D n of elements of U n such that x ∈ U 0 n , y ∈ U D n , and U i-1 n ∩ U i n = ∅ for all i = 1, . . . , D. For n ∈ N let Y n = U 0 n ∪ . . . ∪ U D n . Then Y 1 , Y 2 , Y 3 , .
. . is a sequence of open connected sets containing both x and y. Set

K = ∩ k≥0 ∪ n≥k Y n .
It follows that K ⊂ S 2 is a continuum containing both x and y. We will prove that π Γ (K) = ξ and, therefore, that π -1

Γ (ξ) is connected. Fix z ∈ K. For any k ≥ 0, U k (z) ∩ K = ∅, so we may find n ≥ k such that Y n ∩ U k (z) = ∅. This means that d ε (π Γ (z), ξ) ≤ d ε (π Γ (z), U k (z)) + d ε (U k (z), {U j n , 0 ≤ j ≤ D}) +diam ε {U j n , 0 ≤ j ≤ D} + d ε (ξ, {U j n , 0 ≤ j ≤ D}) e -εk + e -εk + e -εn + e -εn e -εk .
Since k is arbitrary, we obtain that π Γ (z) = ξ.

From [START_REF] Daverman | Decompositions of manifolds[END_REF]Theorem 5], we have that the decomposition of S 2 by the fibers of π Γ is upper-semicontinuous, and the quotient space is homeomorphic, via the map induced by π Γ , to ∂Γ.

The map f induces a cellular map f Γ : Γ ′ → Γ, where Γ ′ is the subgraph on vertices at level ≥ 1. Since it sends vertical rays to vertical rays, it induces a continuous map f Γ : ∂Γ → ∂Γ. Since it sends sufficiently small round balls to round balls [START_REF] Haïssinsky | Pilgrim. Coarse expanding conformal dynamics[END_REF]Prop. 3.2.2], it is an open map; it is easily seen to be at most d-to-1, so it is closed as well. In particular, the local degree function

deg(f Γ , ξ) := inf W {#f Γ -1 (ζ) ∩ W : ζ ∈ f Γ (W ), ξ ∈ W open} is well-defined.

Coincidence of quasi-isometry types

Recall that if X, Y are metric spaces, a function Φ :

X → Y is a quasi-isometry if there exist constants λ ≥ 1, c ≥ 0, b ≥ 0 such that 1 λ |x -x ′ | -c ≤ |Φ(x) -Φ(x ′ )| ≤ λ|x -x ′ | + c
and such that |y -Φ(X)| ≤ b for all y ∈ Y ; here |xx ′ | = d X (x, x ′ ), etc.. For geodesic metric spaces, hyperbolicity is preserved under quasi-isometries.

Recall that a choice of defining data defines a projection π Σ : Σ → S 2 and an identification map Λ : X * → ∪ n≥0 f -n ({b}). The main result of this section is Theorem 6.1 Let f : S 2 → S 2 be a Thurston map without periodic critical points. Then there exist defining data for the associated selfsimilarity complex Σ and the covering graph Γ and a level-preserving quasi-isometry Φ : Σ → Γ such that, for every vertex v ∈ Σ 0 , we have

Λ(v) ∈ Φ(v) and Φ • f Σ = f Γ • Φ.
Remark 6.2 If we admit periodic branch points, then the same proof will show that Σ is quasi-isometric to a covering complex Γ built from a cover of the complement in the sphere of a neighborhood of the set of periodic branch points.

Proof: We first choose the defining data and list some elementary properties. For convenience, we endow S 2 with its ordinary Euclidean length metric.

Choosing the defining data. Choose the basepoint b ∈ S 2 -P to be a regular point. We may choose representative loops γ s , s ∈ S, so that each is simple, peripheral, and has the property that for some common ǫ > 0, the closed ǫ-neighborhood N ǫ (γ s ) is contained in a closed annulus A contained in S 2 -P surrounding on one side at most one point of P . The peripheral condition and the assumption that there are no periodic critical points implies that there exists N 0 such that for all n, deg(f n : A → A) ≤ N 0 for all preimages A of A under f -n . In this situation, if the above degree is say δ, then in the graph Λ, for each vertex v at level n with Λ(v) ∈ A, there is a closed horizontal edge-path of length δ whose edges are labelled by s corresponding and which corresponds to the monodromy at level n induced by γ s .

We may also choose representative arcs λ x ⊂ S 2 -P, x ∈ X, so that each is simple and has the property that for some common ǫ > 0, the closed ǫ-

neighborhood N ǫ (λ x ) is a closed disk D contained in S 2 -P . Thus deg(f n : D → D) = 1 for all preimages D of D under f -n .
For w ∈ X * , let λ xw be the lift of λ x under f |w| which starts at π Σ (w).

Since f is a branched covering, given δ > 0, there is a constant r(δ) > 0 with r(δ) → 0 as δ → 0 such that for any ball B(y, δ) ⊂ S 2 , any preimage of B under f has diameter < r(δ). It follows that we may choose the defining data U 0 for the covering graph Γ to consist of a finite collection of small open balls satisfying the following properties.

Given a set E ⊂ S 2 , we denote by

U n (E) = {U ∈ U n : E ∩ U = ∅}.
1. ∀U ∈ U 0 , P ∩ ∂U = ∅, #P ∩ U ≤ 1, and if equality holds, then the postcritical point in U is at the center of U ;

2. ∀U ∈ U 0 , U 0 (U ) ∩ P ≤ 1; 3. ∀x ∈ X, U 0 (λ x ) ⊂ N ǫ (λ x ); 4. ∀s ∈ S, U 0 (γ s ) ⊂ N ǫ (γ s ); 5. ∀U 0 ∈ U 0 and U 1 ∈ U 1 with U 0 ∩ U 1 = ∅, the union U 0 ∪ U 1 is contained in a ball B such
that the ball 2B of twice the radius meets at most one point of P ;

6. ∀p ∈ S 2 there exists an arc α joining the basepoint b to p such that U 0 (α) is contained in a disk which meets P in at most one point.

7. There is exactly one element of U 0 containing the basepoint b, and this basepoint is the center of the ball.

Basic properties. (BP1)

. Condition 1 implies that every element U ∈ U n has a unique preferred point c U (we call it the center of U ) which is the unique preimage of the center c of the ball f •n (U ) under f n .

(BP2). Conditions 1 and 2 imply the following. Since there are no periodic branch points, there is a uniform upper bound q on the local degree of all iterates of f . Suppose

f n : U i → U i ∈ U 0 , i = 0, 1. If U 0 , U 1 ∈ U n intersect, then at most one of U 0 , U 1 meets a branch point of f n ; say it is U 0 . Then there are q ′ ≤ q -1 elements U 2 , . . . , U q ′ of U n , each a preimage of U 1 , such that f •n : U 0 ∪ U 1 ∪ . . . ∪ U q ′ → U 0 ∪ U 1 is proper and of degree q ′ .
(BP3). From condition 6, we define A 0 to be the finite set of curves α U :

[0, 1] → S 2 joining the center c U = α U (0) of U to b = α U (1)
. Let also A n denote all the lifts of these curves under f n : these are curves joining every center of U ∈ U n to a point of f -n (b). Since every α U is an arc contained in a disk which intersects P in at most one single point, there is some L independent of n and U ∈ U n such that α U is covered by at most L elements of U n .

(BP4). Condition 7 is merely to make the exposition in what follows more convenient, for with this choice we will define Φ : Σ → Γ in a simple way.

Strategy of the proof. Define Φ : V (Σ) → V (Γ) by setting Φ(w) to be the unique set U ∈ U |w| which contains Λ(w). We extend Φ over the edges to all of Σ as follows. Suppose γ s represents a horizontal edge at level 0 (a loop) in Σ. In the sphere, as this edge is traversed, it passes through a sequence of elements of U 0 ; we pick such a sequence, and associate to this a horizontal loop Φ(γ s ) in Γ at level 0, parameterized at constant speed. We define Φ to send the edge of Σ corresponding to γ s to this loop Φ(γ s ). Similarly, each horizontal edge e = e s at level n in Σ 1 corresponds to a lift under f n of a unique edge γ s , s ∈ S, at level zero; hence by lifting the chosen covering for γ s we obtain again a curve Φ(e) in Γ joining the images under Φ of their extremities by definition.

Similarly, for x ∈ X, we associate to the arc λ x an edge-path Φ(λ x ) in Γ corresponding to a covering of the chosen arc α Λ(x) by elements of U 0 concatenated with the vertical edge of Γ 1 between the elements U ∈ U 0 and V ∈ U 1 containing Λ(x). We again extend Φ to the other vertical edges equivariantly by path-lifting.

We also define a coarse inverse Ψ :

V (Γ) → V (Σ) by setting Ψ(U ) = w, where w ∈ V (Σ) is the vertex such that Λ(w) is the endpoint of α U . Note that Ψ • Φ is the identity on V (Σ) since α U is trivial for preimages of U b ∈ U 0 containing b, and Φ • Ψ maps a vertex U ∈ Γ 0 to the endpoint of α U .
To prove that Φ is a quasi-isometry, we will show that both Φ and Ψ are Lipschitz on the set of vertices and that the image of Φ is cobounded. The conclusion will follow since Ψ • Φ is the identity on Σ 0 . To prove that each map is Lipschitz, we will show that the distance between the images of the two extremities of an edge is uniformly bounded.

Φ is Lipschitz. Let us consider the horizontal case first. Suppose v, w ∈ V (Σ), |v| = |w| = n, and v, w are joined by a horizontal edge, so that w = s.v for some s ∈ S. Equivalently, there is a lift γ s of γ s under f n joining π Σ (v) to π Σ (w). Note that f n : γ s → γ s is 1 -1 away from the extremities. The curve γ s is covered by #U 0 (γ s ) elements of U 0 . The lift γ s is covered by at most #U 0 (γ s ) + 1 elements of U n . By the definition of the metric in Γ, this implies that |Φ(v) -Φ(w)| hor ≤ (#U 0 (γ s ) + 1). Note that we may assume that

#U 0 (γ s ) ≤ #U 0 -1. Since |Φ(v) -Φ(w)| Γ ≤ |Φ(v) -Φ(w)| hor , we conclude that |Φ(v) -Φ(w)| Γ ≤ C hor := #U 0 .
Now let us consider the vertical case. Suppose |v| = n + 1, |w| = n, and v and w are joined by a vertical edge in Σ. By definition, w = xv for some x ∈ X, and the points Λ(w) and Λ(v) in S2 are joined by a lift of the arc λ x under f -n starting at Λ(v). The arc λ x is covered by #U 0 (λ x ) elements of U 0 , each contained in the neighborhood N ǫ (λ x ), which is a disk in S 2 -P . It follows that λ x is covered by the same number #U 0 (λ x ) of elements of U n . By the definition of the metric in Γ, this implies that

|Φ(v) -Φ(w)| Γ ≤ #U 0 (λ x ) + 1. Hence |Φ(v) -Φ(w)| Γ ≤ C ver := max{#U 0 (λ x ) : x ∈ X} + 1.
We conclude that Φ is C = max{C hor , C ver }-Lipschitz on V (Σ).

The image of Φ is cobounded in Σ. The proof of this follows along exactly the same lines. For any U ∈ U n , the curve α U ∈ A n joins c U to Φ • Ψ(U ), whose distance is bounded by L according to the basic property (BP3).

Realization of the covering complex Γ. Proving that Ψ is Lipschitz will require some more data. The covering complex Γ comes equipped with a selff Γ : Γ \ B(X, 1) → Γ. We will find a continuous map Γ → S 2 which conjugates f Γ and f where defined. First, if e = (U 1 , U 2 ) denotes an edge with U 1 , U 2 ∈ U 0 , we consider β e to be the geodesic (in the round metric) joining their centers; recall that U j are spherical balls at level 0, so that β e ⊂ U 1 ∪ U 2 . Denote by B 0 this finite set of curves, and let B n denote the lifts of the elements of B 0 under f n .

If e = (U 0 , U 1 ) is a vertical edge of Γ with U 0 ∈ U 0 and U 1 ∈ U 1 , we may find an arc η e joining their centers inside U 0 ∪ U 1 . Let H 0 denote the sets of arcs defined for every such edge, and for each n ≥ 0, define H n to be the set of their lifts under f n . Ψ := Φ -1 is Lipschitz. Let us first consider an edge e joining two sets U 1 , U 2 at the same level n ≥ 0. Let γ = α U 1 * β e * α U 2 . The image f n ( γ) is a closed curve based at b which is homotopically nontrivial in S 2 \ P since γ is not closed. This curve is also the concatenation of three curves belonging to A 0 ∪ B 0 which is a finite set. Let

C hor := max γ ||g(γ)||
be the maximum word length of the corresponding group elements g(γ), where γ varies over this finite collection of loops just constructed. We conclude that

|Ψ( U 1 ) -Ψ( U 2 )| Σ ≤ C hor .
The case of vertically adjacent vertices proceeds similarly in spirit. Suppose 2 U 1 , U 0 are vertically adjacent in Γ, with | U 1 | = | U 0 | + 1 = n + 1, and let e = ( U 1 , U 0 ). The curves α U 1 and α U 2 join their centers to elements π Σ (w) and π Σ (v), w, v ∈ V (Σ). Set

γ = λ w * α U 1 * η e * α U 0
which is a curve joining to points from π Σ (V (Σ)) at the same level n. It follows that γ = f n ( γ) is a loop based at b of the form

γ = λ x * α 1 * η * α 0 .
Hence it belongs to a finite set of curves (see Figure 1).

|Ψ( U 1 ) -Ψ( U 0 )| Σ ≤ C ver .
We conclude that Ψ is max{C hor , C ver }-Lipschitz, and the proof is complete.

Suppose now that f has no periodic critical points, and that the virtual endomorphism φ is contracting. Then Σ is hyperbolic; Theorem 6.1 implies that Γ is also hyperbolic. Hence if ǫ is chosen sufficiently small, the Floyd boundary ∂Γ coincides with the Gromov boundary, and Φ induces a homeomorphism Φ : J → ∂Γ which conjugates the shift f Σ : J → J to the induced map f Γ : ∂Γ → ∂Γ. It follows that

π := Φ -1 • π Γ : S 2 → J
gives a semiconjugacy of branched coverings from f : S 2 → S 2 to f Σ : J → J . The remainder of the proof will use finiteness properties of the dynamical system f Σ : J → J to analyze the possibilities for the fibers of π.

We conclude this section with a technical result needed for the analysis in the next section.

By [START_REF] Haïssinsky | Finite type coarse expanding conformal dynamics[END_REF]Thm. 6.13], f Σ : J → J is a branched covering of degree d. Given E ⊂ S 2 a continuum and a component E of f -1 (E), we define deg(f :

E → E) = min{deg(f : U → U ) : E ⊂ U }
where U is a connected open neighborhood of E. Lemma 6.3 Suppose f Σ ( ξ) = ξ, and set E = π -1 (ξ), E = π -1 ( ξ). Then deg(f :

E → E) = deg(f Σ , ξ).
In particular, the inverse image under f of a fiber of π is a disjoint union of fibers of π, and the degrees of f on these fibers sum to d.

Proof: We first prove deg(f : E → E) ≥ deg(f Σ , ξ). Let U be a small open connected neighborhood of E and U the pullback containing E. Since the decomposition of X by the fibers of π is upper-semicontinuous, and f Σ is a branched covering, we may assume that the images

V = π(U ), V = π( U ) are neighborhoods of ξ, ξ, respectively, that U = π -1 (V ), U = π -1 ( V ), and that deg(f Σ , ξ) = deg(f Σ : V → V ). Since f Σ : J → J is a branched covering, the set of branch values is nowhere dense, so there exists ζ ∈ V which is not a branch value of f Σ . Hence #f -1 Σ (ζ) ∩ V = deg(f Σ , ξ). It follows easily that deg(f Σ , ξ) is a lower bound for deg(f : U → U ).
The proof of the inequality deg(f :

E → E) ≥ deg(f Σ , ξ) is similar. Sup- pose V, V are respectively neighborhoods of ξ, ξ and set U = π -1 (V ), U = π -1 ( V ). Choose ζ ∈ V which is not a branch value of f Σ . Then #f -1 Σ (ζ) = d,
and since f is a branched covering of degree d, this implies that for each z ∈ π -1 (ζ), #f -1 (z) = d also. Hence no two elements of f -1 (z) belong to the same fiber of π. We conclude #f

-1 Σ (ζ) ∩ V = #f -1 (z) ∩ U and so deg(f : U → U ) is a lower bound for deg(f Σ , ξ).
7 Proof of necessity Proposition 7.1 Let f : S 2 → S 2 be a postcritically finite topological cxc map of the sphere. Then its virtual endomorphism is contracting.

Let O be the orbifold modeled on (S 2 , P ), ν : O → N its weight function, and as usual choose a basepoint b ∈ S 2 \ P . Taking an iterate if necessary, we may assume that f (b) = b. The group G = π 1 (O, b) acts naturally as the deck transformation group of a Galois (regular) ramified universal orbifold covering p : O → S 2 such that deg(p, x) = ν(p( x)) for all x ∈ O.

It follows that there exists a lift of "f

-1 ", that is, a map f -: O → O such that f •p• f -= p. Since f (b) = b, we may choose f -so that a given b ∈ p -1 (b) is fixed by f -: ( O, b) f - ←-( O, b) p ↓ ↓ p (O, b) f -→ (O, b)
To see this: at lower-left, replace O with O 1 , so that the bottom arrow becomes an orbifold covering. Lift the orbifold structure on O 1 at lower-left under p to obtain a new orbifold structure O 1 on the underlying space of O so that p : O 1 → O 1 is an orbifold covering. The map f -is then a universal orbifold covering of O 1 .

In this paragraph, we define a metric d on O with respect to which f - becomes a uniform contraction. Let U 0 be a covering as in §5, and endow S 2 with the metric d ε described in § 5. Choose M > 0 so small that (i) any d ε -ball on S 2 of radius 2M is contained in some element of U 0 ; from the choice of U 0 it follows that (ii) any such ball meets P in at most one point. Let D denote the set of Jordan domains U of diameter at most M . Condition (ii) implies that if U ∈ D, then the inverse image p -1 (U ) is a collection { U k } of pairwise disjoint Jordan domains in O invariant under the covering group action of G (their closures might meet, but we don't care), and the restriction of p to any domain U k is proper. Let U be one such component and set, for x, y ∈ U , q U (x, y) = inf diam ε p(K) Proof: (Proposition) Let x, y ∈ O be such that d(x, y) ≤ M , then we may find a continuum K containing {x, y} such that d(x, y) = diam ε p(K). Note that p(K) is contained in an element of U 0 . Therefore, for any n ≥ 0, (p • f n -)(K) is contained in an element of U n and so d( f n -(x), f n -(y)) e -εn . It follows easily that, for any pair of points x, y without any restriction on their distance, we have

d( f n -(x), f n -(y)) e -εn d(x, y) . This implies that lim sup n→∞ sup x,y∈ O d( f n -(x), f n -(y)) d(x, y) 1/n = e -ε < 1. Now, since f (b) = b is
||φ n (g)|| ||g|| 1/n ≤ lim sup n→∞ sup x,y∈ O d( f n -(x), f n -(y)) d(x, y) 1/n < 1.

Proof of sufficiency

From the introduction, recall that the key step is to establish Proposition 8.1 Let f : S 2 → S 2 be a Thurston map with postcritical set P and with hyperbolic biset M. The semiconjugacy π : S 2 → J is injective on P , and no fiber of π separates S 2 .

The main ingredients in the proof of the claim are [HP2, Lemma 6.11, Theorem 6.15], which assert that the dynamical system f Σ : J → J is expanding, irreducible, and of topologically finite type. Finite type implies that there is a positive integer p such that for all ξ ∈ J and all n > 0, deg(f n Σ , ξ) ≤ p. Thus in particular f Σ has no periodic branch points. Irreducible in this context is equivalent to the fact that given any nonempty open set W , there is an integer n with f n Σ (W ) = J .

We remark that these properties imply that f Σ : J → J , as a topological dynamical system, is a so-called finite type topologically coarse expanding conformal (cxc) system; cf. [START_REF] Haïssinsky | Pilgrim. Coarse expanding conformal dynamics[END_REF]. This forces strong constraints on the fibers of π. We prove the claim by a case-by-case analysis of the possibilities. We remark that there is much overlap with the flavor of the arguments in [PL].

A continuum E ⊂ S 2 we call

• special if E ∩ P = ∅; • type III if S 2 -E has at least 3 components that intersect P • type II if S 2 -E has exactly 2 components that intersect P • type I if S 2 -E has exactly 1 component that intersects P
The map f sends fibers of π onto fibers of π. Suppose E = π -1 (ξ) for some ξ ∈ J .

(i) If E is special, then f k (E) is special for all k ≥ 0, and since P is finite, E is eventually periodic. Moreover, there are only finitely many such fibers.

(ii) It is easily verified that if f (E) is not special, then neither is E, and the type of f (E) is at least the type of E.

(iii) If E is type II and non-special then it determines a homotopy class of unoriented simple closed curve in S 2 -P .

(iv) Since distinct fibers are disjoint, there can be only finitely many type III fibers, and only finitely many homotopy classes of curves arise from type II fibers.

(v) If U is a component of S 2 \ E, then π(U ) is a component of J \ {ξ} and is open. This follows from the facts that π is monotone, continuous and onto and that S 2 is locally connected.

The proof of the following technical lemma is straightforward.

Lemma 8.2 Given a continuum E ⊂ S 2 , there is a "thickening" E of E given by an open regular neighborhood of E with the following properties.

• the closure of E is a compact surface with boundary.

• E has the same type as E.

• E ∩ P = E ∩ P .

• for each component E of f -1 (E), we have deg(f : E → E) = deg(f :

E → E)
, where E is the component of f -1 ( E) containing E.

at most one critical value of f . Therefore, f -1 ( U ) is a union of Jordan domains. One of them contains E, hence U = S 2 \ W . It follows that f (U ) ⊂ U . But U contains a component of S 2 \ E, so π(U ) has interior by (v) and f Σ (π(U )) ⊂ π(U ). which contradicts the irreducibility of f Σ .

4. Subcase type II E periodic up to homotopy. We assume that E is type II and f k (E) is homotopic to E rel. P . If deg(f k , E) > 1 then deg(f nk , E) = deg(f nk Σ , ξ) → ∞ as n → ∞, violating f Σ being finite type. So f k | E is injective, and after thickening, we extract a Levy cycle as above.

No preperiodic separating fibers nor type II and type III fibers. Let E be a preperiodic separating fiber. If all its iterates are nonspecial, then we would obtain a periodic separating fiber by (ii) and (iv), which is impossible according to Lemma 8.3. Therefore, there is some n ≥ 1 such that E ′ = f n (E) is special and periodic by (i). By Lemma 8.3 again, E ′ does not separate S 2 and intersects P at a single point, so the inclusion U := S 2 -(P ∪E ′ ) ֒→ S 2 -P is a homotopy equivalence. Therefore f -n (U ) is homotopy equivalent to the connected set S 2f -1 (P ). Hence U is connected, E is nonseparating, and #E ∩ P ≤ 1.

Let us now prove that any type II and type III fibers have to cycle up to homotopy, which will establish their nonexistence.. Since fibers cannot intersect, there can be only finitely many of them up to homotopy, cf. (iv). Let E be of type at least II. If its iterates are non special, then they remain of type at least II by (ii), and hence have to cycle up to homotopy by (iv). Otherwise, one iterate becomes special, hence is preperiodic as well.

In order to complete the proof, we must prove that there are no type I fiber. We have already ruled out preperiodic ones.

No wandering separating type I fiber. All the possible remaining separating type I fibers are nonspecial and hence form a totally invariant set of S 2 . We use the same notation as in Lemma 8.3: given such a fiber E, let W E ⊂ W E contain P , U E = S 2 \ W E and U = S 2 \ W . Therefore, f -1 ( U f (E) ) is a union of Jordan domains disjoint from P . One of them contains E, hence U E . It follows that f (U E ) ⊂ U f (E) . This implies that no preimage of P can enter the sets U E : this contradicts again the irreducibility of f Σ . Therefore, no type I fiber separates the sphere.

Proof: (Proposition 8.1) It follows from the discussion above that fibers of π are connected, do not separate the sphere, and meet P in at most one point.

Proof: (Theorem 1.3) Suppose the virtual endomorphism φ is contracting and f has no periodic branch points. Necessarily #P ≥ 3. For technical reasons, it is convenient later to assume #P > 3 (see the introduction for the case #P = 3).

According to Theorem 6.1, we may consider defining data for Σ and Γ so that both graphs are quasi-isometric. Since φ is contracting, it follows that Γ is hyperbolic and that the quasi-isometry defines a homeomorphism Φ : J → ∂Γ.

The map π = Φ -1 • π Γ : S 2 → J semiconjugates f to f Σ . Moreover, Lemma 5.1 and Proposition 8.1 imply that π is monotone, injective on P and that its fibers do not separate S 2 . By Moore's Theorem [START_REF] Daverman | Decompositions of manifolds[END_REF]Theorem 25.1], the quotient J is homeomorphic to the sphere. The induced map f Σ : J → J is thus an expanding Thurston map.

It remains to construct a homotopy from f to an expanding map g. By [START_REF] Daverman | Decompositions of manifolds[END_REF]Theorems 13.4,25.1], the decomposition G of S 2 by the fibers of the semiconjugacy π : S 2 → J has the property of being strongly shrinkable: there is a one-parameter family of continuous maps h t : S 2 → S 2 , t ∈ [0, 1], called a pseudoisotopy, such that h 0 = id S 2 , h t is a homeomorphism for t ∈ [0, 1), h t | P f = id P f for all t, and the fibers of h 1 and those of π coincide. The induced homeomorphism S 2 → J conjugates f Σ to an expanding Thurston map g : S 2 → S 2 , and the family of maps f t , t ∈ [0, 1] defined as the unique continuous solution of h t • f = f t • h t , with f 1 = g, gives a homotopy through Thurston maps. The proof of the sufficiency in Theorem 1.3 is complete.

Theorem 4. 1

 1 Suppose f has no periodic branch points. If M is hyperbolic, the univalent pullback relation on curves and the pullback relation on arcs have no elements with arbitrarily long nontrivial iterates.

Proposition 4. 3

 3 If a curve C has arbitrarily long nontrivial iterates under the univalent pullback relation, then either C ←֓ C or C ←֓ -C, where -C denotes the curve C with the opposite orientation. This occurs if and only if the matrix A has an eigenvalue λ equal to +1 or -1.

Figure 1 :

 1 Figure 1: Showing |Ψ( U 1 ) -Ψ( U 0 )| Σ = O(1). The disk f (U 1 ) is not shown.

  be the maximum word length of the corresponding group elements g(γ), where γ varies over the finite collection of loops just constructed. Thus

  fixed, we may define the virtual endomorphism φ by choosing the connecting path α (as in the definition given in §3.1) to be the constant path. Hence, g ∈ domφ n if and only if f n -(g( b)) belongs to the orbit G. b, and in this case, Φ(φ n (g)) = f n -(g( b)). From the preceding Lemma 7.2, the metrics || • || and d are comparable, so we obtain

	lim sup n→∞	lim sup ||g||→∞

sometimes the term "section" is used.

Here, it is convenient to index things using zero and one, instead of one and two as above.

where the infimum is over all continua K ⊂ U joining x to y. Now suppose x, y in O. We define the metric d on O by d(x, y) = inf j q U j (x j , x j+1 ) where the infimum is taken over every chain x = x 1 , x 2 , . . . , x ℓ = y joining x to y, with the property that for each j, there exists U j ∈ D and a component U j of p -1 (U j ) such that {x j , x j+1 } ⊂ U j . We note that if d(x, y) ≤ M , then we may find U ∈ D such that d(x, y) = q U (x, y). It follows that the infimum in the definition of d is actually attained.

The function d defines a metric and the group G acts by isometries on ( O, d).

Lemma 7.2 The map Φ : g → g( b) defines a quasi-isometry from G to ( O, d).

Proof: We already know that G is finitely generated and that it acts on O by isometries, properly discontinuously and cocompactly.

We Then Φ(G) is cobounded, S is a finite generating set and there are constants [START_REF] Bridson | Metric spaces of nonpositive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]Prop. 8.19]. The only point to prove is the reverse inequality.

Fix g ∈ G and let us consider x 0 , . . . , x n with x 0 = b and x n = g( b) and let U 1 , . . . , U n be such that

Let N be the integer part of 4 d( b, g( b))/D. For any 1 ≤ k ≤ N , let y k be the first

In what follows, given a fiber E = π -1 (ξ), the symbol E denotes a thickening of E satisfying the conditions of Lemma 8.2. The following paragraphs rule out certain types of fibers.

Lemma 8.3 (no periodic separating fibers) No periodic E separates S 2 , and there are no type II fiber homotopic rel P to one of its iterates. Moreover, if E is periodic and special, then #E ∩ P = 1 and E ∩ P is periodic of the same period as E.

Proof: Taking an iterate if necessary, we may assume that E is fixed. Since f Σ has no periodic branch points, deg(f

and P is forward invariant and finite, we may assume that


Christophe Noll

Gregory Lacraz

Jan Ehses

Josiane Coulaud

Danièle Bailbe

Jean-Louis Paul

Bernard Portha

Françoise Homo-Delarche

Nathalie Janel
email: janel@univ-paris-diderot.fr

Franc ¸oise Homo-Delarche

Pr N Janel

Early reduction of circulating homocysteine levels in Goto

Keywords: homocysteine, cystathionine beta synthase, glutathione, type 2 diabetes, Goto-Kakizaki rats, prediabetes

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List of abbreviations : CBS, Cystathionine beta synthase; DTNB, 5,5'-dithiobis-(2nitrobenzoic acid) ; GK, Goto-Kakizaki; GSH, glutathione; Hcy, Homocysteine; HFS, highfat-sucrose; STZ, streptozotocin; tHcy, total Hcy; ZDF, Zucker diabetic fatty

Introduction

Cardiovascular diseases are the major cause of death in diabetic patients [START_REF] Brand | Diabetes, intermittent claudication, and risk of cardiovascular events[END_REF].

Hyperhomocysteinemia, defined by high plasma homocysteine (Hcy) levels, is now well recognized as an independent risk factor for the development of cardiovascular diseases [START_REF] Welch | Homocysteine and atherothrombosis[END_REF].

Hcy, an intermediate in the sulfur amino acid metabolism [START_REF] Selhub | Homocysteine metabolism[END_REF], is metabolized by remethylation to methionine or transsulfuration to cysteine. Cystathionine beta synthase (CBS), the first enzyme involved in the transsulfuration pathway, catalyzes the condensation of Hcy with serine to form cystathionine. Cystathionine is subsequently hydrolyzed to form cysteine and cysteine, in turn, can be incorporated into protein, or used to synthesize the antioxidant glutathione (GSH) via the γ-glutamyl cycle. GHS is either used as cellular antioxidant or exported and becomes a substrate for γ-glutamyl transpeptidase. The action of γ-glutamyl transpeptidase leads to the formation of cysteinylglycine, which is subsequently cleaved by dipeptidase to form cysteine and glycine. Cysteine can then be transported into the cell and thus be again used for GSH synthesis.

The relationship between diabetes and plasma Hcy levels has been studied in both humans and rodents. Plasma Hcy levels have been found to be higher in type 2 diabetic patients than in healthy subjects and also, among type 2 diabetic patients, in obese than in nonobese individuals [START_REF] Konuko Lu | Plasma total homocysteine concentrations in ğ obese and non-obese female patients with type 2 diabetes mellitus; its relations with plasma oxidative stress and nitric oxide levels[END_REF]. By contrast, reduced serum Hcy levels were described in type 2 diabetic patients without cardiovascular complications or diabetic nephropathy [START_REF] Mazza | Reduced serum homocysteine levels in type 2 diabetes[END_REF]. Chronic hyperglycemia was thus suggested to affect its renal excretion, or accelerate hepatic transsulfuration secondary to insulin disorders [START_REF] Mazza | Reduced serum homocysteine levels in type 2 diabetes[END_REF]. Moreover, subjects with insulin resistance had significantly lower serum Hcy levels compared with non-insulin resistant subjects [START_REF] Rosolová | Unexpected inverse relationship between insulin resistance and serum homocysteine in healthy subjects[END_REF].

Animal models of type 2 diabetes were then used to better understand the relationship between insulin resistance and/or hyperglycemia and circulating Hcy levels. In long-term fed rats with a high-fat-sucrose (HFS) diet, which induces hyperinsulinemia, insulin resistance

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and obesity but no hyperglycemia, plasma Hcy levels were higher than in control rats fed at low-fat, low-carbohydrate diet [START_REF] Fonseca | Effects of a high-fat-sucrose diet on enzymes in homocysteine metabolism in the rat[END_REF]. The Zucker diabetic fatty (ZDF) rat is a spontaneous type 2 diabetic obese animal model characterized by a prediabetic insulin-resistant stage followed by a rapid transition to frank hyperglycemic stage due to relative hypoinsulinemia. In the ZDF rat, plasma Hcy levels were reduced at both prediabetic and diabetic stages, compared to respective control groups [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF]. Thus, conflicting data were observed in 2 rat models of type 2 diabetes with obesity.

The Goto-Kakizaki (GK) rats is another spontaneous and well characterized but nonobese model of type 2 diabetes [START_REF] Goto | Production of spontaneous diabetes by repetition of selective breeding[END_REF]. In this model, a normoglycemic period, which associates low insulinemia but elevated whole body insulin sensitivity precedes onset of hyperglycemia around weaning [START_REF] Movassat | Follow-up of GK rats during prediabetes highlights increased insulin Noll et al. action and fat deposition despite low insulin secretion[END_REF]. The aim of the present study was to analyze Hcy metabolism at 4 crucial steps of the GK rat life: in normoglycemic 2-week-old neonates and at 1 month of age (onset of hyperglycemia), and 2 of 3 months of age (aggravation of insulin resistance) [START_REF] Portha | The GK rat beta-cell: a prototype for the diseased human beta-cell in type 2 diabetes?[END_REF].

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Materials and methods

Rats

All Animal care was conducted in accordance with internal guidelines of the French Agriculture Ministry for animal handling. Rats were housed in a controlled environment with unlimited access to food and water on 12-h light/dark cycle. Number of rats and suffering were minimized as possible. All animal experiments were conducted on age-matched male Goto-Kakizaki (GK) and nondiabetic (control) Wistar rats from our laboratory [START_REF] Portha | The GK rat beta-cell: a prototype for the diseased human beta-cell in type 2 diabetes?[END_REF]. The GK line was established by repeated inbreeding from Wistar rats selected at the upper limit of normal distribution for glucose tolerance [START_REF] Goto | Production of spontaneous diabetes by repetition of selective breeding[END_REF][START_REF] Portha | The GK rat beta-cell: a prototype for the diseased human beta-cell in type 2 diabetes?[END_REF].

Blood, tissue collection and assays

Male rats were weighed, killed by decapitation and blood samples were collected and placed on ice immediately. Basal fed morning glycemia was determined with a glucometer. Plasma was isolated by centrifugation at 2500 g for 15 min at 4°C. Liver were harvested, snap-frozen and stored at -80°C until use. Plasma total Hcy (tHcy), defined as the total concentration of Hcy after quantitative reductive cleavage of all disulfide bonds, and total glutathione (GSH)

were assayed by using the fluorimetric high-performance liquid chromatography method as previously described [START_REF] Fortin | Measurement of homocyst(e)ine in the prediction of atherosclerosis[END_REF]. The inter-and intra-assay coefficients of variation for mean tHcy level were 4.2% and 6.3% respectively and the linearity was from 1 to 100 µM [START_REF] Ducros | Methods for homocysteine analysis and biological relevance of the results[END_REF]. Serum insulin was assayed by ELISA (Rat insulin Elisa, cat 10-1124-01, Mercodia, Uppsala, Sweden) [START_REF] Ehses | IL-1 antagonism reduces hyperglycemia and tissue inflammation in the type 2 diabetic GK rat[END_REF].

CBS Enzyme Activity assays

Determination of CBS activity was assayed on 400 µg of total proteins obtained from liver samples, as described [START_REF] Miller | Folate-deficiency induced homocysteinaemia in rats: disruption of S-adenosylmethionine's co-ordinate regulation of homocysteine metabolism[END_REF]. Proteins were incubated for 1 h at 37°C with 1 mM of

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propargylglycine, 0.2 mM of pyridoxal phosphate, 10 mM of L-serine, 10 mM of DL-Hcy, 0.8 mM of SAM, using DTNB (5,5'-dithiobis-(2-nitrobenzoic acid)) based-assay. All the chemical products were obtained from Sigma (Sigma-Aldrich, France).

Data analysis

Statistical analysis was done with one-way ANOVA followed by Student's unpaired t-test using Statview software. In both cases, Student-Newman-Keuls tests were used for multiple pairwise comparaisons. The results are expressed as mean ± SD. Data were considered significant when p<0.05.

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Results and discussion

Body weight and metabolic parameters in GK rats as a function of age

As previously described [START_REF] Movassat | Follow-up of GK rats during prediabetes highlights increased insulin Noll et al. action and fat deposition despite low insulin secretion[END_REF], male GK rat body weights were significantly lower compared to age-matched Wistar rats (Fig. 1A). Circulating glucose levels were similar in GK and Wistar rats at 2 weeks of age (Fig. 1B), but significantly increased in GK rats from 1 month of age (weaning) (Fig. 1B). Indeed, GK rats become hyperglycemic around weaning, when the diet shifts from lipid-enriched maternal diet to carbohydrate-enriched laboratory chow [START_REF] Movassat | B-cell mass depletion precedes the onset of hyperglycemia in the GK rat: a genetic model of non-insulin-dependent diabetes mellitus[END_REF].

In 2-week-old GK rats, basal insulinemia was significantly decreased (Fig. 1C). However, because GK neonates exhibit a transient increase in whole body insulin sensitivity as previously described [START_REF] Movassat | Follow-up of GK rats during prediabetes highlights increased insulin Noll et al. action and fat deposition despite low insulin secretion[END_REF], they stay normoglycemic until around weaning. Then, GK rats displayed mildly elevated but not statistically significant insulin levels at 1 month of age and the hyperinsulinemia became statistically significant at 2 and 3 months of age, once insulin multi-organ resistance and especially liver insulin resistance has been established [START_REF] Portha | The GK rat beta-cell: a prototype for the diseased human beta-cell in type 2 diabetes?[END_REF][START_REF] Bisbis | Insulin resistance in the GK rat: decreased receptor number but normal kinase activity in liver[END_REF].

Thus, in addition to HFS-fed rats and ZDF rats, GK rats appeared to be a suitable but nonobese model to analyze Hcy metabolism in association with alterations of glucose metabolism.

Hcy metabolism in GK rats as a function of age

Figure 2A shows that GK plasma tHcy levels were only about 45% of that found in agematched Wistar rats in both the prediabetic and diabetic periods. These low plasma tHcy levels are in agreement with those described in both type 1 STZ-induced diabetic rat model [START_REF] Jacob | Effects of streptozotocininduced diabetes and of insulin treatment on homocysteine metabolism in the rat[END_REF] and spontaneous type 2 diabetic obese ZDF rats [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF].

From the prediabetic stage onwards (Fig. 2B), GK rat also showed a falling trend in plasma GSH levels, the effect being only significant at 2 weeks and 3 months of age. GSH is the most abundant low-molecular-weight thiol and plays a key role in the cellular defense against

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oxidative stress. GSH is a reference of antioxidant and its antiatherogenic properties have been demonstrated in humans and rodents [START_REF] Morrison | Serum glutathione in adolescent males predicts parental coronary heart disease[END_REF][START_REF] Robin | Opposite effect of methionine-supplemented diet, a model of hyperhomocysteinemia, on plasma and liver antioxidant status in normotensive and spontaneously hypertensive rats[END_REF]. Here, it should be underlined that decreased plasma GSH levels are indicative of a compromise protection against oxidative stress, which is already present in prediabetic GK animals [START_REF] Lacraz | Diabetic beta-cells can achieve selfprotection against oxidative stress through an adaptive up-regulation of their antioxidant defenses[END_REF].

The hepatic activities of several enzymes involved in the remethylation and transsulfuration pathways that play a role in the removal of Hcy were found to be increased in ZDF rat liver [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF]. Previous results emphasized a direct effect of insulin in repressing CBS expression in both human and rat cultured hepatocytes [START_REF] Ratnam | Hormonal regulation of cystathionine beta-synthase expression in liver[END_REF]. Therefore, to examine whether the lowering of GK rat plasma Hcy levels, reflecting cellular and mainly liver production, could result from increased Hcy catabolism through the first step of the transsulfuration pathway, we measured the CBS enzyme activity. Given that the majority of dietary methionine is metabolized in the liver, which contributes to much of the plasma Hcy levels, we analyzed the hepatic enzyme activity [START_REF] Stead | Characterization of homocysteine metabolism in the rat liver[END_REF]. Fig. 2C shows that the hepatic CBS activity was significantly elevated in GK versus Wistar rats and, once again, the effect was present from the prediabetic stage onwards.

Thus, low plasma tHcy levels were associated with increased hepatic CBS enzyme activity in GK rat. However, increased CBS activity does not imply increased transsulfuration flux. This flux is determined by the rate at which methionine enters into hepatic metabolism, which will be largely determined by dietary methionine consumption. In the experimental mouse hyperhomocysteinemia induced by methionine-enriched diet, increase plasma Hcy levels are accompanied by proportional increased GSH levels, without modification of cysteine and cysteinylglycine levels [START_REF] Likogniani | Thiol-compounds metabolism in mice, rats and humans: comparative study and potential explanation of rodents protection against vascular disease[END_REF]. The major effect of increased CBS activity is to decrease the steady-state Hcy concentration at which transsulfuration occurs, and then contributes to the decrased plasma Hcy level. Therefore, decreased circulating GSH levels may reflect decreased circulating Hcy levels in GK rats.

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Similar data demonstrate low plasma tHcy levels with elevated hepatic CBS activity in 3 different rat models of diabetes, STZ-induced type 1 diabetes [START_REF] Jacob | Effects of streptozotocininduced diabetes and of insulin treatment on homocysteine metabolism in the rat[END_REF] and type 2 diabetes, either obese ZDF [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF] or nonobese GK rats (the present study). Thus, the association between low plasma tHcy levels and high CBS activity does not depend on the type of diabetes and the presence of obesity, and is observed during the short period of prediabetes in both type 2 models.

The repression of hepatic CBS expression and activity by insulin has been demonstrated in STZ-induced type 1 (insulin-dependent) diabetic rats [START_REF] Jacob | Effects of streptozotocininduced diabetes and of insulin treatment on homocysteine metabolism in the rat[END_REF][START_REF] Ratnam | Hormonal regulation of cystathionine beta-synthase expression in liver[END_REF]. In this model, increased hepatic CBS enzyme activity was associated with elevated CBS mRNA expression and insulin treatment normalized hepatic CBS expression and activity. Type 2 diabetes is characterized by relative lack of insulin and/or the resistance to the action of insulin. Insulin resistance is a multisite dysfunction that involves, in particular, the liver [START_REF] Groop | Glucose and free fatty acid metabolism in non-insulin-dependent diabetes mellitus. Evidence for multiple sites of insulin resistance[END_REF]. Insulin signaling in liver is critical in regulating glucose homeostasis and maintaining normal hepatic function. At 1 month of age, GK rats already exhibit increased hepatic glucose production specifically linked to early liver insulin resistance. These data highlight a possible primary role of the liver defect, and not simply a late consequence of chronic hyperglycemia [START_REF] Picarel-Blanchot | Impaired insulin secretion and excessive hepatic glucose production are both early events in the diabetic GK rat[END_REF]. Thus, the lack of functional insulin or decreased insulin sensitivity (insulin resistance) in the liver is probably at play very early, during prediabetes in both the ZDF and GK type 2 models, resulting in increased hepatic CBS activity and low plasma Hcy levels. Indeed, during the prediabetic stage, GK rats are mostly characterized by very low insulin levels, while ZDF rats already exhibit insulin resistance concomitant with hyperinsulinemia [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF]. After diabetes onset, GK rats exhibit progressive hepatic insulin resistance and hyperinsulinemia [START_REF] Bisbis | Insulin resistance in the GK rat: decreased receptor number but normal kinase activity in liver[END_REF], while ZDF rats become hypoinsulinemic in addition to being insulin resistant [START_REF] Wijekoon | Homocysteine metabolism in ZDF (type 2) diabetic rats[END_REF]. Therefore, the lack of insulin or the resistance to insulin action rather than hyperglycemia appear to increase hepatic CBS activity and consequently decrease plasma Hcy levels.

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Because of anti-inflammatory effects of insulin [START_REF] Dandona | Insulin as an anti-inflammatory and antiatherogenic modulator[END_REF], decreased insulin levels in GK rats from end of fetal life onwards [START_REF] Calderari | Is defective pancreatic beta-cell mass environmentally programmed in Goto-Kakizaki rat model of type 2 diabetes?: insights from crossbreeding studies during suckling period[END_REF] might be partly responsible for various neonatal oxidative stress-related signs, like oxidized state of GSH in red blood cells associated with low circulating GSH and elevated chemokine levels, and high islet ROS concentration and alterations in GSH-and thioredoxin-related gene expression [START_REF] Lacraz | Diabetic beta-cells can achieve selfprotection against oxidative stress through an adaptive up-regulation of their antioxidant defenses[END_REF]. We suggest that early insulin deficit may cause systemic and tissue oxidative stress in GK rats. This hypothesis is in agreement with: 1) the situation of prediabetic insulin-resistant (i.e., with a relative lack of insulin) in ZDF rats; 2) the 'unexpected' inverse relationship between insulin resistance and serum Hcy in healthy subjects [START_REF] Rosolová | Unexpected inverse relationship between insulin resistance and serum homocysteine in healthy subjects[END_REF]. Thus, hypohomocysteinemia associated with low systemic levels of GSH (the latter has antiatherogenic properties in humans and rodents [START_REF] Morrison | Serum glutathione in adolescent males predicts parental coronary heart disease[END_REF][START_REF] Robin | Opposite effect of methionine-supplemented diet, a model of hyperhomocysteinemia, on plasma and liver antioxidant status in normotensive and spontaneously hypertensive rats[END_REF]) rather than hyperhomocysteinemia might be a risk factor for CVD in insulin resistance and diabetes in humans and rodents.

By contrast, hyperhomocyteinemia in type 1 and type 2 diabetic patients appears to be dependent on the presence of nephropathy [START_REF] Bostom | Hyperhomocysteinemia in chronic renal disease[END_REF][START_REF] Ozmen | Association between homocysteinemia and renal function in patients with type 2 diabetes mellitus[END_REF]. GK rat does not spontaneously develop kidney disease. However, secondary injurious mechanisms such as hypertension induces progressive nephropathy [START_REF] Janssen | hypertension sureimposed on type II diabetes in Goto Kakizaki rats induces progressive nephropathy[END_REF]. By contrast, long-term (from 6 months to 2 years) HFS-fed rats, which are obese, normoglycemic, hyperinsulinemic, and insulin resistant, display hyperhomocysteinemia and lower hepatic CBS enzyme activity and mRNA expression than normally fed rats [START_REF] Fonseca | Effects of a high-fat-sucrose diet on enzymes in homocysteine metabolism in the rat[END_REF]. These HFS-fed rats exhibit hypertension, and their hyperhomocysteinemia probably result from kidney alterations which develop rapidly after a few weeks of diet [START_REF] Roberts | HMG-CoA reductase, cholesterol 7alpha-hydroxylase, LDL receptor, SR-B1, and ACAT in diet-induced syndrome X[END_REF][START_REF] Altunkaynak | The effects of high-fat diet on the renal structure and morphometric parametric of kidneys in rats[END_REF]. Altered CBS activity in HSF-fed rat kidney may cause hyperhomocysteinemia and diminish hepatic CBS activity [START_REF] Hamelet | Effects of catechin on homocysteine metabolism in hyperhomocysteinemic mice[END_REF]. Accordingly, hyperhomocysteinemia in obese and/or diabetic patients might more probably reflect kidney alteration.

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In conclusion, the spontaneous nonobese type 2 diabetic GK rat, characterized by an early deficit in functional beta-cell mass, shows decreased levels of circulating Hcy and a falling trend in its major byproduct antioxidant GSH from the prediabetic stage onwards.

Concomitant increased hepatic CBS activity emphasizes the role of insulin in Hcy and antioxidant GSH metabolism.

Fig. 1 .

 1 Fig. 1. Age-dependent changes in body weight, and in basal plasma glucose and insulin levels in GK and Wistar rats. Postnatal growth (A). Glucose (B) and insulin (C) levels were determined.*p<0.05, **p<0.005, ***p<0.0005 versus age-matched Wistar group.

Fig. 2 .

 2 Fig. 2. Age-dependent changes in plasma Hcy and GSH levels, and relative hepatic CBS activity. Hcy (A) and GSH (B) levels were determined in plasma. (C) CBS activity assay was performed on extracts from liver of individual rats. CBS activity values are normalized from age-matched Wistar rats. *p<0.05, *p<0.005, *p<0.0005 versus age-matched Wistar group.

ACCEPTED MANUSCRIPTNoll et al.

Acknowledgments

Christophe Noll and Gregory Lacraz are supported by a fellowship from the Ministère de l'Enseignement supérieur et de la Recherche.