Pierre-Antoine Guihéneuf

Emmanuel Militon

Parabolic isometries of the fine curve graph of the torus

In this article we finish the classification of actions of torus homeomorphisms on the fine curve graph initiated by Bowden, Hensel, Mann, Militon, and Webb in

This is made by proving that if f ∈ Homeo(T 2 ), then f acts elliptically on C † (T 2 ) if and only if f has bounded deviation from some v ∈ Q 2 \ {0}. The proof involves some kind of slow rotation sets for torus homeomorphisms.

Introduction

The fine curve graph C † (S) of a closed surface S was introduced by Bowden, Hensel, and Webb [START_REF] Bowden | Quasi-morphisms on surface diffeomorphism groups[END_REF] to give a counterpart of the classical curve graph adapted to the study of the group of all homeomorphisms of S. More precisely, the classical curve graph C(S) has vertex set the isotopy classes of essential simple closed curves on S, with edges between pairs of isotopy classes that can be realized disjointly (a slight modification is needed for genus 1 surfaces). Note that the natural action of a homeomorphism on curves quotients down to an action of the mapping class group Map(S) on the curve graph C(S) by isometries. The Gromov hyperbolicity (or equivalently δ-hyperbolicity) of C(S), showed by Masur and Minsky [MM99, MM00], then implies numerous geometric and algebraic properties of the mapping class group Map(S) (e.g. [BBF15, BKMM12, BM08, DGO16, Mah11]).

In this paper we will focus on the case of the torus S = T2 . Let us give the precise definition of the fine curve graph in this context. Definition 1. The fine curve graph on the torus T 2 is the graph C † (T 2 ) whose vertices are essential1 simple loops. There is an edge between two vertices α and β if and only if the loops α and β have at most one intersection point.

As a consequence of the Gromov hyperbolicity of the classical curve graphs for punctured surfaces, it was proved in [START_REF] Bowden | Quasi-morphisms on surface diffeomorphism groups[END_REF] that the fine curve graph C † (S) is Gromov hyperbolic. This enables the authors to use large scale geometry techniques to study Homeo(S) via its action on C † (S). As an application, they prove that, for any closed surface S of genus ≥ 1, stable commutator length and fragmentation norm on Homeo 0 (S) are unbounded, answering a question posed by Burago, Ivanov, and Polterovich [START_REF] Burago | Conjugation-invariant norms on groups of geometric origin, Groups of diffeomorphisms in honor of Shigeyuki Morita on the occasion of his 60th birthday[END_REF].

In the same way as the mapping class group Map(S) acts on C(S) by isometries, the whole homeomorphism group Homeo(S) acts on C † (S) by isometries. Gromov has classified isometries of Gromov hyperbolic spaces [Gro87, paragraph 8], [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF], according to the asymptotic translation length, defined for an isometry g of a Gromov hyperbolic space X as

|g| X = lim n→+∞ 1 n d X x, g n (x) .
It is a standard exercise to see that this limit exists and is independent of x. This independence immediately implies that the asymptotic translation length is a conjugacy invariant of isometries of X. Gromov classification is then as follows: for g an isometry of a Gromov hyperbolic space, g is • Hyperbolic if the asymptotic translation length is positive;

• Parabolic if the asymptotic translation length is zero but g has no finite diameter orbits, and

• Elliptic if g has finite diameter orbits.

There is an equivalent reformulation of this trichotomy in terms of fixed points on the Gromov boundary of X, but we do not require this point of view in the present work. While there is no mapping class acting parabolically on 2 C(S), the situation is much richer for the action of homeomorphisms on C † (S): in [BHM + 22], the authors prove that there are homeomorphisms of T 2 acting parabolically on C † (T 2 ). They also initiate a classification of actions of homeomorphisms on C † (T 2 ) in terms of rotational behaviour: they give a criterion of hyperbolicity in terms of rotation set, as well as examples of parabolic and elliptic homeomorphisms. In the present article, we complete their work to give a complete classification of actions of homeormorphisms of T 2 in terms of rotational behaviour.

Definition 2. Let v ∈ R 2 \{0}. We say that f ∈ Homeo(T 2 ) has bounded deviation from direction v if there exists ρ ∈ R 2 and a lift f : R 2 → R 2 of f such that | f n (x)-x-nρ, v | is bounded from above, uniformly in x ∈ R 2 and n ∈ N.

Observe that this definition does not depend on the chosen lift f of f . Recall the classification of homotopy classes for torus homeomorphisms: if f ∈ Homeo(T 2 ), then f of f 2 is homotopic either to the identity, or to a Dehn twist (defined in Section 1.1), or to a linear Anosov automorphism.

A homeomorphism homotopic to a linear Anosov automorphism has unbounded deviation in any direction, and a homeomorphism homotopic to a Dehn twist has unbounded deviation in any direction but possibly one. To see this, consider one point of R 2 and some nontrivial integer translate of it, and look at the deviation between these two points (at some point one has to use the fact that the eigendirections of linear Anosov automorphisms have irrational slope).

Note that if f ∈ Homeo(T 2 ) is homotopic to identity and has bounded deviation from two non-collinear directions, then it has bounded deviation from any direction.

The main theorem of this article is the following.

Theorem A. Let f ∈ Homeo(T 2 ). Then f acts elliptically on C † (T 2 ) if and only if f has bounded deviation from some v ∈ Q 2 \ {0}.
Combined with the results of [BHM + 22], this theorem implies a complete classification of the action of homeomorphisms on C † (T 2 ) in terms of rotational behaviour.

We denote by Homeo 0 (T 2 ) the connected component of identity in the group Homeo(T 2 ); it coincides with the set of homeomorphisms that are homotopic to identity. We denote the rotation set of f by ρ( f ). It is a compact convex subset of the plane capturing the rotational behaviour of the homeomorphism (see Section 1.1 for precise definitions).

Corollary 3. Let f ∈ Homeo 0 (T 2 ). Then

• f acts hyperbolically on C † (T 2 ) if and only if ρ( f ) has nonempty interior;

• f acts parabolically on C † (T 2 ) if and only if ρ( f ) is a segment of irrational direction, or ρ( f ) is a segment of rational direction not passing through a rational point, or f is a pseudo-rotation3 with unbounded deviation from any v ∈ Q 2 \ {0};

• f acts elliptically on C † (T 2 ) if and only if ρ( f ) is a segment of rational direction passing through a rational point, or f is a pseudo-rotation with bounded deviation from some v ∈ Q 2 \ {0}.

Let us explain how to deduce this corollary from Theorem A. The first point is [BHM + 22, Theorem 1.3]. Hence it suffices to distinguish homeomorphisms that act elliptically from those that do not.

By Passeggi and Sambarino [START_REF] Passeggi | Deviations in the Franks-Misiurewicz conjecture[END_REF], any homeomorphism whose rotation set is a segment of rational slope not passing by a rational point, has unbounded deviation, and hence by Theorem A acts parabolically.

Moreover, by Dávalos [START_REF] Dávalos | On annular maps of the torus and sublinear diffusion[END_REF], any homeomorphism whose rotation set is a segment of rational direction v passing through a rational point has bounded deviation in v ⊥ , and hence by Theorem A acts elliptically.

If we suppose that the Franks-Misiurewicz conjecture [FM90] holds, the above corollary implies the following improvement of the second point: f acts parabolically on C † (T 2 ) if and only if ρ( f ) is a segment of irrational direction, or f is a pseudo-rotation with unbounded deviation from some v ∈ Q 2 \ {0}. In particular any f whose rotation set is a segment with rational direction acts elliptically on C † (T 2 ).

Remark that there are both pseudo-rotations with bounded displacement in some rational direction (like actual rotations) and pseudo-rotations with unbounded displacement in any rational direction (see for [START_REF] Koropecki | Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion[END_REF] for rational pseudo-rotations and [START_REF] Kocsard | A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus[END_REF] for irrational ones). Hence one cannot distinguish completely the possible types of actions of homeomorphisms on C † (T 2 ) only in terms of rotation sets. Note that Jäger [J 09], Jäger-Tal [START_REF] Jäger | Irrational rotation factors for conservative torus homeomorphisms[END_REF] and Kocsard [Koc21] gives criteria of semi-conjugation to a rotation for torus homeomorphisms with bounded displacements.

One can give a statement similar to Corollary 3 from the rotation viewpoint:

Corollary 4. Let f ∈ Homeo 0 (T 2 ). Then

• If ρ( f ) has nonempty interior, then f acts hyperbolically on C † (T 2 );

• If ρ( f ) is a segment with irrational slope, then f acts parabolically on C † (T 2 );

• If ρ( f ) is a segment with rational slope passing through a rational point, then f acts elliptically on C † (T 2 );

• If ρ( f ) is a segment with rational slope not passing through a rational point (a case that should never hold according to Franks-Misiurewicz conjecture [START_REF] Franks | Rotation sets of toral flows[END_REF]), then f acts parabolically on C † (T 2 );

• If f is a pseudo-rotation with unbounded deviation from any v ∈ Q 2 \ {0}, then f acts parabolically on C † (T 2 );

• If f is a pseudo-rotation with bounded deviation from some v ∈ Q 2 \ {0}, then f acts elliptically on C † (T 2 ).

Theorem A also allows to give a complete classification of the action of homeomorphisms on C † (T 2 ) for the ones that are not homotopic to identity.

As proved in [BHM + 22], any homeomorphism having an iterate homotopic to an Anosov linear automorphism acts hyperbolically on C † (T 2 ). Thus it remains to classify the actions in the case of a homeomorphism having an iterate homotopic to a Dehn twist.

Corollary 5. Let f ∈ Homeo(T 2 ) such that f r is homotopic to a Dehn twist.

• f acts hyperbolically on C † (T 2 ) if and only if ρ( f r ) has nonempty interior;

• f acts parabolically on C † (T 2 ) if and only if ρ( f r ) is a single number, and f has unbounded displacement in the vertical direction;

• f acts elliptically on C † (T 2 ) if and only if ρ( f r ) is a single number, and f has bounded displacement in the vertical direction.

Note that by Addas-Zanata, Tal, and Garcia [START_REF] Addas-Zanata | Dynamics of homeomorphisms of the torus homotopic to Dehn twists[END_REF], if ρ( f r ) is reduced to a single rational number, then f has bounded displacement in the vertical direction and hence acts elliptically on C † (T 2 ). To our knowledge, the question whether the second case is nonempty (i.e. if there exists homeomorphisms homotopic to Dehn twists acting parabolically on

C † (T 2 )) is open.
Remark that by [START_REF] Passeggi | Rational polygons as rotation sets of generic homeomorphisms of the two torus[END_REF], on a open and dense subset of Homeo 0 (T 2 ), the rotation set is a polygon with rational vertices, hence the set of parabolic elements of Homeo 0 (T 2 ) is included in a closed set with empty interior.

A last comment: in [LRW22] Le Roux and Wolff prove that any automorphism of a variant of the fine curve graph is realized by some homeomorphism of the surface. They suggest that the same result holds for our definition of the fine curve graph, hence in some sense our classification of actions of homeomorphisms on the fine curve graph covers the whole automorphism group of the ifne curve graph.

Some open questions

1. Can we get a similar classification for higher genus surfaces4 ? 2. Are there some torus homeomorphisms homotopic to identity satisfying lim inf d n < +∞ and lim sup d n = +∞, where d n is the image under a lift of f n to R 2 of a fixed fundamental domain?

3. Are there some torus homeomorphisms homotopic to identity acting parabolically but not properly on C † (T 2 )?

4. More generally, what are the sets of possible good limit values? Can they be classified?

A potential master's student of the first author should start thinking about the last three questions soon.

Rotation sets for torus homeomorphisms

Let f ∈ Homeo 0 (T 2 ) and fix a lift f : R 2 → R 2 .

The rotation set of f is the set

ρ( f ) = v ∈ R 2 ∃(x k ) k ∈ R 2 , (n k ) k → +∞ : f n k (x k ) -x k n k -→ k→+∞ v .
A theorem of Misiurevicz and Ziemian [START_REF] Misiurewicz | Rotation sets for maps of tori[END_REF] states that it is a compact convex subset of R 2 (see also Lemma 8). Some basic properties are straightforward consequences of the definition: for any k ∈ Z, ρ( f k ) = kρ( f ), and ρ is a conjugacy invariant: if g ∈ Homeo 0 (T 2 ) and g is a lift of g to R 2 , then ρ(g f g-1 ) = ρ( f ). It depends on the lift of f in the following way: any other lift of f can be written

f + v, with v ∈ Z 2 . Then ρ( f + v) = ρ( f ) + v.
We will be inspired by an equivalent formulation of the rotation set in Section 2 to define good limit values: fix a fundamental domain D ⊂ R 2 of the torus (e.g. D = [0, 1] 2 ), then (see [START_REF] Misiurewicz | Rotation sets for maps of tori[END_REF])

ρ( f ) = lim n→+∞ f n (D) n ,
where the limit holds in the Hausdorff topology (and in particular, the result states that the limit does exist). Now, let f ∈ Homeo(T 2 ) homotopic to a Dehn twist. By this, we mean that there exists a basis of the torus in which f is homotopic to the linear automorphism

1 k 0 1 for some k ∈ Z * .
Fix a lift f : R 2 → R 2 , and denote by p 2 : R 2 → R the projection on the second coordinate (according to the basis used to define the Dehn twist).

Following [START_REF] Doeff | Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist[END_REF], we can define the rotation set of f as

ρ( f ) = v ∈ R ∃(x k ) k ∈ R 2 , (n k ) k → +∞ : p 2 f n k (x k ) -x k n k -→ k→+∞ v .
This set is a segment of R. As for homeomorphisms isotopic to the identity, it follows from the definition that for any g ∈ Homeo 0 (T 2 ), we have ρ( f ) = ρ(g f g-1 ). Moreover, two lifts of f to R 2 have rotation sets which differ by an integral translation of R.

Outline

One way of Theorem A is easy to prove: if the homeomorphism f has bounded deviation from a rational direction, then f acts elliptically on C † (T 2 ). We write down the proof of this implication.

Proof of the "if" part in Theorem A. Let us say that f has bounded deviation from v ∈ Q 2 \ {0}. Without loss of generality, we can suppose that v = (p, q), where either p and q are relatively prime integers, or (p, q) ∈ {(0, 1), (1, 0)}. Let α : R/Z → T 2 be the loop defined by t → t(q, -p). When γ and γ are isotopic loops of T 2 , we denote by C γ (γ ) the number of lifts of γ that are met by a given lift γ of γ . This number does not depend on the chosen lift γ of γ . Note that (as already said before) a homeomorphism homotopic to a linear Anosov automorphism has unbounded deviation in any direction, and a homeomorphism homotopic to a Dehn twist has unbounded deviation in any direction but possibly one. In the last case, this direction is such that the loops (f n (α)) n∈N are all homotopic one to each other.

As f has bounded deviation from direction v = (p, q), there exists ρ ∈ R 2 and a lift f :

R 2 → R 2 of f such that | f n (x) -x -nρ, v | is bounded uniformly in x and n.
This implies that the sequence (C α+nρ (f n (α)) n≥0 is bounded, where α + nρ is the loop t → α(t) + nρ. Hence, as the loop α + nρ is either disjoint from α or equal to α, the sequence (C α (f n (α))) n≥0 is bounded. But, by Lemma 4.5 of [BHM + 22], for any n ≥ 0,

C α (f n (α)) + 1 ≥ d C † (T 2 ) (α, f n (α))
and the orbit of α under f in C † (T 2 ) is bounded.

To prove the direct implication, we suppose that f has unbounded deviation from any rational direction; we want to prove that, in this case, f does not act elliptically on

C † (T 2 ).
The first step is to define good limit values (Section 2), that are analogs of rotation sets capturing sublinear speeds: instead of dividing by the time, one divides by the diameter of the iterate of the fundamental domain. An important property is that good limit values are convex, as stated in Lemma 8.

We then state a (non) ellipticity criterion in terms of good limit values. This criterion (Proposition 9) improves a criterion given in [BHM + 22] and relies on branched covering maps of the torus by square tiled surfaces.

The next step is done in Section 4. We set a dichotomy for possible shapes of good limit values: either one of them contains a segment of irrational slope, or there exists a rational line containing any good limit value. In the first case, the criterion set in the previous section applies and shows that the action of the homeomorphism is non elliptic. It then remains to treat the second case.

The final argument is given in Section 5. We show that if any good limit value is contained in a single rational line, and if f has unbounded deviation from any rational direction, then (a modified version of) some good limit value contains a segment of irrational slope. This allows to apply once again the parabolicity criterion.

Good limit values

In this section we define good limit values, that could also be called "slow rotation sets": they capture the rotational behaviour in the case when the rotation speed is sublinear.

For any n ≥ 0, we denote by d n the diameter of f n (D), where D is the fundamental domain [0, 1] 2 of the torus T 2 . The following lemma implies that there is a subsequence of (d n ) n which tends to +∞. In the sequel we will need a more precise result which is the second part of this lemma. Lemma 6. If f ∈ Homeo(T 2 ) is not homotopic to a linear Anosov automorphism and has unbounded deviation in some direction, then

sup n∈N d n = +∞.
More precisely, if there exists v ∈ R 2 , ρ ∈ ρ( f ), (n k ) going to infinity, and a sequence of points

(x k ) such that for any k ∈ N, | f n k (x k ) -x k -n k ρ, v | ≥ k, then lim k→+∞ sup x,y∈D f n k (x) -f n k (y), v = +∞.
(2.1)

Let us explain this statement. If f ∈ Homeo(T 2 ) has no iterate homotopic to identity, then sup n∈N d n = +∞, so the first part is relevant only in the case f ∈ Homeo 0 (T 2 ). If f is homotopic to a Dehn twist about the horizontal direction, then (2.1) holds for any v / ∈ R(0, 1), so the lemma is only relevant in the case v ∈ R(0, 1), in which case one can define

f n k (x k ) -x k -n k ρ, v := p 2 f n k (x k ) -x k -n k ρ.
In the case where f is homotopic to a linear Anosov automorphism, then (2.1) holds for any direction v, except possibly the stable direction of the automorphism (which is irrational).

Proof. We prove the lemma in the case f ∈ Homeo 0 (T 2 ), the case where f is homotopic to a Dehn twist is identical.

Suppose that f has unbounded deviation in some direction v. In particular, there exists ρ ∈ ρ( f ), (n k ) going to infinity, and a sequence of points (x k ) such that for any k,

| f n k (x k )-x k -n k ρ, v | ≥ k.
Taking the image of each x k under an integral translation if necessary, we can suppose that the points x k all belong to D. Without loss of generality, by taking a subsequence if necessary, we can suppose that each of those scalar products are positive.

If the set { ρ, v | ρ ∈ ρ( f )} is a nontrivial segment, then the conclusion of the lemma is straightforward. If not, then the quantity of (2.1) does not depend on the choice of ρ ∈ ρ( f ).

Suppose that the last limit of the lemma does not hold. Then, by considering a subsequence if necessary, it holds that

sup k∈N sup x,y∈D 2 f n k (x) -f n k (y), v = R < +∞.
Then for any y ∈ D, applying this to x = x k , we have (considering that n 0 = 0)

f n k (y) -y -n k ρ, v = f n k (y) -f n k (x k ), v + f n k (x k ) -x k -n k ρ, v + x k -y, v ≥ k -2R.
Observe that the left side of the above inequality does not change if we replace the point y by one of its integral translate so that the inequality actually holds for any y ∈ R 2 . By choosing k ≥ 2R + 1, we get f n k (y) -y -n k ρ, v ≥ 1. As this holds for any y ∈ R 2 , one can iterate: for any z ∈ R 2 and any ∈ N, we have f n k (z) -z -n k ρ, v ≥ . In particular, this implies that there exists ρ ∈ ρ( f ) \ (ρ + Rv ⊥ ), a contradiction. Fix f ∈ Homeo(T 2 ) and a point x0 ∈ int(D). For any subset A of R 2 , any λ > 0 and any v ∈ R 2 , let us denote λA

+ v = {λa + v | a ∈ A}. For any n ≥ 0, let A n = 1 d n f n (D) -f n (x 0 ) .
More generally, fix a sequence (a k ) k≥0 of positive real numbers as well as a sequence (n k ) k≥0 of integers with n k → +∞. For any k ≥ 0, define

B k = 1 a k f n k (D) -f n k (x 0 ) . (2.2)
As x0 ∈ D, observe that, for any n ≥ 0, 0 ∈ A n . Moreover, by definition of (d n ), for any n ≥ 0, the set A n is contained in the closed unit disc of R 2 and has diameter 1. Recall that the set of compact subsets of the closed unit disc, endowed with the Hausdorff topology, is compact.

We endow the set of closed subsets of R 2 with the following topology, which resembles Hausdorff convergence on any (large) ball. Let φ be the map that to any closed subset F of R 2 associates the compact subset F ∪ {∞} of the Alexandroff compactification of R 2 (which is homotopic to S 2 ). The topology on closed subsets of R 2 we consider is then the initial topology associated to φ and Hausdorff topology on the Alexandroff compactification of R 2 . The set of closed subsets of R 2 endowed with this topology is compact.

Definition 7. We call good limit value of the sequence (A n ) any limit value A ∞ of this sequence such that there exists a subsequence (A n k ) k≥0 which converges to A ∞ with lim k→+∞ d n k = +∞.

Note that a good limit value of the sequence (A n ) is a limit value of the sequence (B k ) associated to some sequence (n k ) and with a k = d n k .

By Lemma 6 -combined with the compactness of the set of compact subsets of the unit disk endowed with Hausdorff topology -if f has unbounded deviation in some direction, then it has at least one good limit value. More generally, for any sequences (a k ) k≥0 of positive real numbers and (n k ) k≥0 of integers with n k → +∞, the sequence (B k ) admits a limit value (not necessarily compact).

The following lemma is a direct adaptation of a result of Misiurevicz and Ziemian [START_REF] Misiurewicz | Rotation sets for maps of tori[END_REF] asserting that rotation sets are convex.

Lemma 8. Suppose that lim k→+∞ a k = +∞. Then any limit value B ∞ of the sequence

(B k ) is a convex subset of R 2 .
Note that this implies that any good limit value A ∞ of the sequence (A n ) is a convex subset of the closed unit disc.

Proof. Let ξ 1 and ξ 2 be two points of a limit value B ∞ of the sequence (B k ) k . Then, extracting a subsequence if necessary, there exist sequences

(p k ) k and (q k ) k of points of D such that lim k→+∞ 1 a k f n k (p k ) -f n k (x 0 ) = ξ 1 and lim k→+∞ 1 a k f n k (q k ) -f n k (x 0 ) = ξ 2 .
Let ξ = λξ 1 + (1 -λ)ξ 2 , with 0 < λ < 1 and let us prove that ξ ∈ B ∞ . For any k ≥ 0, let

z k = λ f n k (p k ) + (1 -λ) f (q k )
. By Lemma 3.3 of the article [MZ89] by Misiurewicz and Ziemia, the set f n k (D) is √ 2-quasi-convex, so that for any k ≥ 0 there exists a point

r k ∈ D such that d( f n k (r k ), z k ) ≤ √ 2. Then the sequence 1 a k f n k (r k ) -f n k (x 0 )
k has the same limit as the sequence

1 a k z k -f n k (x 0 )
k and this limit is ξ. Hence the point ξ belongs to B ∞ and the set B ∞ is convex.

A non-ellipticity criterion

Let f ∈ Homeo(T 2 ) and x 0 ∈ T 2 . The goal of this section is to prove the following criterion, which generalizes Section 6 of [BHM + 22].

Proposition 9 (Criterion of non-ellipticity). Suppose that the following holds:

1. lim k→+∞ a k = +∞.

2. There exists a sequence (w k ) k of vectors of R 2 such that the sequence

(B k + w k ) k≥0 = 1 a k f n k (D) -f n k (x 0 ) + w k k≥0
of compact subsets of R 2 converges to some closed subset B ∞ for the topology defined before Definition 7.

3. The set B ∞ contains a nontrivial segment of irrational slope.

Then f does not act as an elliptic isometry of C † (T 2 ).

Recall that by [BHM + 22, Theorem 1.3], f acts hyperbolically on C † (T 2 ) if and only if int(ρ( f )) = ∅. Hence this criterion can be used to prove that some homeomorphism acts parabolically on C † (T 2 ).

To prove this proposition, we need to introduce some notation and three lemmas.

For any integer m > 0 we let

T 2 m = R/mZ × R/mZ.
This space can be seen as a cover of T 2 = R 2 /Z 2 of degree m 2 via the projection T 2 m → T 2 . We denote by p m : R 2 → T 2 m the projection. We endow T 2 m with the translation surface structure which makes the map

T 2 = R/Z × R/Z → T 2 m (x, y) → (mx, my)
an isomorphism, where T 2 is endowed with the usual translation surface structure. This means that, in comparison to the usual euclidean metric on R 2 , distances are multiplied by 1 m on both coordinates (hence the diameter of T 2 m is √ 2). The first lemma we need is a purely topological lemma. We state it in the case of the torus, which is the case we need, but it is valid on any surface.

Lemma 10. Let γ 1 and γ 2 be two simple paths [0, 1] → T 2 which are homotopic with fixed extremities in T 2 . Then there exists a nonempty open set U ⊂ T 2 such that, for any point p of U , the two paths γ 1 and γ 2 are homotopic with fixed extremities in T 2 \ {p}.

Proof. Define the equivalence relation on T 2 whose equivalence classes are the singletons of points outside γ 1 , and γ 1 . This is the equivalence relation that shrinks the path γ 1 to a point. We denote by T 1 the quotient of T 2 by this equivalence relation and by q 1 the point that is the image of γ 1 in T 1 .

The space T 1 is still a 2-torus and the image α 1 of γ 2 in T 1 is a path. As γ 2 is simple, the path α 1 has only self intersections at the point q 1 and the autointersection points cannot be transverse. Hence the path α 1 is a homotopically trivial loop; it is a union of simple loops based at q 1 which do not meet each other except at q 1 , some of them homotopically trivial and some of them homotopically non trivial (and there is a finite number of such last ones).

The complement of any homotopically trivial simple loop has one component which is homeomorphic to a disk, which we call the interior of such a loop. Take the closure C of the union of the interiors of the homotopically trivial simple loops appearing in the decomposition of α 1 . Observe that C contains the point q 1 , we can shrink C to a point q 2 to obtain a new torus T 2 .

We call α 2 the image of the path α 1 in the torus T 2 . The path α 2 is the concatenation of a finite number of homotopically non trivial simple loops. Moreover, the loop α 2 has auto-intersections only at the point q 2 , none of which are transverse (too see this, use the fact that γ 1 is a simple loop). We then write α 2 as a concatenation of loops β 1 , β 2 , . . . , β r , each of which is a homotopically trivial loop which cannot be written as a concatenation of nontrivial homotopically trivial loops. Fix a lift q2 of the point q 2 to R 2 and denote by α2 , β1 , β2 , . . . , βr the respective lifts of α 2 , β 1 , β 2 , . . . , β r based at q2 . Observe that the loop α2 is the concatenation of the loops β1 , β2 , . . . , βr and that each of the latter loop is simple. Observe that the interior of each βi is disjoint from its translates under the group of deck transformations. Hence the interior of βi projects injectively to the torus T 2 (because no transverse autointersection is allowed and no deck transformation has a fixed point). We call this projection the interior of the loop β i , and we denote it by I i . Observe also that, for i = j, either I i is contained in I j or I i contains I j or I i and I j are pairwise disjoint. An induction on r shows that the complement of the union of the closures of the I i 's is nonempty. It suffices to take as a set U the preimage in T 2 of this subset of T 2 to prove the lemma.

Remark 11. There is a shorter argument for the proof of this lemma, which uses the folkloric dual function: the curve γ 1 γ -1 2 is homologous to 0. This allows to define a dual function on the complement of its image in the torus, and any point p in which the dual function is equal to 0 suits the conclusion of the lemma.

The three lemmas that follow are essentially proved in [BHM + 22].

Lemma 12. Let α and β be two curves in C † (T 2 ) and m ≥ 1. Then, for any respective lifts α and β of α and β in T 2 m (via the cover map T 2 m → T 2 ), we have

d C † (T 2 m ) (α, β) ≤ d C † (T 2 ) (α, β).
Proof. This is a straightforward consequence of Lemma 6.3 in [BHM + 22].

Lemma 13. Fix K > 0. There exists a square-tiled surface Σ(K) such that, for any m > 0 and any p ∈ T 2 m , there exists a branched covering map f m,p : Σ(K) → T 2 m , which is branched only at p, with the following properties.

1. For any two essential simple closed curves α and β of T 2 m with d C † (T 2 m ) (α, β) ≤ K and which do not meet the point p, there exist lifts of α and β to Σ(K) which are disjoint.

2. The map f m,p is a local isomorphism of translation surfaces outside f -1 m,p ({p}).

Proof. The proof is almost identical to the proof of Lemma 6.4 in [BHM + 22].

The following lemma is Lemma 6.5 in [BHM + 22].

Lemma 14. Let ξ ∈ R \ Q and Σ be a square-tiled surface. There exists L > 0 such that any line segment in Σ of slope ξ and of length greater than L meets any horizontal closed curve.

As a consequence of the above Lemma, we obtain the following corollary.

Corollary 15. Fix ξ and Σ as in the above lemma and take L = 2L , where L is given by the above lemma. Then there exists ε > 0 such that the following property holds. For any line segment S which is ε-close, for the Hausdorff distance, to a line segment of slope ξ and of length greater than L, any path which is homotopic to S with fixed extremities in the complement of singular points in Σ meets any horizontal closed curve.

Proof. Consider the set K consisting of compact connected subsets of Σ(K) which are either segments of length L and slope ξ or a union of two segments of Σ of slope ξ with a common extremity which is a singularity of Σ and whose total length (i.e. the sum of the lengths of those segments) is equal to L. By Lemma 14, any element of K meets any horizontal closed curve of Σ. Moreover, the set K is compact for the Hausdorff topology.

For any element S of K, there exists ε S > 0 such that any line segment S which is ε S -close to S meets any horizontal curve. By compactness of K, we can take ε S = ε independent of S . Observe that any such segment S as above has a nonzero algebraic intersection number with any horizontal curve so that Corollary 15 holds.

Proof of Proposition 9. We prove this proposition by contradiction. Suppose that f acts elliptically on C † (T 2 ). We denote by α the curve t → (t, 0) on T 2 and by α its lift t → (t, 0) to R 2 . Then there exists K > 0 such that, for any n ≥ 0

d C † (T 2 ) α, f n (α) ≤ K .
Let K = K +1 and observe that, for any essential simple closed curve

β which is disjoint from α, d C † (T 2 ) α, f n (β) ≤ K.
Apply Lemma 13 to obtain a square tiled surface Σ(K). Take θ ∈ R \ Q such that the set B ∞ contains a nontrivial segment with irrational slope θ. Apply Corollary 15 with this slope and the surface Σ(K). This corollary gives a number L > 0. Fix M > 0 in such a way that the set M B ∞ contains an irrational segment S ∞ with length > L and slope θ. Take a compact subset C whose interior contains this segment S ∞ . Then the sequence of subsets

1 a k M f n k (D) -f n k (x 0 ) + M a k M w k k
converges to the subset M B ∞ for the Hausdorff topology. Take an integer k 0 sufficiently large so that the set

1 a k 0 M f n k 0 (D) -f n k 0 (x 0 ) + M a k 0 M w k 0 ∩ C is ε-close to the set M B ∞ ∩ C
, where ε is given by Corollary 15 and observe that the set

1 a k 0 M f n k 0 (D) -f n k 0 (x 0 ) ∩ (C -M w k 0 ) is ε-close to the set M (B ∞ -w k 0 ) ∩ (C -M w k 0 ). Fix m = a k 0
M . Hence there exist two points x, y ∈ D such that the point f n k 0 (x) -f n k 0 (x 0 ) is mεclose to one extremity of the segment m(S ∞ -M w k 0 ) and the point f n k 0 (y) -f n k 0 (x 0 ) is mε-close to the other extremity of this segment. Moreover, we choose those points x and y outside any lift of α. In particular, the point p m ( f n k 0 (x) -f n k 0 (x 0 )) is ε-close to one extremity of the segment S = p m (m(S ∞ -M w k 0 )) and the point p m ( f n k 0 (y) -f n k 0 (x 0 )) is ε-close to the other extremity of this segment. Observe that the length of the segment S is greater than L. Denote by S the line segment in T 2 m joining these two points that remains ε-close to S and let S = S + p m ( f n k 0 (x 0 )). Observe that the segment S is ε-close to a segment of the same slope and same length as S.

Take a simple path γ contained in D whose extremities are the points x and y and that does not meet any lift of α. Fix an essential simple closed curve β of T 2 homotopic to α that is disjoint from α and has a lift to R 2 containing the path γ. Observe that the path p m ( f n k 0 (γ)) is homotopic with fixed extremities to S in T 2 m (because those paths admit lifts to R 2 with the same endpoints). By Lemma 10, there exists a point p of T 2 m , which neither belongs to any lift of α nor to any lift of β, such that those two paths are still homotopic with fixed extremities in T 2 m \ {p}. By Lemma 13, there exists a covering map Σ(K) → T 2 m which is ramified only at the point p. By Corollary 15, any lift of the curve p m ( f n k 0 ( β)) to the surface Σ(K) meets any horizontal curve of Σ(K). However, the curve p m ( f n k 0 ( β)) is a lift of the curve f n k 0 (β) to T 2 m so that, by Lemma 12,

d C † (T 2 m ) p m ( f n k 0 ( β)), p m (α) ≤ K.
Hence the curves p m ( f n k 0 ( β)) and p m (α) must admit disjoint lifts to Σ(K), which is a contradiction as the latter curve is horizontal.

Possible directions of good limit values

The following proposition gives two possibilities for the possible shapes of the good limit values of the sequence (A n ) n .

Proposition 16. Let f ∈ Homeo 0 (T 2 ). One of the following holds.

1. There exists a good limit value of the sequence (A n ) n which contains a nontrivial segment with irrational slope.

2. There exists a line with rational slope which contains any good limit value of the sequence (A n ) n .

Proof. Suppose that 1. does not hold. Then any good limit value is a convex set with empty interior (any convex set with nonempty interior contains a nontrivial segment with irrational slope). Hence, any good limit value is a segment of rational slope, contained in the closed unit disk, and having diameter 1. We argue by contradiction by supposing that there are at least two different rational directions containing a good limit value. Call these directions θ 1 , θ 2 ∈ P(R 2 ).

We endow P(R 2 ) with a distance δ making it homeomorphic to the circle. For θ ∈ P(R 2 ), denote L θ the line of direction θ passing by 0. As a first step, we state that if d n is large enough, then A n is very close to a rational segment. This follows from a simple compactness argument.

Claim 17. For any ε > 0, there exists R > 0 such that if d n > R, then there exists θ ∈ P(R 2 ) rational such that the following holds:

∀x ∈ A n , d(x, L θ ) ≤ ε. (P n,θ,ε )
Proof. Suppose the claim is false. Then there exists ε 0 > 0 and a subsequence (n k ) such that d n k → +∞, and that for any rational direction θ ∈ P(R 2 ), there exists x ∈ A n k with d(x, L θ ) > ε 0 . By taking a subsequence of (n k ) k if necessary, one can suppose that d n k ≥ k and that the sets A n k converge towards some compact set for Hausdorff topology. By the discussion at the beginning of the proof of the proposition, this set can only be a segment with rational slope θ 0 , i.e. contained in some line L θ 0 . Hence, if k is large enough, for any x ∈ A n k we have d(x, L θ 0 ) ≤ ε 0 . This is a contradiction.

It allows to define the following.

Definition 18. A main direction of A n for ε > 0 is a direction θ such that (P n,θ,ε ) holds. An R-excursion is any integer interval [n 1 , n 2 ] on which any n ∈ [n 1 , n 2 ] satisfies d n ≥ R.
The following says that if R is large enough, then on any R-excursion, the main directions of A n cannot vary a lot.

Lemma 19. For any δ 0 > 0, there exists ε > 0 and R > 0 satisfying: if [n 1 , n 2 ] is an R-excursion, then for any n, n ∈ [n 1 , n 2 ] and any θ, θ such that (P n,θ,ε ) and (P n ,θ ,ε ) hold, we have δ(θ, θ ) < δ 0 .

Note that if the conclusion of the lemma holds for some ε > 0, then it holds for any 0 < ε < ε.

Proof. Suppose it is false. Then there exists δ 0 > 0 such that for any k > 0, there is a k-

excursion [n k 1 , n k 2 ] and two directions α k 1 , α k 2 with δ(α k 1 , α k 2 ) ≥ δ 0 such that (P n k 1 ,α k 1 ,1/(2k)
) and (P n k 2 ,α k 2 ,1/(2k) ) hold. By extracting a subsequence, one can suppose that (α k 1 , α k 2 ) converge towards (α 1 , α 2 ), and more precisely that δ(α k 1 , α 1 ) ≤ 1/(2k) and δ(α k 2 , α 2 ) ≤ 1/(2k). In this case, for k large enough, (P n k 1 ,α 1 ,1/k ) and (P n k 2 ,α 2 ,1/k ) hold. We now prove that for n ∈ [n k 1 , n k 2 ], the set of main directions cannot vary a lot between times n and n + 1. Indeed, calling

K f = 2 max d( f , Id R 2 ), d( f -1 , Id R 2 ) , we have d H f n (D) -f n (x 0 ), f n+1 (D) -f n+1 (x 0 ) ≤ K f Hence, d H f n (D) -f n (x 0 ) d n , f n+1 (D) -f n+1 (x 0 ) d n ≤ K f d n . (4.1) But we also have |d n -d n+1 | ≤ K f , so |1 -d n+1 /d n | ≤ K f /d n . The fact that -given a compact subset A of the unit disc -the map R + λ → λA is 1-Lipschitz for Hausdorff distance d H implies that d H f n+1 (D) -f n+1 (x 0 ) d n , f n+1 (D) -f n+1 (x 0 ) d n+1 = d H d n+1 d n f n+1 (D) -f n+1 (x 0 ) d n+1 , f n+1 (D) -f n+1 (x 0 ) d n+1 ≤ 1 - d n+1 d n ≤ K f d n .
Combined with (4.1), by triangle inequality, this gives

d H (A n , A n+1 ) = d H f n (D) -f n (x 0 ) d n , f n+1 (D) -f n+1 (x 0 ) d n+1 ≤ 2K f d n .
Using the fact that d n tends to infinity, we deduce that the Hausdorff distance between

A n and A n+1 is in O(1/k) (recall that n ∈ [n k 1 , n k 2 ]
). Hence, if (θ n ) and (θ n ) are such that (P n,θn,1/k ) and (P n+1,θ n ,1/k ) hold, then δ(θ n , θ n ) tends to 0 as k tends to infinity. This implies for any η > 0, the directions α 1 and α 2 can be joined by an η-chain consisting of accumulation points of sequences (θ n k ), where

n k ∈ [n k 1 , n k 2 ], and
(P n k ,θn k ,1/k ) holds.
This proves that the set of θ ∈ P(R 2 ) satisfying the following property contains an interval containing both α 1 and α 2 : there exists a sequence

n k ∈ [n k 1 , n k 2 ] such that (P n k ,θn k ,1/k ) holds, d n k ≥ k and θ n k → θ.
This is a contradiction as such an interval has to contain an irrational direction.

We are now ready to end the proof of Proposition 16. The idea is that the images of the fundamental domain cannot grow in two different directions on two R-excursions of the same length. We first build such two R-excursions, and then use them to get to a contradiction.

Let δ 0 < δ(θ 1 , θ 2 )/3, and consider

α = min{∠(θ, θ ) | δ(θ 1 , θ) < δ 0 , δ(θ 2 , θ ) < δ 0 }.
Take ε > 0 such that sin α ≥ 6ε.

Consider R associated to ε and δ 0 as in Lemma 19. Increasing R if necessary, one can suppose that R ≥ K f , with

K f = 2 max d( f , Id R 2 ), d( f -1 , Id R 2 ) .
We now consider two R-excursions of the same length, one with main directions close to θ 1 , the other with main directions close to θ 2 ; we moreover suppose that on the first one, the diameter increases a lot. More precisely, we first prove that there exists n 1 ≤ n 2 such that:

• [n 1 , n 2 ] is an R-excursion; • d n 2 ≥ 4R/ε; • d n 1 ≤ 2R;
• for any n ∈ [n 1 , n 2 ], there exists θ ∈ P(R 2 ) such that δ(θ 1 , θ) < δ 0 and (P n,θ,ε ) holds.

Indeed, by Lemma 19, for any R > R, as by hypothesis there is at least two different rational directions containing a good limit value, there exists an infinite number of Rexcursions. Moreover, still by Lemma 19, there exists an infinite number of R -excursions [n 1 , n 2 ] such that for any n ∈ [n 1 , n 2 ] and any θ such that (P n,θ,ε ) holds, we have δ(θ, θ 1 ) < δ 0 . So we can consider such a 4R/ε-excursion [ñ 1 , n 2 ], and n 1 the minimal integer such that [n 1 , n 2 ] is a R-excursion. Trivially, [n 1 , n 2 ] satisfies the two firsts points and the last point. But, as we have already seen,

|d n 1 -d n 1 -1 | ≤ K f , so d n 1 ≤ d n 1 -1 + K f ≤ 2R.
As, for any n, |d n+1 -d n | ≤ K f , observe that there are arbitrarily long R-excursions with one element n of the excursion satisfying (P n,θ 2 ,ε ). As a consequence, using Lemma 19, one can similarly find n 1 ≤ n 2 such that: d n 2 , leading to a contradiction.

f n 1 (D) f n 1 (D) f n 2 -n 1 f n 2 -n 1 f n 2 (D) f n 2 (D) + w f n 2 (D)
• [n 1 , n 2 ] is an R-excursion; • n 2 -n 1 = n 2 -n 1 ; • d n 1 ≤ 2R;
• for any n ∈ [n 1 , n 2 ], there exists θ ∈ P(R 2 ) such that δ(θ 2 , θ) < δ 0 and (P n,θ,ε ) holds.

Indeed, consider an R + (n 2 -n 1 )K f -excursion [ñ 1 , ñ 2 ], consider n 1 the minimal integer such that [n 1 , ñ 2 ] is an R-excursion, and set n 2 = n 1 + (n 2 -n 1 ); the hypothesis on the size of the excursion ensures that [n 1 , n 2 ] is a R-excursion.

The contradiction then comes as follows (see Figure 1). Take

x 1 ∈ D such that f n 2 (x 1 ) -f n 2 (x 0 ) ≥ d n 2 /2. Note that f n 1 (x 1 ) -f n 1 (x 0 ) ≤ 2R. There exists v 0 , v 1 ∈ Z 2 such that f n 1 (x 0 ) -v 0 and f n 1 (x 1 ) -v 1 both belong to the fundamental domain f n 1 (D). Hence, v 0 -v 1 ≤ v 0 -f n 1 (x 0 ) -v 1 -f n 1 (x 1 ) + f n 1 (x 0 ) -f n 1 (x 1 ) ≤ d n 1 + d n 1 ≤ 4R.
Now, take any θ such that δ(θ, θ 2 ) ≤ δ 0 , and consider a unit vector u θ orthogonal to L θ .

dist : = d f n 2 -n 1 f n 1 (x 0 ) -v 0 -f n 2 -n 1 f n 1 (x 1 ) -v 1 , L θ = f n 2 (x 0 ) -f n 2 (x 1 ), u θ + v 0 -v 1 , u θ ≥ f n 2 (x 0 ) -f n 2 (x 1 ) sin ∠ f n 2 (x 0 ) -f n 2 (x 1 ), θ -4R ≥ d n 2 2 sin α -εd n 2 (this is here we use d n 2 ≥ 4R/ε) ≥ d n 2 2 6ε -εd n 2 ≥ 2εd n 2 .
As the two points f n 1 (x 0 ) -v 0 and f n 1 (x 1 ) -v 1 belong to f n 1 (D), and as (P n 2 ,θ,ε ) holds for some θ such that δ(θ, θ 2 ) ≤ δ 0 , then dist has to be smaller than εd n 2 for some θ such that δ(θ, θ 2 ) ≤ δ 0 . Hence, d n 2 ≥ 2d n 2 . But then one can apply the exact same argument, permuting θ 1 with θ 2 , to deduce that d n 2 ≥ 2d n 2 . This is a contradiction.

Rational case and end of the proof of Theorem A

We now state the last result we need to prove Theorem A.

Proposition 20. Let f ∈ Homeo(T 2 ). Suppose that any good limit value of the sequence (A n ) is contained in the horizontal axis. Then either f has bounded deviation in the vertical direction, or there exists a sequence (a n ) of positive real numbers tending to infinity, and a sequence (w n ) of vectors of R 2 , such that some limit value of the sequence

(B n + w n ) n = 1 a n f n (D) -f n (x 0 ) + w n n contains B(0, 1).
Let us first show how this proposition implies Theorem A.

Proof of Theorem A. Let f ∈ Homeo 0 (T 2 ), and suppose that in any rational direction, f has no bounded deviation. We want to prove that f does not act elliptically. Apply Proposition 16. In the first case given by this proposition, Proposition 9 implies that f does not act elliptically on C † (T 2 ).

Suppose now that the second case given by Proposition 16 holds: There exists a line with rational slope which contains any good limit value of the sequence (A n ) n . By conjugating with an element of SL 2 (Z) if necessary, we do not lose generality by supposing that this direction is horizontal.

Apply Proposition 20. As f has unbounded deviation in the vertical direction, there exists a sequence (a k ) of positive real numbers tending to infinity, and a sequence (w k ) of vectors of R 2 , such that some limit value of the sequence (B n + w k ) k contains B(0, 1).

The parabolicity criterion (Proposition 9) applies and shows that f does not act elliptically on C † (T 2 ). Now, suppose that f ∈ Homeo(T 2 ) has an iterate homotopic to a linear Anosov homeomorphism A. Then f has unbounded deviation from any direction, and by [BHM + 22, Theorem 5.3], f acts hyperbolically on C † (T 2 ). Note that we could also use our ellipticity criterion (Proposition 9) to conclude that f does not act elliptically: consider (0, 0), (1, 1) ∈ D, then f n (1, 1) -f n (0, 0) = A n (1, 1); using the fact that these vectors tend to some irrational direction of P(R 2 ) together with the quasi-convexity of fundamental domains we conclude that some limit set B ∞ contains a nontrivial segment of irrational slope.

Finally, suppose that f ∈ Homeo(T 2 ) has an iterate homotopic to a Dehn twist. Conjugating by an element of SL 2 (Z) if necessary, we can suppose that f n is homotopic to a Dehn twist about the horizontal direction. Suppose that f has unbounded displacement in the vertical direction. Then there is some k ∈ Z * such that f n (0, 1)-f n (0, 0) = (nk, 1). Hence any good limit value contains a nontrivial horizontal interval. If there is a good limit value which is not included in the horizontal axis, as such a limit value is convex and contains a nontrivial horizontal interval, then Proposition 9 applies and f does not act elliptically on C † (T 2 ). If not, then any good limit value is included in the horizontal axis, and Proposition 20 allows to once again apply Proposition 9 to prove that f does not act elliptically on C † (T 2 ).

Proof of Proposition 20. In this proof, for a set A and r ≥ 0, we denote B(A, r) = {x | d(x, A) ≤ r}.

Suppose that any good limit value of the sequence (A n ) n is included in the horizontal axis, and that f has unbounded displacement in the vertical direction.

Applying the same idea as in the proof of Theorem A (3 paragraphs above), by considering the iterates of (0, 0) and (1, 1), we can see that under these conditions, an iterate of f cannot be isotopic to a linear Anosov automorphism, or a Dehn twist in a direction that is not horizontal (adapting the points (0, 0) and (1, 1) in the latter case if necessary). So some iterate of f is isotopic to the identity, or to a Dehn twist. Hence, replacing f with an iterate of it if necessary, we can suppose that f is homotopic to

1 k 0 0 1
for some k 0 ∈ Z.

Claim 21. Suppose that any good limit value of the sequence (A n ) n is contained in the horizontal axis. Suppose also that for any sequence (a n ) of positive real numbers tending to infinity, and any sequence (w n ) of vectors of R 2 , any limit value of the sequence (B n + w n ) n does not contain B(0, 1).

Then there exists C > 0 and, for any n ≥ 0, a line L n passing through 0 and of direction θ n such that:

• θ n tends to the horizontal direction (1, 0);

• f n (D) -f n (x 0 ) ⊂ B(L n , C).
If moreover f has unbounded deviation in the vertical direction then, up to taking a subsequence, we can moreover suppose the following:

• the projection of f n (D) on the vertical axis has length h n tending to infinity. Let n ∈ [m, 2m] ∩ N. Then

f n 1 D 0 + (0, n) = f n 1 (D 0 ) + w n ,
with w n = (k 0 n 1 n, n).

Let γ 1 be a path included in f n 1 (D 0 ) linking x - 1 to x + 1 , and γ 2 a path included in f n 2 (D) linking x - 2 to x + 2 . We want to prove that the paths γ 1 + w n and γ 2 intersect for any n ∈ [m, 2m]. We first define an affine shear mapping A with linear part of the form 1 0 -tan θ n 2 1 for some angle θ n 2 , such that the abscissa of Ax is 0, and that the points Ax - 2 , Ax + 2 are on the horizontal axis {y = 0}. Note that it forces the angle θ n 2 to be close to θ n 2 , in the sense that θ n 2 -θ n 2 θ n 2 (because of the bound by C, and the fact that h n 2 goes to infinity).

Let us write Ax - 1 = (a - 1 , b - 1 ), Ax + 1 = (a + 1 , b + 1 ), Ax - 2 = (-M, 0) and Ax + 2 = (M, 0). (5.3) note that a - 1 + k 0 n 1 n is the abscissa of A(x - 1 + w n ) and a + 1 + k 0 n 1 n is the abscissa of A(x + 1 + w n ). The same estimates hold for any point of γ 1 + w n . We also have (see Figure 3 for the notations and the configuration.) The last inequality comes from the fact that, by the hypothesis (5.1) on x, we have that d(P, Q) ≥ d(P, R)/2. This implies that

p 2 (x - 1 -x) ≤ p 2 (P -Q) + 2C = -d(P, Q) sin θ n 1 + 2C ≤ - d n 1 -2C 2 sin θ n 1 + 2C.
b - 1 = p 2 A(x - 1 -x) ≤ - d n 1 -2C 2 sin θ n 1 + 2C + | tan θ n 2 p 1 (x - 1 -x)| ≤ - d n 1 2 sin θ n 1 + 3C + | tan θ n 2 ||a - 1 | ≤ - d n 1 2 sin θ n 1 + 3C + 2| tan θ n 2 |d n 1 ,
where the last inequality is a consequence of (5. (5.4)

Using the fact that n ≥ m ≥ 5C and that (because θ n 1 ≥ 0) p 2 (x + 1 ) ≥ -2C (see Figure 3), we get b + 1 + n ≥ C.

(5.5)

The estimates (5.3), (5.4) and (5.5) allow to apply the following lemma: Then the paths γ 1 and γ 2 meet.

Figure 2 :

 2 Figure 2: End of proof of Proposition 20 in the case of k 0 = 0. In the case k 0 = 0 there is a shear appearing in the translates of D.

RFigure 3 :

 3 Figure 3: End of proof of Proposition 20: estimation of p 2 (x - 1 -x). The length d(P, R) of the red segment is bigger than d n 1 -2C.

  2). Because we have supposed| tan θ n 2 |d n 1 ≤ C, we get b - 1 ≤ -d n 1 2 sin θ n 1 + 5C.Moreover,h n 1 ≤ p 2 (R -P ) + 2C = sin θ n 1 d(R, P ) + 2C ≤ sin θ n 1 d n 1 + 2C + 2C ≤ sin θ n 1 d n 1 + 4C, h n 1 ≥ 20m, 6C ≤ m and n ≤ 2m, b - 1 + n ≤ -10m + 8C + 2m = 8(C -m) ≤ -C.

Lemma 22 .

 22 Let M, C ∈ R + , and γ 2 be a path of R 2 linking the points (-M, 0) and(M, 0) of R 2 , that is included in [-M, M ] × [-C, C]. Let γ 1 be a path of R 2 linking the points (a - 1 , b - 1 ) and (a + 1 , b + 1 ) of R 2 , that is included in (-M, M ) × R, with b - 1 < -C and b + 1 > C.

  Figure 1: Proof of Proposition 16. If f n 2 (D) and f n 2 (D) have more or less the same size but different main directions, then it forces f n 2 (D) to have interior (light red shape): the shape in deep red is impossible for f n 2 (D) (recall that n 2 -n 1 = n 2 -n 1 ) because of the shape of the integer translate f n 2 (D) + w of f n 2 (D). This forces the relation d n 2 d n 2 , but a symmetric argument implies that d n 2

  The fact thatd n 2 d n 1 implies that max(|a - 1 |, |a + 1 |) ≤ max d(x, x - 1 ), d(x, x + 1 ) ≤ d n 1 ≤ M/2. (5.2) Hence, because k 0 n 1 m ≤ 6k 0 n 1 C d n 2 , max |a - 1 + k 0 n 1 n|, |a + 1 + k 0 n 1 n| ≤ M ;

I.e. non contractible.

It follows from [MM99, Proposition 4.6] and the Nielsen-Thurston classification [Thu88].

We call pseudo-rotation a homeomorphism f ∈ Homeo0(T 2 ) whose rotation set is reduced to a single point.

The authors have a strategy for a characterization of homeomorphisms isotopic to identity acting hyperbolically on C † (Sg): they should be the ones with nonempty interior homological rotation set, and the ones with a pseudo-Anosov mapping class when removing some periodic orbit. It should be the subject of a future work.

Acknowledgments

The first author was supported by a PEPS-JCJC grant. The second author was supported by the ANR project Gromeov ANR-19-CE40-0007.

The authors warmly thank Alejandro Kocsard and Roberta Shapiro for their useful comments about the first version of this article.

Proof. For any n ∈ N, let x n , y n ∈ D such that d( f n (x n ), f n (y n )) = d n . Let L n be the line passing by f n (x n ) and f n (y n ).

Let b n = max{d( f n (z), L n ) | z ∈ D} and z n ∈ D be such that d( f n (z n ), L n ) = b n . If sup n b n = C/2 < +∞, the two first points of the claim are proved, by setting L n = L n -f n (x 0 ): in this case the distance of any point of

Otherwise, there exists a subsequence

because of the definition of x n and y n (for a, b ∈ R 2 , we denote by [a, b] the affine segment between points a and b). Let also

Then the set

k, and such that the segment between these points contains 0; this set also contains the point

By Lemma 8, we deduce that any limit value of the sequence B n k + w k contains a quarter of disk (centered at 0) of radius 10. By modifying a bit the sequence (w k ) to (w k ), i.e. by applying a translation to each set B n k + w k , we get a limit value of the sequence (B n + w n ) n containing B(0, 1).

For the last point of the claim, define

the "diameter of f n (D) in the vertical direction". By Lemma 6, using that f has unbounded deviation in the vertical direction, we have a subsequence (n k ) along which lim e n k = +∞. This proves the last point. Now, up to increasing the constant C of Claim 21 if necessary, suppose C ≥ 2. Let m = 5 C and n 1 ∈ N such that h n 1 ≥ 20m and |θ n 1 | ≤ 1/100 (by Claim 21). We denote by p 1 the projection on the first (horizontal) coordinate, and p 2 the projection on the second (vertical) one.

Let

(we identify L n 1 with R), the other inequality can be treated identically. Similarly, we suppose that θ n 1 > 0.

Proof. It suffices to define the path α 2 by concatenating (-∞, -M ) × {0}, γ 2 and (M, +∞) × {0}. This is a Jordan loop of the Alexandroff compactification S 2 of R 2 , which is isotopic -with an isotopy with support included in [-M, M ] × [-C, C]to the path R × {0}. It is then easy to see that the points (a - 1 , b - 1 ) and (a + 1 , b + 1 ) lie in different connected components of S 2 \ α 2 , and hence that α 2 and γ 2 intersect. But it is also easy to see that any intersection point cannot belong to (-∞, -M ) × {0} or (M, +∞) × {0}; this implies that γ 1 and γ 2 intersect.

From this we deduce that for any n ∈ [m, 2m], the paths γ 1 + w m and γ 2 intersect. Hence, for any n ∈ [m, 2m] ∩ N, we have

This implies that there exists two points z 1 , z 2 ∈ f n 2 -n 1 (D) such that |p 1 (z 1 )-p 1 (z 2 )| ≤ 1 and |p 2 (z 1 ) -p 2 (z 2 )| ≥ 3C. This contradicts the fact that n 2 -n 1 satisfies the two first points of Claim 21: if θ n 2 -n 1 is small enough such a property is incompatible with


Dr Yaniss Belaroussi
email: yaniss.belaroussi@chu-bordeaux.fr

MD, MSc Romain Hustache-Castaing

MD Jean-Michel Maury

PhD Laurent Lehot

Arnaud Rodriguez

MD Xavier Demant

Hadrien Roze

PhD Geoffrey Brioude

Xavier -Benoit D'journo

PhD Gabrielle Drevet

MSc Arnaud Meng

Rodriguez

MD Hadrien Rozé

MD Xavier-Benoit D'journo

MD Francois Tronc

PhD Simone Mathoulin-Pélissier

PhD Jacques Jougon

PhD Pascal- Alexandre Thomas

PhD Matthieu Thumerel

Prediction of survival after lung transplantation at one year (SALTO cohort) using information available at different key time-points

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Manuscript including tables

The Author(s) 2023. Published by Oxford University Press on behalf of the European Association for Cardio-Thoracic Surgery. All rights reserved. Background: Lung transplantation is the final treatment option for end-stage lung disease. In this study, we evaluated the individual risk of 1-year mortality at each stage of the lung transplantation process.

Methods: This was a retrospective analysis of patients undergoing bilateral lung transplantation between January 2014 and December 2019 in three French academic centers.

Patients were randomly divided into development and validation cohorts. Three multivariable logistic regression models of 1-year mortality were applied (A) at recipient registration, (B) the graft allocation, and (C) after surgery. The 1-year mortality was predicted for individual patients assigned to three risk groups at time points A-C.

Results:

The study population consisted of 478 patients with a mean (SD) age of 49.0 (14.3) years. The 1-year mortality rate was 23.0%. There were no significant differences in patient characteristics between the development (n = 319) and validation (n = 159) cohorts. The models analyzed recipient, donor, and intraoperative variables. The discriminatory power (area under the receiver operating characteristic curve) was 0.67 (0.62 -0.73), 0.70 (0.63 -0.77), and 0.82 (0.77 -0.88), respectively, in the development cohort, 0.74 (0.64-0.85), 0.76 (0.66 -0.86) and 0.87 (0.79 -0.95), respectively, in the validation cohort. Survival rates were significantly different among the low-(< 15%), intermediate-(15%-45%), and high-risk (> 45%) groups in both cohorts.

Conclusions: Risk prediction models allow estimation of the 1-year mortality risk of individual patients during the lung transplantation process. These models may help caregivers identify high-risk patients at times A-C, and reduce the risk at subsequent time-points. The median survival of bilateral lung transplantation recipients is 8.8 years (conditional on survival to 1 year after transplant) [START_REF] Chambers | The International Thoracic Organ Transplant Registry of the International Society for Heart and Lung Transplantation: Thirty-sixth adult lung and heart-lung transplantation Report-2019; Focus theme: Donor and recipient size match[END_REF] , and predicting 1-year survival is challenging [START_REF] Foroutan | Predictors of 1-year mortality in adult lung transplant recipients: a systematic review and meta-analysis[END_REF] . Some predictive scores have been developed to estimate 1-year survival after double lung transplantation. However, these previous studies considered only donor characteristics 6 , recipient-and donor-specific characteristics 7 , or postoperative events [START_REF] Huppmann | The Munich-LTX-Score: predictor for survival after lung transplantation[END_REF] . None of the models described in the literature integrated all stages of the lung transplantation process.

Bilateral lung transplantation involves a number of steps, including recipient registration on a waiting list [START_REF] Orens | International Guidelines for the Selection of Lung Transplant Candidates: 2006 Update-A Consensus Report From the Pulmonary Scientific Council of the International Society for Heart and Lung Transplantation[END_REF][START_REF] Weill | A consensus document for the selection of lung transplant candidates: 2014-An update from the Pulmonary Transplantation Council of the International Society for Heart and Lung Transplantation[END_REF] , donor lung selection [START_REF] Reyes | Guidelines for Donor Lung Selection: Time for Revision? The Annals of Thoracic Surgery[END_REF] , treatment allocation [START_REF] Egan | Development of the New Lung Allocation System in the United States[END_REF] , and the surgical procedure itself

MATERIAL AND METHODS

Ethics Statement

The study received Institutional Review Board (Société Française de Chirurgie Thoracique et Cardio-Vasculaire) approval (IRB00012919). Patients provided informed consent for data collection before the lung transplantation procedure. This study was approved for retrospective data collection (CNIL MR-004 2223379-08/25/2021).

Settings and study population

The

Primary outcome

The primary endpoint was 1-year mortality, defined as death from any cause within 365 days after lung transplantation surgery.

Variables

Recipient, donor, and intraoperative variables were collected from the national database of the Agence de la Biomédecine, and from medical reports. Operative reports were screened for important intraoperative variables (intraoperative events, pleural adherence, and difficulty in performing anastomosis). An intraoperative event was defined as one of the following: major bleeding (> 1,000 mL or a major hemorrhagic event during the operation), hemodynamic failure (systolic blood pressure < 50 mmHg) treated with high-dose catecholamines or extracorporeal assistance, cardiac arrest, or visual pulmonary edema.

Development and validation cohorts

The study population was randomly assigned to the development and validation cohorts (ratio of 2:1) using R Software. The two cohorts were compared according to baseline recipient-, donor-, and intraoperative-specific variables, to ensure the reliability of randomization. A predictive model was constructed based on the development cohort data set and then applied to the validation cohort.

Statistical analysis

Categorical variables are presented as number (percentage) and continuous variables as the mean (standard deviation, SD). Continuous variables were compared with Student's t test.

Categorical variables were compared with the chi-squared test and Fisher's exact test according to the expected number of subjects. Statistical analyses were performed using RStudio (version 4.0.3; R Development Core Team, Vienna, Austria). The model with the lowest Akaike information criterion (AIC) was considered the most accurate. The above procedure was performed for each of the three models. Variables retained in model A were force-entered into the multivariable regression model during the construction of models B and C. A maximum ratio of one parameter for five events was used in the final models [START_REF] Van Smeden | No rationale for 1 variable per 10 events criterion for binary logistic regression analysis[END_REF] . The interactions between selected factors and pathology were tested.

Variable linearity was tested using polynomial regression. The collinearity of the variates in the models were assessed using the VIF function (Variance Inflation Factor).

The predictive variables for each model were allocated scores corresponding to their linear (β) coefficients. All patients were assigned a final score corresponding to the sum of their βcoefficients in all three models. The predicted 1-year probability of mortality was calculated for each patient at each time point (A-C), and associated with the final score according to a mathematical formula. Therefore, each patient had three probabilities of mortality, expressed as percentages.

The models were then applied to the validation cohort. The areas under the receiver operator characteristic curves (AUCs) were used to determine the discriminatory power of the predictive models. We performed bootstrap calculation of the AUC within 1000 re-samplings for validation [START_REF] Fernandez-Felix | Bootstrap internal validation command for predictive logistic regression models[END_REF] We investigated the calibration and the overall performance of the models by analyzing Brier's score, calibration slope and calibration-in-the-large [START_REF] Steyerberg | Assessing the performance of prediction models: a framework for traditional and novel measures[END_REF] . The data steering committee defined thresholds for dividing the patients into three groups according to the predicted probability of mortality: low risk (< 15%), intermediate risk (15-45%), and high risk (> 45%).

The predictive accuracy of each model was determined by comparison of the observed and predicted events. Survival analysis was performed using Kaplan-Meier method. We performed decision curve analyses using 'dcurves' package.

In all analyses, p < 0.05 was taken to indicate statistical significance.

RESULTS

The study population consisted of 478 patients (Figure 1). The mean (SD) age of the recipients was 49.0 (14.3) years and 42.9% were women. Chronic obstructive pulmonary disease (COPD, 36.8%) and cystic fibrosis (25.3%) were the most common diagnoses. More than 10% of the population (12.6%) had at least one previous thoracic surgery, 5.9% had a history of cancer, 17.6% had diabetes mellitus, 4.2% had a psychiatric disease, and 4.2% had a history of venous thromboembolism. A total of 75 patients were managed in the intensive care unit (ICU) and 25 (5.2%) required extracorporeal membrane oxygenation (ECMO) before lung transplantation.

The mean (SD) age of the donors was 47.7 (15.7) years and 46.0% were women. The main cause of death was vascular events (56.3%) and the mean (SD) duration of mechanical ventilation before organ retrieval was 2.47 (2.0) days. Among the donors, 37.7% were smokers and 5.5% had a history of pulmonary disease.

Cytomegalovirus (CMV) mismatch (donor-positive [D + ] and recipient-negative [R -]) occurred in 13.4% of cases and sex-matched lung transplantation was performed in 67.2% of cases.

The mean (SD) operative duration was 373.0 (86.6) minutes and the mean (SD) ischemic time for the second implanted lung was 381.7 (66.2) minutes. Intraoperative assistance was used in 70.7% of procedures, consisting of cardiopulmonary bypass and ECMO in 54.8% and 45.2% of cases, respectively. Intraoperative assistance was used in 80% of the cases in the early period (2014-2015) versus 45% in the late period (2018-2019). The mean (SD) arterial oxygen partial pressure to fractional inspired oxygen ratio (PaO2/FiO2 ratio) at the end of the procedure was 247.8 (115.40) mmHg. A total of 108 patients died within 1 year after lung transplantation (Figure 2).

The main recipient, donor, and intraoperative variables were comparable between the development and validation groups (Table S1). Mean (SD) follow-up time was 35 months (25). The 1-year mortality rate was higher in the development than validation cohort, although the difference was not significant (25.1% vs. 17.6%, p = 0.0848).

Univariable regression analysis

The data steering committee selected variables considered as relevant for analysis. Finally, 147 variables were included in the univariable regression analysis (Supplementary Material Table S2).

Recipient variables at registration, donor variables, intraoperative variables, and variables reflecting recipient status at the end of the procedure (impossibility of weaning off assistance, PaO2/FiO2 ratio at the end of the procedure, norepinephrine) were tested in multivariable models (Supplementary Material Table S3). Three multivariable models were finally obtained (Table 1) for each time point and applied to all patients (Supplementary Material S4). The collinearity of the variates in the models were low (Variance Inflation Factor inferior to 5).

The distributions of scores A-C were similar between the development and validation cohorts (p = 0.982, p = 0.611, and p = 0.690, respectively).

The correlations between the individual risk of mortality estimated using models A-C and scores A-C are shown in Figure 3.

The probability of mortality was calculated using the equations shown in Supplementary Material S5.

With regard to discriminatory power, the AUCs (95%CI) for models A-C were 0.67 (0.62 -0.73), 0.70 (0.63 -0.77), and 0.82 (0.77 -0.88), respectively, in the development cohort, 0.74 (0.64-0.85), 0.76 (0.66 -0.86) and 0.87 (0.79 -0.95), respectively, in the validation cohort.

When using the bootstrap calculation for validation, mean AUCs were 0.69, 0.73 and 0.83 for model A, B and C respectively. When investigating calibration in both cohorts, results were interesting (Supplementary Material S6 -S9), and no interactions with diagnosis were observed. Both models had low collinearity between the covariates (Supplementary Material S10). Decision curve analyses showed a benefit in using the models when the probability of death is higher than 10% approximately at timepoint A and B and when the probability of death is higher than 5% at timepoint C (Supplementary Material S11-S13).

In both cohorts, the high-risk group had significantly more events in all models (Table 2), and the proportion of patients who died within 1 year increased with the predicted probability of death. The distribution of predicted probability of death at 1 year was similar between the validation and development cohorts; however, those who had an unfavorable event at 1 year had a significantly higher probability of death in all models (Figure 4).

DISCUSSION

The SALTO study represents the first attempt to estimate the probability of death (and by extension, survival) 1 year after lung transplantation at three different time-points before the postoperative period. This approach was based on three scores: score A (4 predictive variables), score B (8 predictive variables based on recipient-and donor-specific characteristics), and score C (13 predictive variables based on recipient, donor, and procedure characteristics). The three scores for each patient together comprised the absolute risk, i.e., survival probability, calculated prior to the postoperative period.

The outcomes of the study population were somewhat lower to those reported in the Registry of the International Society for Heart and Lung Transplantation (77% in our population vs 81%) 3,4 .

Our models included variables considered as prognostic factors in other studies, such as imaging abnormalities [START_REF] Oto | Feasibility and Utility of a Lung Donor Score: Correlation With Early Post-Transplant Outcomes[END_REF] , donor oxygenation ratio [START_REF] Thabut | Outcomes after lung transplantation[END_REF] , need for assistance at the end of the procedure, and mean pulmonary arterial pressure [START_REF] Hayes | Impact of pulmonary hypertension on survival in patients with cystic fibrosis undergoing lung transplantation: An analysis of the UNOS registry[END_REF] . Surprisingly, other variables such as ICU hospitalization, ECMO, sex, CMV mismatch, and graft ischemia time were not included in the models. For example, it has been reported that patients supported by ECMO before lung transplantation have poorer short-term outcomes [START_REF] Bittner | Outcome of Extracorporeal Membrane Oxygenation as a Bridge to Lung Transplantation and Graft Recovery[END_REF][START_REF] Bermudez | Outcomes of intraoperative venoarterial extracorporeal membrane oxygenation versus cardiopulmonary bypass during lung transplantation[END_REF] , but pre-operative ECMO or intraoperative assistance were not included in predictive models in the present study. The high rate of intraoperative assistance or the size of the population could explain that preoperative ECMO or intraoperative assistance were not included in the predictive models. However, the effect of intraoperative event could be captured by the intraoperative event variable in model C. Moreover, if a variable was not included in our models, it does not mean it is not a prognostic variable but other variables better fit with our predictive models. The small number of patients included in the study (n = 479) and careful patient selection may have impacted the outcomes.

The study power may have limited the detection of factors that are associated with posttransplant mortality, illustrated by the absence of variables previously identified as prognostic factors such as ischemic time. Although variable selection can be discussed, the purpose of this study was not to examine the prognostic factors themselves, but rather to achieve the most accurate prediction of 1-year survival; stepwise backward selection was adequate for this purpose [START_REF] Sanchez-Pinto | Comparison of variable selection methods for clinical predictive modeling[END_REF] . We tested four randomization sequences, which all yielded the same combinations of variables. The variables included in our models differed from those described in predictive scores in previous studies 7, [START_REF] Huppmann | The Munich-LTX-Score: predictor for survival after lung transplantation[END_REF][START_REF] Oto | Feasibility and Utility of a Lung Donor Score: Correlation With Early Post-Transplant Outcomes[END_REF] . Our selected variables could be considered as predictive factors. However, our study design did not permit exploring causal inferences. We cannot assert that the variables included in the model cause death. They may help to predict the outcome.

As an example (supplementary materialsurvival probability calculator), a subject with COPD and mean pulmonary arterial pressure of 18 mmHg would have an estimated probability of death at time A of 11.4% (low risk). With donor characteristics (PaO2/FiO2 ratio of 280, atelectasis and lung condensation on computed tomography (CT)), this probability would increase to 20.3% (intermediate risk). With an intraoperative event (major hemorrhagic event) requiring the transfusion of 6 units of red blood cells and 3.5 μg/kg/min norepinephrine at the end of the procedure, the estimated probability would increase to 51.8%. This prediction corresponds to the worst-case scenario; the reverse situation, i.e., changing from high to low risk, was also possible, but less likely.

Of the three models tested, model C showed the best discrimination and calibration, which was reasonable as predictions should be more accurate closer to the endpoint. However, this could also be affected by the clinical course, as illustrated by the 251 patients showing a change in risk group over the follow-up period. This approach could facilitate decision-making during the entire process of lung transplantation, from recipient registration to surgery. Despite these encouraging results, this study had some limitations. First, the small number of subjects limited the number of parameters that could be analyzed [START_REF] Van Smeden | No rationale for 1 variable per 10 events criterion for binary logistic regression analysis[END_REF] . The resulting AUC for models A and B could be improved with a larger population. On the other hand, the small sample size made it possible to collect complete data in a large proportion of cases. Second, as some data were collected retrospectively, the analyses of certain variables could have lacked accuracy, and some factors could not be integrated into our study, such as the preoperative Karnofsky Performance Status score 7 and human leukocyte antigen (HLA) mismatch [START_REF] Brugière | Relative impact of human leukocyte antigen mismatching and graft ischemic time after lung transplantation[END_REF] . Three centers participated in the study, and survival may have varied according to the protocols used at those centers. In those three centers, the number of intra-operative assistance is rather high (70%). In the early 2010s, in those centers, intra-operative assistance was frequently used to benefit oxygenation and to reduce the occurrence of reperfusion damage [START_REF] Rozé | Cardiopulmonary bypass during a second-lung implantation improves postoperative oxygenation after sequential double-lung transplantation[END_REF] . The number of intra-operative assistance reported in the European centers differs from a center to another (for instance, Atchade et al. reported 75% of intra-operative assistance [START_REF] Atchade | ECMO support as a bridge to lung transplantation is an independent risk factor for bronchial anastomotic dehiscence[END_REF] whereas 28% of the patients required intraoperative ECMO support in the Ius et al study [START_REF] Ius | Five-year experience with intraoperative extracorporeal membrane oxygenation in lung transplantation: Indications and midterm results[END_REF] ). Finally, further prospective studies are needed for external validation before using SALTO prediction models in clinical practice.

SALTO scores are clinically interesting as they could be used to predict changes in risk group between time points A-C. For example, when managing a patient with a high risk of mortality at time A, the goal is to decrease this risk at time B and/or C. Better graft allocation or a specific surgical procedure could be discussed during the meetings pertaining to patient registration for transplant. Ultimately, the aim is to achieve the lowest probability of death at time C. Primary diagnostic indications for transplantation and donor characteristics have changed over the past two decades (e.g., the median donor age increased by 10 years between 2000 and 2018) [START_REF] Chambers | The International Thoracic Organ Transplant Registry of the International Society for Heart and Lung Transplantation: 37th adult lung transplantation report -2020; focus on deceased donor characteristics[END_REF] and would change in the future since proportion of cystic fibrosis patients is decreasing among lung transplantation population. SALTO models could be considered as an actualization of the prediction approach and do not aim to substitute to lung allocation score.

We aimed to provide a new insight of the lung transplantation process. Variables that are not captured by UNOS or ISHLT registries, particularly intraoperative variables, were collected in order to update prediction abilities. In comparison with previous studies 6,7 based on UNOS data in North American population, our models are more specific for the European population.

As medical knowledge and techniques evolve, additional models should be used to corroborate our findings. The coefficients of the models (bêta) should be adjusted based on the inclusion of additional consecutive patients, which could involve the application of artificial intelligence systems [START_REF] Sanchez-Pinto | Comparison of variable selection methods for clinical predictive modeling[END_REF] . The results of this study can be considered as primary data; additional data could be used to improve the models automatically through artificial intelligence.

However, prospective studies are needed for external validation. Predictive models can be updated as new data becomes available, which can help refine the models and improve their accuracy over time. This can lead to more personalized and effective treatments for lung transplant patients.

Lung transplantation is a complex procedure with many factors that can affect outcomes such as donor and recipient characteristics and surgical procedure. Predictive models can be used to identify patients who are at high risk of poor outcomes and who may require more aggressive care or closer monitoring. However, it is important to note that predictive models should always be used in conjunction with clinical judgement and should not be relied on as the sole basis for clinical decision-making.

MANUSCRIPT

  The data underlying this article will be shared on reasonable request to the corresponding author.KEYWORDS: Lung transplantation, predictive model, survivalDownloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023 Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023 Take home message: The models are clinically relevant as they provide individualized predictions of mortality according to each patient's unique situation. Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023

  ABBREVIATIONS

13 .

 13 This study was performed to predict survival after lung transplantation at one-year (SALTO) at three time points: (A) recipient registration on the waiting list; (B) graft allocation; and (C) after the surgical procedure. Three risk groups were distinguished at each time point to help physicians reduce the risk at subsequent time points. Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023

  study population consisted of patients aged ≥ 15 years who underwent lung transplantation between January 2014 and December 2019 at one of three French academic centers. Patients undergoing single lung transplantation, or simultaneous solid organ transplantation were excluded, as were those with previous lung transplantation. Patient data were encrypted and pseudonymized to comply with international data privacy requirements. The study was designed in accordance with the Transparent Reporting of a Multivariable Prediction Model for Individual Prognosis or Diagnosis (TRIPOD) guidelines and Strengthening the Reporting of Observational Studies in Epidemiology (STROBE).

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  Three predictive models were constructed to estimate the 1-year probability of survival at registration on the transplantation list (based on recipient variables; model A), at the time of graft allocation (based on recipient and donor variables; model B), and at the end of the lung transplantation procedure (based on recipient, donor, and intraoperative variables; model C).A data steering committee, including two surgeons and an epidemiologist, overviewed the operative reports. Sixteen reports lacked data and were excluded from the development process for model C. However, these patients were included during the development of models A and B.A univariable logistic regression model was applied to the development cohort to identify potential factors predicting 1-year mortality. Variables with a p-value < 0.25 in the exploratory analysis were incorporated into a multivariable model using backward stepwise regression. Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023

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Figure 5

 5 Figure 5 presents a summary of the mortality prediction for each patient and time point (A-C),

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  Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023 Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023 6. Russo MJ, Davies RR, Hong KN, Iribarne A, Kawut S, Bacchetta M, et al. Who is the high-risk recipient? Predicting mortality after lung transplantation using pretransplant risk factors. J Thorac Cardiovasc Surg. 2009 Nov;138(5):1234-1238.e1. 7. Grimm JC, Valero V, Magruder JT, Kilic A, Dungan SP, Silhan LL, et al. A novel risk score that incorporates recipient and donor variables to predict 1-year mortality in the current Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023

  FIGURES

Figure 1 .

 1 Figure 1. Flow chart of the study population.

Figure 2 .

 2 Figure 2. Kaplan Meier survival curve of overall population.

Figure 3 .

 3 Figure 3. Probability of death at 1 year according to the calculated scores.

Figure 4 .

 4 Figure 4. Predicted probability of death in models A-C. Part 1: red, patients who died within 1

Figure 5 .

 5 Figure 5. Risk group according to calculated model scores: 16 patients had no score for model

  Figure 1

Figure 2

 2 Figure 2

  Figure 3

  Figure 4

Figure 5

 5 Figure 5

Table 1 . Multivariable logistic regression analysis of predictors of 1-year mortality

 1

					R I P T
				S C
		M A N U
	TABLES	C E P T E D
	A C
		Model A		Model B		Model C
	Variable	β	p-value	β	p-value	β	p-value
	Recipient information
	Diagnosis

Table 1 . Multivariable logistic regression analysis of predictors of 1-year mortality

 1

	Cystic fibrosis	Ref A	Ref	Ref	Ref	Ref	Ref
	Emphysema/COPD	0.9727	0.064	0.8013	0.127	1.6249	0.011
	Pulmonary hypertension	0.6159	0.477	0.3806	0.671	0.5588	0.604
	Fibrosis	1.6869	0.002	1.6815	0.002	1.8299	0.005
	Other Psychiatric disease History of venous thromboembolism	1.0019 1.4226 1.7798	0.117 0.021 0.006	0.9733 1.4650 1.8994	0.126 0.024 0.005 R I P T 1.2658 1.6031 1.8738	0.114 0.041 0.019
	Mean pulmonary artery pressure (mmHg) Not measured < 20 20-40 > 40	1.1070 Ref 0.9115 1.0695 M A N 0.102 Ref 0.058 0.107 U 1.1787 Ref 0.9861 1.0929 S C 0.089 Ref 0.046 0.111	0.8225 Ref 1.1035 1.1193	Ref 0.059 0.150 0.344
	Donor information PaO2/FiO2 ratio B Atelectasis Imaging C E P T E D --	--	-0.0220 0.4420	0.101 0.118	-0.0208 0.4456	0.193 0.181
	Normal ≥ 1 abnormality C A C	--	--	Ref 0.3995	Ref 0.172	Ref 0.5794	Ref 0.084
	Positive blood culture	-	-	0.9316	0.052	1.2666	0.023
	Procedure
	Intraoperative information:

Table 1 . Multivariable logistic regression analysis of predictors of 1-year mortality

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	Major bilateral pleural adherence	-	-	-	-	1.1028	0.041
	≥ 1 intraoperative event	-	-	-	-	0.3877	0.306
	Red blood cell transfusion, units	-	-	-	-	0.1025	0.008
	At the end of the procedure:
	Norepinephrine rate (μg/kg/min)	-	-	-	-	1.1028	0.003
	Need of assistance	-	-	-	-	1.0159	0.022
	COPD, chronic obstructive pulmonary disease.

A Ref: reference. B PaO2/FiO2 divided by 10. C Excluding atelectasis.

Table 2 .

 2 One-year mortality by risk group Downloaded from https://academic.oup.com/ejcts/advance-article/doi/10.1093/ejcts/ezad167/7143385 by Universite de Bordeaux user on 13 May 2023

			Development Cohort			Validation Cohort
		Number of	OR (95% CI)	p-value	Number of	OR (95% CI)	p-value
		events			events
		(n/total)			(n/total)
	Model A
	Risk group			<0.001			0.009
	Low-risk Intermediate-risk High-risk	13/92 55/210 12/17	Ref 2.16 (1.11-4.18) 14.58 (4.41-48.27)	Ref 0.023 <0.001	3/52 21/95 4/12 R I P T 4.64 (1.31-16.38) Ref 8.17 (1.53-43.52)	Ref 0.017 0.014
	For 10% increase in predicted mortality risk		1.69 (1.37-2.08)	<0.001 S C	1.53 (1.15-2.03)	0.004
	Model B Risk group Low-risk	13/92	<0.001 Ref A N Ref M U	4/52	Ref	0.019 Ref
	Intermediate-risk High-risk For 10% increase in predicted mortality risk Model C C E P T E D 33/166 1.8 (0.92-3.54) 34/61 11.31 (4.43-28.85) 1.72 (1.44-2.06)	0.086 <0.001 <0.001	15/72 9/35	2.91 (0.92-9.2) 6.46 (1.64-25.5) 1.43 (1.11-1.84)	0.068 0.008 0.005
	Risk group Low-risk A C	11/132	Ref	<0.001 Ref	2/64	Ref	<0.001 Ref
	Intermediate-risk	33/129	3.78 (1.82-7.87)	<0.001	8/57	5.06 (1.03-24.92)	0.046
	High-risk	31/47	21.31 (8.99-50.52)	<0.001	15/33	25.83 (5.4-123.67)	<0.001
	For 10% increase in		1.74 (1.5-2.02)	<0.001		1.67 (1.36-2.05)	<0.001
	predicted mortality risk

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ACKNOWLEDGMENTS

The authors would like to thank the Agence de Biomédecine for their support with the data collection and cleaning.