Barbara Dembin
email: bdembin@lpsm.paris

Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous *

Keywords: Regularity, percolation, isoperimetric constant Ams msc 2010: primary 60K35, secondary 82B43

 is Lipschitz continuous on all intervals [p 0 , p 1 ] ⊂ (p c (d), 1).

Introduction

The study of isoperimetric problems in the discrete setting is more recent than in the continuous setting. In the continuous setting, we study the perimeter to volume ratio; in the context of graphs, the analogous problem is the study of the size of edge boundary to volume ratio. This can be encoded by the Cheeger constant. For a finite graph G = (V (G), E(G)), we define the edge boundary ∂ G A of a subset A of V (G) as

∂ G A = e = x, y ∈ E(G) : x ∈ A, y / ∈ A .
We denote by |B| the cardinal of the finite set B. The isoperimetric constant of G, also called Cheeger constant, is defined as

ϕ G = min |∂ G A| |A| : A ⊂ V (G), 0 < |A| ≤ |V (G)| 2 .
This constant was introduced by Cheeger in his thesis [START_REF] Cheeger | A lower bound for the smallest eigenvalue of the laplacian[END_REF] in order to obtain a lower bound for the smallest eigenvalue of the Laplacian. The isoperimetric constant of a graph gives information on its geometry. Let d ≥ 2. We consider an i.i.d. supercritical bond percolation on the graph (Z d , E d ) having for vertices Z d and for edges E d the set of pair of nearest neighbors in Z d for the Euclidean norm. Every edge e ∈ E d is open with a probability p > p c (d), where p c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C p [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]. In this paper, we want to study how the geometry of C p varies with p through its Cheeger constant. However, if we minimize the isoperimetric ratio over all possible subgraphs of C p without any constraint on the size, one can prove that ϕ Cp = 0 almost surely. For that reason, we shall minimize the isoperimetric ratio over all possible subgraphs of C p given a constraint on the size. There are several ways to do it. We can for instance study the Cheeger constant of the graph C n = C p ∩ [-n, n] d or of the largest connected component C n of C n for n ≥ 1. As we have ϕ Cp = 0 almost surely, the isoperimetric constants ϕ Cn and ϕ Cn go to 0 when n goes to infinity. Roughly speaking, by analogy with the full lattice, we expect that subgraphs of C n that minimize the isoperimetic ratio have an edge boundary size of order n d-1 and a size of order n d .

In [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF], Biskup, Louidor, Procaccia and Rosenthal defined a modified Cheeger constant ϕ Cn and proved that n ϕ Cn converges towards a deterministic constant in dimension 2. In [START_REF] Gold | Isoperimetry in supercritical bond percolation in dimensions three and higher[END_REF], Gold proved the same result in dimension d ≥ 3. Instead of considering the open edge boundary of subgraphs within C n , they considered the open edge boundary within the whole infinite cluster C p , this is more natural because C n has been artificially created by restricting C p to the box [-n, n] d . They also added a stronger constraint on the size of subgraphs of C n to ensure that minimizers do not touch the boundary of the box [-n, n] d . Moreover, they proved that the subgraphs achieving the minimum, properly rescaled, converge towards a deterministic shape that is the Wulff crystal. Namely, it is the shape solving the continuous anisotropic isoperimetric problem associated with a norm β p corresponding to the surface tension in the percolation setting. The quantity n ϕ Cn converges towards the solution of a continuous isoperimetric problem.

This modified Cheeger constant was inspired by the anchored isoperimetric profile ϕ n (p). This is another way to define the Cheeger constant of C p , that is more natural in the sense that we do not restrict minimizers to remain in the box [-n, n] d . It is defined as follows:

ϕ n (p) = min |∂ Cp H| |H| : 0 ∈ H ⊂ C p , H connected, 0 < |H| ≤ n d ,
where we condition on the event {0 ∈ C p }. We say that H is a valid subgraph if 0 ∈ H ⊂ C p , H is connected and |H| ≤ n d . We also define the open edge boundary of H as:

∂ o H = e ∈ ∂H, e is open
where ∂H is the edge boundary of

H in (Z d , E d ). Note that if H ⊂ C p , then ∂ Cp H = ∂ o H.
We need to introduce some definitions to be able to define properly a limit shape in dimension d ≥ 2. In order to build a continuous limit shape, we shall define a continuous analogue of the cardinal of the open edge boundary. In fact, we will see that the cardinal of the open edge boundary may be interpreted in term of a surface tension I, in the following sense. Given a norm τ on R d and a subset E of R d having a regular boundary, we define I τ (E) as

I τ (E) = ∂E τ (n E (x))H d-1 (dx) ,
where H d-1 denotes the Hausdorff measure in dimension d -1 and n E (x) is the normal unit exterior vector of E at x. The quantity I τ (E) represents the surface tension of E for the norm τ . At the point x, the tension has intensity τ (n E (x)) in the direction of n E (x). We denote by L d the d-dimensional Lebesgue measure. We can associate with the norm τ the following isoperimetric problem:

minimize I τ (E) L d (E) subject to L d (E) ≤ 1 .
We use the Wulff construction to build a minimizer for this anisotropic isoperimetric problem (see [START_REF] Wulff | Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen[END_REF]). We define the set W τ as

W τ = v∈S d-1 x ∈ R d : x • v ≤ τ (v) ,
where • denotes the standard scalar product and S d-1 is the unit sphere of R d . Taylor proved in [START_REF] Taylor | Unique structure of solutions to a class of nonelliptic variational problems[END_REF] that the set W τ properly rescaled is the unique minimizer, up to translations and modifications on a null set, of the associated isoperimetric problem. We need to build an appropriate norm β p for our problem that will be directly related to the cardinal of the open edge boundary.

In [START_REF] Dembin | Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher[END_REF], Dembin proves the existence of the limit of nϕ n (p) and that it converges towards the solution of the continuous isoperimetric problem associated with the norm β p . Theorem 1. Let d ≥ 2, p > p c (d) and let β p be the norm that will be properly defined in section 2. Let W p be a dilate of the Wulff crystal W βp for the norm β p such that L d (W p ) = 1/θ p where θ p = P(0 ∈ C p ). Then, conditionally on the event {0 ∈ C p },

lim n→∞ nϕ n (p) = I p (W p ) θ p L d (W p ) = I p (W p ) a.s.. Remark 1.1.
Actually, the same result holds when we condition on the event {0 ∈ C p0 } for any p 0 ∈ (p c (d), p].

In this paper, we aim to study the regularity properties of the anchored isoperimetric profile. This was first studied by Garet, Marchand, Procaccia, Théret in [START_REF] Garet | Continuity of the time and isoperimetric constants in supercritical percolation[END_REF], they proved that the modified Cheeger constant in dimension 2 is continuous on (p c (2), 1]. The aim of this paper is the proof of the two following theorems. The first theorem asserts that the anchored isoperimetric profile is Lipschitz continuous on every compact interval [p 0 , p 1 ] ⊂ (p c (d), 1).

Theorem 2 (Regularity of the anchored isoperimetric profile). Let d ≥ 2. Let p c (d) < p 0 < p 1 < 1. There exits a positive constant ν depending only on d, p 0 and p 1 , such that for all p, q ∈ [p 0 , p 1 ], conditionally on the event {0 ∈ C p0 },

lim n→∞ n|ϕ n (q) -ϕ n (p)| ≤ ν|q -p| .
Remark 1.2. Actually, the Cheeger constant is also continuous at 1, this is not a consequence of Theorem 2 but it comes from the fact that the map p → β p is continuous on (p c (d), 1]. This result is a corollary of Theorem 4 in [START_REF] Rossignol | Existence and continuity of the flow constant in first passage percolation[END_REF].

Remark 1.3. We did not manage to obtain here that the anchored isoperimetric profile is Lipschitz continuous on [p 0 , 1] for a technical reason that is due to a coupling we use in the proof of Theorem 2. However, this restriction is likely irrelevant.

The second theorem studies the Hausdorff distance between two Wulff crystals associated with norms β p and β q .

Theorem 3 (Regularity of the anchored isoperimetric profile). Let d ≥ 3. Let p c (d) < p 0 < p 1 < 1. There exits a positive constant ν depending only on d, p 0 and p 1 , such that for all p, q ∈ [p 0 , p 1 ],

d H ( W βp , W βq ) ≤ ν |q -p| ,
where d H is the Hausdorff distance between non empty compact sets of R d .

The key element to prove these two theorems is to prove the regularity of the map p → β p . We recall that it is already known that the map p → β p is continuous on (p c (d), 1].

Theorem 4 (Regularity of the flow constant). Let p c (d) < p 0 < p 1 < 1. There exists a positive constant κ depending only on d, p 0 and p 1 , such that for all p ≤ q in [p 0 , p 1 ],

sup x∈S d-1 |β p (x) -β q (x)| ≤ κ|q -p| .
The proof of this theorem will strongly rely on an adaptation of the proof of Zhang in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF].

Remark 1.4. In this paper, we choose to work on the anchored isoperimetric profile instead of the modified Cheeger constant because the norm we use is the same for all dimensions d ≥ 2. The existence of the modified Cheeger constant in dimension 2 uses another norm specific to this dimension (see [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF]). In [START_REF] Gold | Isoperimetry in supercritical bond percolation in dimensions three and higher[END_REF], Gold proved the existence of the modified Cheeger constant for d ≥ 3 with the same norm β p . Actually, we believe that his proof also holds in dimension 2 up to using similar combinatorial arguments as in [START_REF] Dembin | Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher[END_REF]. Therefore, Theorem 2 may be shown for the modified Cheeger constant in dimension d ≥ 2 using the same ingredients as in this paper.

Here is the structure of the paper. In section 2, we define the norm β p . We prove that the map p → β p is Lipschitz continuous in section 3. Finally, we prove Theorems 2 and 3 in section 4.

Definition of the norm β p

We introduce now many notations used for instance in [START_REF] Rossignol | Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation[END_REF] concerning flows through cylinders. Let A be a non-degenerate hyperrectangle, that is to say a rectangle of dimension d -1 in R d . Let -→ v be one of the two unit vectors normal to A. Let h > 0, we denote by cyl(A, h) the cylinder with base A and height 2h defined by

cyl(A, h) = {x + t - → v : x ∈ A, t ∈ [-h, h]} . The set cyl(A, h) \ A has two connected components, denoted by C 1 (A, h) and C 2 (A, h). For i = 1, 2, we denote by C i (A, h) the discrete boundary of C i (A, h) defined by C i (A, h) = x ∈ Z d ∩ C i (A, h) : ∃y / ∈ cyl(A, h), x, y ∈ E d .
We say that the set of edges

E cuts C 1 (A, h) from C 2 (A, h) in cyl(A, h) if any path γ from C 1 (A, h) to C 2 (A, h) in cyl(A,
τ p (A, h) = min {|E| o,p : E cuts C 1 (A, h) from C 2 (A, h) in cyl(A, h)} .
Note that it is a random quantity as |E| o,p is random, and that the cutsets in this definition are anchored at the border of A. This quantity is related to the fact that graphs that achieve the infimum in the definition of ϕ n (p) try to minimize their open edge boundary. We refer to section 3 in [START_REF] Dembin | Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher[END_REF] for more detailed explanations on the construction of this norm β p . To build a norm upon this quantity, we use the fact that the quantity τ p (A, h) properly renormalized converges towards a deterministic constant when the size of the cylinder goes to infinity.

The following proposition is a corollary of Proposition 3.5 in [START_REF] Rossignol | Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation[END_REF].

Proposition 1 (Definition of the norm β p ). Let d ≥ 2, p > p c (d), A be a non-degenerate hyperrectangle and -→ v one of the two unit vectors normal to A. Let h be a height function such that lim n→∞ h(n) = ∞. The limit

β p ( - → v ) = lim n→∞ E[τ p (nA, h(n))] H d-1 (nA)
exists and is finite. Moreover, the limit is independent of A and h and the homogeneous extension of β p to R d is a norm.

As the limit does not depend on A and h, in the following for simplicity, we will take

h(n) = n and A = S( - → v ) where S( - → v ) is a square isometric to [-1, 1] d-1 × {0} normal to - → v . We will denote by B(n, - → v ) the cube cyl(nS( - → v ), n) and by τ p (n, - → v ) the quantity τ p (nS( - → v ), n).
3 Regularity of the map p → β p Let p 0 > p c (d) and let q > p ≥ p 0 . Our strategy is the following, we easily get that β p ≤ β q by properly coupling the percolations of parameters p c (d) < p < q. The second inequality requires more work. We denote by E n,p the random cutset of minimal size that achieves the minimum in the definition of τ p (n, -→ v ). By definition, as E n,p is a cutset, we can bound by above τ q (n, -→ v ) by the number of edges in E n,p that are q-open, which we expect to be at most

τ p (n, - → v ) + C(q -p)|E n,p |
P p N n,p > αn d-1 ≤ C 1 exp(-C 2 n d-1
) .

Remark 3.1. The proof is going to be simpler than the proof of Theorem 2 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF], because passage times in our context can take only values 0 or 1, i.e., to each edge we associate an i.i.d random variable of distribution G p = pδ 1 + (1 -p)δ 0 whereas Zhang considers in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF] more general distributions. Our setting is equivalent to bond percolation of parameter p by saying that an edge is closed if its passage time is 0, and open if its passage time is 1.

Let us briefly explain the idea behind that theorem. Let p ≥ p 0 . We work on bond percolation of parameter p (equivalently on first passage percolation with distribution G p = pδ 1 +(1-p)δ 0 ). We aim at bounding the size of the smallest minimal cutset that cuts the set

C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n) in B(n, - → v ).
To do so we do a renormalization at a scale t in order to build a "smooth" minimal cutset. The collection (B t (u)) u∈Z d is a partition of

Z d into boxes of size t and Bt (u) = v * ∼u B t (u) where v * ∼ u if u -v ∞ = 1.
We will need the following Lemma that controls the probability that a p-atypical event occurs in a cube. We will prove this lemma after proving Theorem 5 Lemma 1 (Uniform decay of the probability an atypical event occurs). Let p 0 > p c (d).

There exist positive constants C 1 (p 0 ) and C 2 (p 0 ) depending only on p 0 and d such that for all p ≥ p 0 , for all u ∈ Z d , for all t ≥ 1,

P [a p-atypical event occurs in B t (u)] ≤ C 1 (p 0 ) exp(-C 2 (p 0 )t) . (1)
We would like to highlight the fact that in Lemmas 6 and 7 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF], Zhang proves the same result but with constants C 1 and C 2 depending on p. Obtaining a decay that is uniform for p ∈ [p 0 , 1] is the key element to adapt this proof and show that the constant α in the statement of the Theorem 5 does depend only on p 0 and d.

As the original proof is very technical, the adaptation of the proof is also technical.

Adaptation of the proof of Theorem 1 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF] to get Theorem 5 using Lemma 1. We keep the same notations as in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]. The following adaptation is not self-contained. Let p 0 > p c (d) and -→ v ∈ S d-1 . In [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF], the author bounds the size of the smallest minimal cutset that cuts a given set of vertices V from infinity. However, his construction of a linear cutset in section 2 of [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF] is not specific to the set B(k, m) and can be defined in the same way for any set of vertices. In particular we can replace B(k, m) by

C 1 (nS( - → v ), n) and ∞ by C 2 (nS( - → v ), n)
(as it is done by Zhang in Theorem 2 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]). We denote by C(n) the set that corresponds to C(k, m) defined in Lemma 1 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]:

C(n) = {v ∈ Z d : v is connected to C 1 (nS( - → v ), n) by an open path } .
We denote by G(n) the event that ]). On this event, the exterior edge boundary ∆ e C(n

C(n) ∩ C 2 (nS( - → v ), n) = ∅ (it corresponds to G(k, m) in [ 12
) of C(n) is a closed cutset that cuts C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n).
We denote by A the set of t-cubes that intersect ∆ e C(n).

By Zhang construction, we can extract from A a set of cubes Γ t such that Γ t is * -connected and the union Γt of the 3t-cubes in Γ t (the cubes in Γ t and their * -neighbors) contains a cutset of null capacity that cuts the set The set . Thus, we obtain that

C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n).
E = { x, y ∈ B(n, - → v ) : x ∈ C 1 (nS( - → v ), n) } cuts the set C 1 (nS( - → v ), n) from the set C 2 (nS( - → v ), n) in B(n, - → v )
τ p (n, - → v ) ≤ |E| ≤ c d n d-1 .
We denote by E n,p the cutset that achieves the infimum in τ p (n, -→ v ) and such that |E n,p | = N n,p (E n,p corresponds to W (k, m) in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]). For a configuration ω, we denote by e 1 , . . . , e J(ω)

the p-open edges in E n,p . We have

J(ω) = τ p (n, - → v )(ω) ≤ c d n d-1
. We denote by σ(ω)

the configuration which coincides with ω except in edges e 1 , . . . , e J(ω) that are closed for σ(ω). Thus, the set E n,p (σ(ω)) is a p-closed (for the configuration σ(ω)) cutset that cuts

C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n) in B(n, - → v ).
Note that the set of edges E n,p (σ(ω)) is determined by the configuration ω whereas we consider its capacity for σ(ω). We recall that all the edges outside B(n, -→ v ) are closed so that the event G(n) occurs in the configuration σ(ω) and we can use the construction of section 2 in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]: Γt contains a p-closed (for σ(ω)) Lemma 4 in [12]). By taking the intersection of this cutset with the box B(n, -→ v ), we obtain the existence of a closed cutset

cutset Γ that cuts C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n) (see
that cuts C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n) in B(n, - → v ).
We now change σ(ω) back to ω. For i ∈ {1, . . . , J(ω)}, the passage time of e i changes from 0 to 1. We write Γ(ω) when we consider the edge set Γ with its edges capacities determined by the configuration ω. The set Γ(ω) exists as an edge set, it is still a cutset but it is no longer closed, all edges in Γ(ω) except the e i are closed. Therefore, |Γ(ω)| o,p ≤ J(ω), but by definition of E n,p , we have

J(ω) = |E n,p (ω)| o,p ≤ |Γ(ω)| o,p ≤ J(ω) and so |Γ(ω)| o,p = J(ω).
Moreover, for each ω, by definition of N n,p (ω), we get that |Γ(ω)| ≥ N n,p (ω).

Note that for the t-cubes B t (u) ∈ Γ t such that Bt (u) intersects the boundary of B(n, -→ v ),

we cannot be sure that there exists a t-cube in Bt (u) where a p-atypical event occurs, but the number of such cubes is at most C d,t n d-1 . Thus, if the number of t-cubes in Γ t is greater than βn d-1 , then the number of t-cubes in Γ t that do not intersect the boundary of B(n, -→ v ) and that do not contain any edge among e 1 , . . . , e J is greater than (β -C d,t -c d )n d-1 . All these t-cubes have at least one * -neighbor with a blocked or disjoint property. This leads to small modifications of constants in the proof of [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]. We insist on the fact that the remainder of the proof is the same except that we use Lemma 1, i.e., a uniform decay for p ∈ [p 0 , 1] of the probability of a p-atypical event instead of using the control in [START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF].

Let us now prove Lemma 1. We need to adapt some existing proofs in order to obtain a decay which is uniform in p. Let us first introduce some useful definitions. We say that B t (u) has a p-blocked property if there is a p-open cluster C in Bt (u) with vertices in B t (u) and in the boundary of Bt (u), but without vertices in a t-cube of Bt (u). We say that a p-atypical event occurs in B t (u) if it has a p-blocked property or a p-disjoint property (see Figure 1).

Bt (u)

B t (u)
Proof of Lemma 1. First, note that if B t (u) has a p-disjoint property and Bt (u) has a pcrossing cluster, then one of the two disjoint cluster is different from the p-crossing cluster. Therefore, there is a p-open cluster of diameter greater than t different from the p-crossing cluster, so the event T t,3t (p) occurs in the box Bt (u). Similarly, let us assume that B t (u) has a p-blocked property and Bt (u) and all of its sub-boxes (i.e, boxes B t (v) such that B t (v) ⊂ Bt (u)) have a p-crossing cluster. We denote by C the p-open cluster in the definition of the p-blocked property. Thus, there is at least one cluster among C and the p-crossing clusters of the sub-boxes that are disjoint from the p-crossing cluster of Bt (u) and so the event T t,3t (p) occurs in the box Bt (u). Thus,

P[a p-atypical event occurs in B t (u) ] ≤ P[ Bt (u) does not have a p-crossing cluster] + 3 d P[B t (u) does not have a p-crossing cluster] + P [T t,3t (p)] (2)
As the event {B t (u) doesn't have a p-crossing cluster} is non-increasing in p, we have

P[B t (u) doesn't have a p-crossing cluster] ≤ P[B t (u) doesn't have a p 0 -crossing cluster] .
The probability for a box B t (u) not to have a p 0 -crossing cluster is decaying exponentially fast with t d-1 , see for instance Theorem 7.68 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]. Therefore, there exist positive constants c 1 (p 0 ) and c 2 (p 0 ) such that

P[B t (u) does not have a p-crossing cluster] ≤ c 1 (p 0 ) exp(-c 2 (p 0 )t d-1 ) . (3)
It remains to prove that there exist positive constants κ(p 0 ) and µ(p 0 ) depending only on p 0 such that for all p ≥ p 0 , for all positive integers m and N

P[T m,N (p)] ≤ κN 2d exp(-µm) . (4)
In dimension d ≥ 3, we refer to the proof of Lemma 7.104 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]. The proof of Lemma 7.104 requires the proof of Lemma 7.78. The probability controlled in Lemma 7.78 is clearly non decreasing in the parameter p. Thus, if we choose δ(p 0 ) and L(p 0 ) as in the proof of Lemma 7.78 for p 0 > p c (d), then these parameters can be kept unchanged for some p ≥ p 0 . Thanks to Lemma 7.104, we obtain

∀p ≥ p 0 , P(T m,N (p)) ≤ d(2N + 1) 2d exp m L(p 0 ) + 1 -1 log(1 -δ(p 0 )) ≤ d.3 d 1 -δ(p 0 ) N 2d exp - -log(1 -δ(p 0 )) L(p 0 ) + 1 m .
We get the result with

κ = d.3 d 1 -δ(p 0 ) and µ = -log(1 -δ(p 0 )) L(p 0 ) + 1 > 0 .
In dimension 2, the result is obtained by Couronné and Messikh in the more general setting of FK-percolation, see Theorem 9 in [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF]. We proceed similarly as in dimension d ≥ 3, the constant appearing in this theorem first appeared in Proposition 6. The probability of the event considered in this proposition is clearly increasing in the parameter of the underlying percolation which have parameter 1 -p, it is an event for the subcritical regime of the Bernoulli percolation. Let us fix a p 0 > p c (2) = 1/2, then 1 -p 0 < p c (2) and we can choose the parameter c(1 -p 0 ) and keep it unchanged for some 1 -p ≤ 1 -p 0 . In Theorem 9, we get the expected result with c(1 -p 0 ) for a p ≥ p 0 and g(n) = n. The result follows.

We have now the key ingredients to prove that the map p → β p is Lipschitz continuous.

Proof of Theorem 4. Let p c < p 0 < p 1 < 1, -→ v ∈ S d-1 , and p, q such that p 0 ≤ p < q ≤ p 1 .

First, we fix a cube B(n, -→ v ) and we couple the percolations of parameters p and q in the standard way, i.e., we consider the i.i.d. family (U (e)) e∈E d distributed according to the uniform law on [0, 1] and we say that an edge e is p-open (resp. q-open) if U (e) ≥ p (resp. U (e) ≥ q). Thanks to this coupling, we easily obtain that τ p ( -→ v , n) ≤ τ q ( -→ v , n) and by dividing by (2n) d-1 , taking the expectation and letting n go to infinity we conclude that

β p ( - → v ) ≤ β q ( - → v ) . (5)
Let E n,p be a random cutset of minimal size that achieves the minimum in the definition of τ p (n, -→ v ). We consider now another coupling. The idea is to introduce a coupling of the percolations of parameter p and q such that if an edge is p-open then it is q-open and E n,p is independent of the q-state of any edge. Unfortunately, we cannot find such a coupling but we can introduce a coupling that almost has this property. To do so, for each edge we consider two independent Bernoulli random variables U and V of parameters p and (q -p)/(1 -p).

We say that an edge e is p-open if U (e) = 1 and that it is q-open if U (e) = 1 or V (e) = 1. Indeed,

P[{U = 1} ∪ {V = 1}] = p + (1 -p) q -p 1 -p = q .
Let δ > 0. We have,

P τ q (n, - → v ) > τ p (n, - → v ) + q -p 1 -p + δ αn d-1 , N n,p < αn d-1 ≤ P τ q (n, - → v ) -τ p (n, - → v ) > q -p 1 -p + δ |E n,p | ≤ E P E n,p = E, #{e ∈ E : (U (e), V (e)) = (0, 1)} > q -p 1 -p + δ |E| ≤ E P[E n,p = E] P #{e ∈ E : V (e) = 1} > q -p 1 -p + δ |E| ≤ exp(-2δ 2 n d-1 ) (6)
where the sum is over sets

E that cut C 1 (nS( - → v ), n) from C 2 (nS( - → v ), n) in B(n, - → v
) and where we use in the last inequality Chernoff bound and the fact that |E n,p | ≥ n d-1 (uniformly in -→ v ). Finally, using inequality (6) and Theorem 5, we get

E[τ q (n, - → v )] ≤ E[τ q (n, - → v )1 Nn,p<αn d-1 ] + E[τ q (n, - → v )1 Nn,p≥αn d-1 ] ≤ E[τ p (n, - → v )] + q -p 1 -p + δ αn d-1 + |B(n, - → v )| e -2δ 2 n d-1 +C 1 e -C2n d-1 ≤ E[τ p (n, - → v )] + q -p 1 -p + δ αn d-1 + C d (2n) d e -2δ 2 n d-1 +C 1 e -C2n d-1 ,
where C d is a constant depending only on d. Dividing by (2n) d-1 and by letting n go to infinity, we obtain

β q ( - → v ) ≤ β p ( - → v ) + q -p 1 -p + δ α 2 d-1 (7)
and by letting δ go to 0,

β q ( - → v ) ≤ β p ( - → v ) + κ(q -p) (8)
where κ = α/((1 -p 1 )2 d-1 ). Combining inequalities ( 5) and ( 8), we obtain that sup

- → v ∈S d-1 |β q ( - → v ) -β p ( - → v )| ≤ κ|q -p| .
4 Proof of Theorems 2 and 3

Proof of Theorem 2. Let p c < p 0 < p 1 < 1 and p, q ∈ [p 0 , p 1 ]. We recall that W p denotes the Wulff crystal for the norm β p such that L d (W p ) = 1/θ p . In this section we aim to prove that the map p → I p (W p ) is Lipschitz continuous on [p 0 , p 1 ]. Notice that as the map p → θ p is non-decreasing, for p < q we have

L d (W p ) ≥ L d (W q ) . (9)
Moreover, the map p → θ p is infinitely differentiable, see for instance Theorem 8.92 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]. Therefore, there exists a constant L depending on p 0 , p 1 and d such that for all p, q ∈ [p 0 , p 1 ],

|θ p -θ q | ≤ L|q -p| . (10)
Let us compute now some useful inequalities. For any set E ⊂ R d with Lipschitz boundary, by Theorem 4, we have

|I p (E) -I q (E)| = ∂E (β p (n E (x)) -β q (n E (x))) H d-1 (dx) ≤ ∂E |β p (n E (x)) -β q (n E (x))| H d-1 (dx) ≤ κ|q -p|H d-1 (∂E) . (11)
We recall that the map p → β p is uniformly continuous on [p 0 , p 1 ]. We denote by β min and β max its minimal and maximal value, i.e., for all -→ v ∈ S d-1 and p ∈ [p 0 , p 1 ], we have

β min ≤ β p ( - → v ) ≤ β max .
Together with inequality ( 9) and the fact that the Wulff crystal is a minimizer for an isoperimetric problem, we get

I p (W p ) ≤ I p (W p0 ) = ∂Wp 0 β p (n Wp 0 (x))H d-1 (dx) ≤ β max H d-1 (∂W p0 ) . (12)
We also have

H d-1 (∂W p ) = ∂Wp H d-1 (dx) ≤ ∂Wp β p (n Wp (x)) β min H d-1 (dx) ≤ I p (W p ) β min
and so together with inequality (12), we get

H d-1 (∂W p ) ≤ H d-1 (∂W p0 ) β max β min . (13)
Finally, we obtain combining inequalities ( 9), ( 11) and (13), I p (W p ) ≥ I q (W p ) -κ|q -p|H d-1 (∂W p ) ≥ I q (W q ) -κ|q -p|H d-1 (∂W p0 )

β max β min . ( 14
)
As L d (W p ) = L d (W q ) • θ q /θ p = L d (W q (θ q /θ p ) 1/d ) and as W p is the minimizer for the isoperimetric problem associated with the norm β p , we have

I p (W p ) ≤ I p θ q θ p 1/d
W q ≤ θ q θ p (d-1)/d I p (W q ) ≤ θ q θ p I p (W q ) and so using inequalities [START_REF] Taylor | Unique structure of solutions to a class of nonelliptic variational problems[END_REF], [START_REF] Wulff | Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen[END_REF], ( 12) and (13) I p (W p ) ≤ θ q θ p I q (W q ) + κ|q -p|H d-1 (∂W q )

≤ 1 + L θ p0 |q -p| I q (W q ) + κ|q -p|H d-1 (∂W p0 ) β max β min ≤ I q (W q ) + β max .

Note that y/β * p (y) (resp. y/β * q (y)) is in the unit sphere for the norm β * p (resp. β * q ). Let x ∈ S d-1 . Using the definition of β * , we obtain

1 β max ≤ x • x β p (x) ≤ β * p (x) .
Finally, we have

d H (B β * p , B β * q ) ≤ sup y∈S d-1 1 β * p (y) - 1 β * q (y) ≤ sup y∈S d-1 1 β * q (y)β * p (y)
β * p (y) -β * q (y)

≤ sup y∈S d-1 (β max ) 2 β * p (y) -β * q (y) ≤ κ(β max ) 2 (β min ) 2 |q -p| . (18)
The result follows.

  h) contains at least one edge of E. We call such a set a cutset. For any cutset E, let |E| o,p denote the number of p-open edges in E. We shall call it the p-capacity of E. Define

  and there exists a constant c d depending only on d but not on -→ v such that |E| ≤ c d n d-1

Figure 1 -

 1 Figure 1 -On the left a box with a disjoint property, on the right a box with a blocked property

  Finally, combining inequalities (2), (3) and (4), we get P[a p-atypical event occurs in B t (u)] ≤ c 1 (p 0 ) exp(-c 2 (p 0 )(3t) d-1 ) + 3 d c 1 (p 0 ) exp(-c 2 (p 0 )t d-1 ) + κ(p 0 )(3t) 2d exp(-µ(p 0 )t) .

H d- 1 (

 1 ∂W p0 ) (14) and (15) together with Theorem 1, conditionally on the event {0 ∈ C p0 }, we getlim n→∞ n| ϕ n (q) -ϕ n (p)| = |I p (W p ) -I q (W q )| ≤ ν|q -p|(16)where we setν = β max H d-1 (∂W p0 ) L θ p0 + κ β min 1 + L θ p0 .Proof of Theorem 3. Let p c < p 0 < p 1 < 1 and p, q ∈ [p 0 , p 1 ]. We consider β * p the dual norm of β p , defined by∀x ∈ R d , β * p (x) = sup{x • z : β p (z) ≤ 1} .Then β * p is a norm. The Wulff crystal W βp associated with β p is in fact the unit ball associated with β * p . Note that the supremum in the definition of β * p is always achieved for a z such that β p (z) = 1. Let x ∈ S d-1 . Let y ∈ S d-1 be the direction that achieves the supremum for β * p (x), thus we haveβ * p (x) = x • y β p (y)and so using Theorem 2,β * p (x) -β * q (x) ≤ x • y β p (y) -x • y β q (y) ≤ x 2 y 2 β p (y)β q (y) |β p (y) -β q (y)| ≤ κ (β min ) 2 |q -p|where β min was defined in the proof of Theorem 2. We proceed similarly for β * q (x) -β * p (x). Finally, we obtain supx∈S d-1 |β * p (x) -β * q (x)| ≤ κ (β min ) 2 |q -p| . (17)We recall the following definition of the Hausdorff distance between two subsets E and F ofR d : d H (E, F ) = inf{r ∈ R + : E ⊂ F r and F ⊂ E r }where E r = {y : ∃x ∈ E, y -x 2 ≤ r}. Thus, we haved H ( W β * p , W βq ) ≤ sup y∈S d

  where C is a constant. We next need to get a control of |E n,p | which is uniform in p ∈ [p 0 , 1] of the kind c d n d-1 where c d depends only on p 0 . In[START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF], Zhang obtained a control on the size of the smallest minimal cutset corresponding to maximal flows in general first passage percolation, but his control depends on the distribution G of the variables (t(e)) e∈E d associated with the edges. We only consider probability measures G p = pδ 1 + (1 -p)δ 0 for p > p c (d), but we need to adapt Zhang's proof in this particular case to obtain a control that does not depend on p anymore. More precisely, let us denote by N n,p the total number of edges in E n,p . We have the following control on N n,p .

Theorem 5 (Adaptation of Theorem 2 in

[START_REF] Zhang | Limit theorems for maximum flows on a lattice[END_REF]

). Let p 0 > p c (d). There exist constants C 1 , C 2 and α that depend only on d and p 0 such that for all p ∈ [p 0 , 1], for all n ∈ N * ,

Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016)


A Glutsyuk
email: aglutsyu@ens-lyon.fr

E Shustin
email: shustin@post.tau.ac.il

On polynomially integrable planar outer billiards and curves with symmetry property

Keywords: points, 15 1, 6

We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards.

1 Introduction, main result and plan of the paper

Introduction and main result

Let C ⊂ R 2 be a smooth closed strictly convex curve. Let S denote the complement of the ambient plane R 2 to the closure of the interior of the curve C. The (planar) outer billiard is a dynamical system T : S → S defined as follows. Pick a point A ∈ S. There are two tangent rays to C issued from the point A. Let R denote the right tangent ray: the other tangent ray is obtained from R by rotation around the point A of angle between zero and π. Let P denote the tangency point of the ray R with the curve C. By definition, the image T (A) is the point of the ray R that is symmetric to A with respect to the point P .

Outer billiards were introduced by J.Moser in [START_REF] Moser | Is the solar system stable?[END_REF] as a toy model for planetary motion. A mechanical interpretation of outer billiard as an impact oscillator was given by Ph. Boyland [START_REF] Boyland | Dual billiards, twist maps and impact oscillators[END_REF]. For a survey on outer billiards see [24,[START_REF] Tabachnikov | Geometry and billiards[END_REF][START_REF] Tabachnikov | Dual billiards[END_REF]. It is well-known that the outer billiard map preserves the standard Euclidean area, and hence, it is a symplectomorphism. The famous Birkhoff Conjecture deals with convex bounded planar billiards having a smooth boundary: the classical (Birkhoff) billiards with the usual reflection law, the angle of incidence equals the angle of reflection. Recall that a caustic of a planar billiard Ω is a curve C such that each tangent line to C reflects from the boundary of the billiard to a line tangent to C. A billiard is called Birkhoff caustic-integrable, if a neighborhood of its boundary is foliated by closed caustics. The Birkhoff Conjecture states that the only integrable billiard is an ellipse.

The Birkhoff Conjecture motivated an analogous conjecture for the outer billiards. Namely, let C ⊂ R 2 be a smooth closed strictly convex curve, S ⊂ R 2 be the exterior component of its complement. We say that the corresponding outer billiard map T : S → S is integrable, if some neighborhood U of the curve C in S admits a smooth T -invariant function U → R (called a first integral) that has no critical points on C. It is known that every elliptic outer billiard is integrable. S.L.Tabachnikov's conjecture [27, p. 101] states that the converse is also true: if an outer billiard is integrable, then it is an ellipse. We prove this conjecture for polynomial first integrals. Definition 1.1 A planar outer billiard is polynomially integrable, if there exists a non-constant polynomial f (x, y) that is invariant under the outer billiard mapping.

Remark 1.2 Polynomiality of the integral is a very strong restriction of the general Tabachnikov's Conjecture. At the same time, the condition of just non-constancy of a polynomial integral does not forbid it to have critical points on the billiard boundary, while the definition of general integrability forbids them.

The main result of the paper is the following theorem.

Theorem 1.3 Let a planar outer billiard generated by a C 4 -smooth strictly convex closed curve C be polynomially integrable. Then C is an ellipse.

A particular case of Theorem 1.3 under some non-degeneracy assumptions was proved by S.L.Tabachnikov [27, theorem 1]. These assumptions imply in particular that the complex projective Zariski closure of the curve C is non-singular. See the end of Subsection 1.2 for more details.

We also prove the following more general theorem on non-convex curves generating multivalued outer billiard mappings, see the next definition. Definition 1.4 A C k -smoothly immersed curve C ⊂ R 2 generates a polynomially integrable multivalued outer billiard, if there exists a non-constant polynomial f (x, y) such that for every P ∈ C and every A, B ∈ T P C symmetric with respect to the point P one has f (A) = f (B). The latter polynomial is called an integral of the multivalued outer billiard constructed on the curve C. Theorem 1.5 Let C ⊂ R 2 be a C 4 -smoothly immersed image of either an interval, or a circle, that does not lie in a line. Let C generate a polynomially integrable multivalued outer billiard. Then the curve C lies in a conic. Theorem 1.3 follows from Theorem 1.5. For the proof of Theorem 1.5 we consider the complexification C 2 of the real plane and the ambient projective plane CP 2 ⊃ C 2 . Let C ∞ denote the infinity line:

C ∞ = CP 2 \ C 2 .
It is well-known that the polynomial integral f is constant on C (cf. [START_REF] Tabachnikov | On algebraically integrable outer billiards[END_REF], see Proposition 2.1 below). This implies that the projective complex Zariski closure γ of the curve C is an algebraic curve lying in a level curve of the complex polynomial f ; say, in the zero level curve Γ = {f = 0}, provided that f | C ≡ 0. It suffices to show that each nonlinear irreducible component α of the curve γ is a conic. S.Tabachnikov's arguments [START_REF] Tabachnikov | On algebraically integrable outer billiards[END_REF] together with their generalization due to M.Bialy and A.E.Mironov [6, section 6] imply that all the singular and inflection points (if any) of the curve γ lying in α are contained in the infinity line C ∞ (Theorem 2.2 in Section 2). The curve γ has the so-called relative symmetry property: for every point t ∈ γ ∩ C 2 the central symmetry of the affine line T t γ with respect to the point t permutes the points of its intersection with a bigger complex algebraic curve Γ ⊃ γ (follows from definition and analyticity). We study the local branches of the curve α at points of the intersection γ ∩ C ∞ . Each local branch is a nonlinear irreducible germ of analytic curve, thus it admits a local holomorphic parametrization by small complex parameter t, t → (t q , ct p (1 + o(1))), p, q ∈ N, p > q ≥ 1, c = 0 in local affine chart centered at the base point. We show that the relative symmetry property (and even its weaker, local version) implies that each local branch transverse to C ∞ is quadratic: p = 2q (Theorem 3.6 and its corollary). This is done in Section 3 via asymptotic analysis of the relative symmetry property. Finally, α has the two following properties: all its singular and inflection points (if any) lie in the infinity line C ∞ ; each its local branch transverse to C ∞ is quadratic. This together with the next new algebro-geometric theorem proved in Section 4 will imply Theorem 1.5.

Theorem 1.6 Let a nonlinear irreducible algebraic curve α ⊂ CP 2 have neither singular, nor inflection points in an affine chart C 2 ⊂ CP 2 . Let each of its local branches at every point in α ∩ C ∞ that is transverse to C ∞ (if any) be subquadratic: p ≤ 2q in its above parametrization. Then α is a conic.

The proof of Theorem 1.5 will be given in Subsection 4.3.

Remark 1.7 It is well-known that polynomial integrability of a Birkhoff billiard Ω is equivalent to the existence of a non-trivial first integral of the billiard flow in a neighborhood in T R 2 | Ω of the zero section of the bundle T R 2 | ∂Ω that is analytic in the speed. In more detail here analyticity is required to be uniform so that the convergence radius of its Taylor series in the speed be uniformly bounded from below. A direct reformulation of this assertion for the outer billiards would state that the existence of an analytic first integral of an outer billiard in a neighborhood of its boundary implies the existence of a polynomial integral. This is unknown, but this would reduce the Tabachnikov's Conjecture for analytically integrable outer billiards to the polynomial case and hence, together with the present results, would prove it. To our opinion, the analytic Tabachnikov's Conjecture seems to be of a level comparable to that of the general smooth case.

Historical remarks

The classical Birkhoff Conjecture was studied by many mathematicians starting from the famous paper by H.Poritsky [START_REF] Poritsky | The billiard ball problem on a table with a convex boundary -an illustrative dynamical problem[END_REF], where he stated it in print and proved it under the additional assumption that the billiard in each closed caustic near the boundary has the same closed caustics, as the initial billiard (see also [START_REF] Amiran | Caustics and evolutes for convex planar domains[END_REF]). Here we will mention just few of most known results. In 1993 M.Bialy [START_REF] Bialy | Convex billiards and a theorem by E. Hopf[END_REF] proved that if the phase cylinder of the billiard is foliated (almost everywhere) by continuous curves which are invariant under the billiard map, then the boundary curve is a circle (see also [START_REF] Wojtkowski | Two applications of Jacobi fields to the billiard ball problem[END_REF]). See also Bialy's papers [START_REF] Bialy | Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane[END_REF][START_REF] Bialy | On totally integrable magnetic billiards on constant curvature surface[END_REF] for similar results on billiards in constant curvature and on magnetic billiards respectively. D.V.Treschev's experiences [START_REF] Treschev | Billiard map and rigid rotation[END_REF][START_REF] Treschev | On a Conjugacy Problem in Billiard Dynamics[END_REF][START_REF] Treschev | A locally integrable multi-dimensional billiard system[END_REF] provide a numerical evidence of existence of the so-called locally integrable billiards, where a germ of the second iterate of the billiard map is conjugated to a rigid rotation (in two and higher dimensions). Recently V.Kaloshin and A.Sorrentino have proved a local version of the Birkhoff Conjecture [START_REF] Kaloshin | On local Birkhoff Conjecture for convex billiards[END_REF]: an integrable deformation of an ellipse is an ellipse (see [START_REF] Avila | An integrable deformation of an ellipse of small eccentricity is an ellipse[END_REF] for the case of ellipses with small eccentricities). Birkhoff Conjecture motivated the so-called Algebraic Birkhoff Conjecture, which deals with the polynomially integrable planar billiards, where the billiard geodesic flow has a first integral that depends polynomially on the speed components and is non-constant on the unit level hypersurface of the norm of the speed. It states that the only polynomially integrable convex billiard with smooth boundary is an ellipse. The study of the Algebraic Birkhoff Conjecture and its appropriate generalization to (not necessarily convex) billiards with piecewise smooth boundary on surfaces of constant curvature was started by S.V.Bolotin in [START_REF] Bolotin | Integrable Birkhoff billiards[END_REF][START_REF] Bolotin | Integrable billiards on surfaces of constant curvature[END_REF], see also a survey in [19, chapter 5, section 3]. It was proved for polynomial integrals of degrees up to 4 in [START_REF] Bialy | On fourth-degree polynomial integrals of the Birkhoff billiard[END_REF]. Its complete proof for billiards on any surface of constant curvature was recently obtained as a result of the three following papers: two joint papers by M.Bialy and A.E.Mironov [START_REF] Bialy | Angular billiard and Angular Birkhoff Conjecture[END_REF][START_REF] Bialy | Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane[END_REF]; a very recent paper [START_REF] Glutsyuk | On polynomially integrable Birkhoff billiards on surfaces of constant curvature[END_REF] (see also its short version [START_REF] Glutsyuk | On two-dimensional polynomially integrable Birkhoff billiards on surfaces of constant curvature[END_REF]) of the first author of the present article. For more detailed surveys on Birkhoff Conjecture and its algebraic version see [START_REF] Bialy | Angular billiard and Angular Birkhoff Conjecture[END_REF][START_REF] Glutsyuk | On polynomially integrable Birkhoff billiards on surfaces of constant curvature[END_REF][START_REF] Kaloshin | On local Birkhoff Conjecture for convex billiards[END_REF] and references therein.

For outer billiards a particular case of Theorem 1.3 was proved by S.L.Tabachnikov [27, theorem 1] under the following additional assumptions:

(i) the complex Zariski closure of the curve C in CP 2 is a non-singular algebraic curve;

(ii) the gradient ∇F of the polynomial integral F does not vanish identically on C.

Tabachnikov introduced a powerful method that allowed him to show that under assumptions (i) and (ii) all the singular and inflection points (if any) of the complex projective Zariski closure of the curve C lie in the infinity line. M.Bialy and A.E.Mironov introduced a modified version of Tabachnikov's method that allowed them to prove the same result in the general case (their private communication, see Theorem 2.2 below and its proof) and to prove a similar statement in the context of the Algebraic Birkhoff Conjecture in constant curvature [START_REF] Bialy | Angular billiard and Angular Birkhoff Conjecture[END_REF][START_REF] Bialy | Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane[END_REF].

Remark 1.8 The second part of the proof of the Algebraic Birkhoff Conjecture given in [START_REF] Glutsyuk | On polynomially integrable Birkhoff billiards on surfaces of constant curvature[END_REF][START_REF] Glutsyuk | On two-dimensional polynomially integrable Birkhoff billiards on surfaces of constant curvature[END_REF] uses results of [START_REF] Bialy | Angular billiard and Angular Birkhoff Conjecture[END_REF][START_REF] Bialy | Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane[END_REF] and techniques elaborated in the present article and in the previous paper of the first author [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF].

Complexification of the curve C: singularities and inflection points

The following proposition is a version of a result from [START_REF] Tabachnikov | On algebraically integrable outer billiards[END_REF].

Proposition 2.1 Let C ⊂ R 2 be a C 1 -smoothly immersed image of either an interval, or a circle. Let C generate a polynomially integrable multivalued outer billiard with the integral f (x, y). Then one has f | C ≡ const.

Proof For every point P ∈ C the restriction of the polynomial integral f to the affine tangent line T P C is invariant under the central symmetry with respect to the point P , by definition. Hense, its derivative at P vanishes, as does the derivative at 0 of an even function. Thus, the polynomial f has zero derivative along a vector tangent to C at P . Therefore, f | C ≡ const, by connectivity. This proves the proposition. 2

Theorem 2.2 Let C ⊂ R 2 be a C1 -smoothly immersed image of either an interval, or a circle, that does not lie in a line and generates a polynomially integrable multivalued outer billiard. Its complex projective Zariski closure γ ⊂ CP 2 is an algebraic curve. Each nonlinear irreducible component of the curve γ contains neither singular, nor inflection points of the affine curve γ ∩ C 2

A particular case of Theorem 2.2 under assumptions (i) and (ii) from Subsection 1.2 was proved by S.L.Tabachnikov in [START_REF] Tabachnikov | On algebraically integrable outer billiards[END_REF] (for convex curves C, but his argument works without convexity assumption). M. Bialy and A. E. Mironov have extended his proof to the general case by using their ideas from [6, section 6]. This proof of Theorem 2.2 due to Tabachnikov, Bialy, and Mironov is given below. Proof of Theorem 2.2. Let f be a polynomial integral normalized so that f | C ≡ 0. Then f | γ ≡ 0 by definition, and this implies that γ is an algebraic curve. Let α ⊂ γ be a nonlinear irreducible component of the curve γ. Let Ψ be an irreducible polynomial vanishing on α. Then ,y). The function F is multivalued algebraic, and any two of its holomorphic leaves differ by multiplication by m-th root of unity. Its branching locus is contained in the curve {g = 0}.

f = g(x, y)Ψ m (x, y), m ∈ N, g| α ≡ 0. Set 1 F (x, y) = f 1 m (x, y) = g 1 m (x, y)Ψ(x
For every P ∈ α ∩ C 2 and for every two points A, B ∈ T P α symmetric with respect to the point P one has f (A) = f (B): this equality holds in the real domain with P ∈ C (by definition) and extends analytically to the complex domain. This implies that for every point P ∈ α such that g(P ) = 0 and P is not a singular point of the curve α each leaf of the function F holomorphic on a neighborhood of the point P has local symmetry property: for every A, B ∈ T P α symmetric with respect to the point P and close enough to it one has

F (A) = F (B).
Consider the (multivalued) vector field

V = F y ∂ ∂x -F x ∂ ∂y
, which is tangent to the level curves of the function F and does not vanish identically on α. The above local symmetry property is equivalent to the statement that the function

U (x, y, ε) = F (x + εF y , y -εF x ) (2.1)
is even in ε for all P = (x, y) ∈ α \ (Sing(α) ∪ {g = 0}). Equivalently, its Taylor series in ε should contain only even powers of the variable ε. Set

H(F ) = F xx F 2 y -2F xy F x F y + F yy F 2 x .
One has

H(F )| α ≡ const (2.2)
locally for each leaf of the function F over α\(Sing(α)∪{g = 0}). Indeed, for every P = (x, y) ∈ α \ (Sing(α) ∪ {g = 0}) the Taylor coefficient at ε 3 of the function U (x, y, ε), which should vanish, equals dH(F ) dV up to constant factor, see [27, lemma 2] and [6, section 6]. Thus, the latter derivative vanishes, and hence,

H(F )| α ≡ const, since V is tangent to α.
Let Ψ(x, y, z) denote the homogeneization of the polynomial Ψ: Ψ(x, y) = Ψ(x, y, 1). Recall that Hess( Ψ)(x, y, z) is the classical Hesse determinant of the polynomial Ψ; it is a homogeneous polynomial in (x, y, z). One has

Hess( Ψ)(x, y, 1) ≡ -(d -1) 2 H(Ψ)(x, y) along the curve {Ψ = 0}, (2.3) since Hess( Ψ)(x, y, 1) = (d(d -1)(Ψ xx Ψ yy -Ψ 2 xy )Ψ -(d -1) 2 H(Ψ))(x, y),
see [31, chapter IV, section 6.2] and [6, formula (23)]. Recall that H(F ) = g 3 m H(Ψ) on the curve α = {Ψ = 0} (as in [27, lemma 2]). Therefore, H(F ) vanishes over all the singular and inflection points of the affine curve α ∩ C 2 , by formula (2.3) and since the polynomial Hess( Ψ)(x, y, 1) vanishes at these points. For every regular point A ∈ α ∩ C 2 that is singular for the curve γ one has g(A) = 0, hence, H(F )(P ) → 0, as P → A along the curve α. This together with (2.2) implies that α ∩ C 2 should be a straight line, as soon as it contains either a singular, or an inflection point of the curve γ. Thus, since α ∩ C 2 is not a line by assumption, it contains neither singular, nor inflection points of the curve γ. Theorem 2.2 is proved. 2

3 Relative symmetry property and quadraticity Definition 3.1 Recall that a local branch of an analytic curve γ at a point A ∈ γ is an irreducible component of its germ at A.

In the present section we study nonlinear local branches of the complex projective Zariski closure γ of the curve C at points of the intersection γ ∩ C ∞ and show that each branch transverse to C ∞ is quadratic (Theorem 3.6 and its corollary, see Subsection 3.1). Theorem 3.6 is stated in a more general context, for an irreducible germ b of analytic curve at a point A ∈ C ∞ satisfying the so-called local relative symmetry property introduced in Subsection 3.1. In the proof of Theorem 3.6 given in Subsection 3.3 we use preparatory results given partly in [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF] on the asymptotics of the intersection points of the tangent line T P b with a given irreducible germ a of analytic curve at A, as P → A (Propositions 3.8, 3.10 and their corollaries in Subsection 3.2).

Relative symmetry property

Definition 3.2 Let C 2 ⊂ CP 2 be a fixed affine chart. Let γ ⊂ CP 2 be an irreducible algebraic curve distinct from a line. We say that γ has relative symmetry property, if there exists an algebraic curve Γ ⊂ CP 2 containing γ such that for every t ∈ γ ∩ C 2 the intersection T t γ ∩ Γ ∩ C 2 is symmetric with respect to the point t as a subset of the affine complex line T t γ ∩ C 2 : it is invariant under the central symmetry x → -x in an affine coordinate x on T t γ centered at t.

We will also deal with the local version of the relative symmetry property. To state it, let us introduce the following definition. Consider an irreducible nonlinear germ b of analytic curve in CP 2 at a given point A. Let us choose affine coordinates (z, w) centered at A so that the tangent line T A b be the z-axis. Then one can find a local bijective parametrization of the germ b by a complex parameter t ∈ (C, 0) of the type

t → (t q , c b t p (1 + o(1))), q = q b , p = p b ∈ N, 1 ≤ q < p, c b = 0; (3.1) q = 1 if and only if b is a smooth germ.
In the case, when b is a germ of line, it is parameterized by t → (t, 0), and we set q b = 1, p b = ∞.

z(t) = t q b ; w(t) = c b t p b (1 + o(1)
). Then for every t small enough the intersection T t b ∩ a consists of q a points ξ 1 , . . . , ξ qa whose coordinates have the following asymptotics, as t → 0:

z(ξ j ) = O(t p b ) = o(t q b ) = o(z(t)), w(ξ j ) = (1 -r b )w(t)(1 + o(1)). (3.2)
(Recall that q a = 1, if a is a germ of line.)

Proof It suffices to prove just the second asymptotic formula in (3.2). Indeed, one has z(ξ j ) = O(w(ξ j )), by transversality. This together with the second formula in (3.2) implies the first one:

z(ξ j ) = O(w(t)) = O(t p b ).
For every t small enough the tangent line T t b intersects the z-axis at a point P t with the coordinate

z(P t ) = νz(t)(1 + o(1)) = νt q b (1 + o(1)), ν = r b -1 r b , (3.3
w(Q t ) = ν ν -1 w(t)(1 + o(1)) = (1 -r b )w(t)(1 + o(1)). (3.4)
Indeed, the triangle with the vertices P t , t, (z(t), 0) is "complex-similar" to the triangle P t Q t O (O is the origin), since their sides opposite to the vertex P t lie in parallel affine complex lines. That is, in the new affine coordinates centered at P t the second triangle is obtained from the first one by multiplication by the complex number z(Pt) z(Pt)-z(t) = ν ν-1 (1 + o(1)), see (3.3). This implies (3.4). Let now a be an arbitrary irreducible germ of holomorphic curve at the origin that is transverse to b. Every family of points ζ(t) of the intersection T t b ∩ a has w-coordinate asymptotically equivalent to w(Q t ), by transversality and since the line

T t b = Q t ζ(t)
z(ξ j ) = r b -1 r b z(t)(1 + o(1)) = r b -1 r b t q b (1 + o(1)) for 1 ≤ j ≤ q a , (3.6) z(t) = O((z(ξ j )) ra-1 r b -1 ) = o(z(ξ j )) for j > q a . (3.7)
(Points satisfying (3.7) exist only when a is nonlinear.) Case 2): r a = r b . One has

z(ξ j ) = ζ qa j z(t)(1 + o(1)) = ζ qa j t q b (1 + o(1)), (3.8)
where ζ j are the roots of the polynomial

R pa,qa,c (ζ) = cζ pa -rζ qa + r -1; r = p a q a > 1, c = c a c b .
(3.9) (In the case, when b = a, one has c = 1, and the above polynomial has the double root 1 corresponding to the tangency point t.) Case 3): r a < r b . One has

z(ξ j ) = O((z(t)) r b ra ) = o(z(t)).
Proof All the statements of the proposition were proved in loc. cit. except for the statement saying that in Case 1) one has exactly q a intersection points with asymptotics (3.6) and exactly p a -q a intersection points with asymptotics (3.7) (no points satisfying (3.7), if a is linear). Let us prove the latter statement. Thus, we consider that r a > r b . The case, when a is linear, was treated in [13, proposition 2.10, p.250], see also formula (3.3) above. Thus, we consider that a is nonlinear. Let τ denote the value of the local parameter of the curve a at a point of intersection ξ(t) ∈ T t b ∩ a. The obvious analytic equality

G(t, τ ) = w(t) + w (t) z (t) (z(ξ(t)) -z(t)) -w(ξ(t)) = 0 (3.10)
has asymptotic form G(t,τ ) is generated by the three monomials: (1 -r b )t p b , r b t p b -q b τ qa , -cτ pa . It consists of two edges: the first one with the vertices (p b , 0) and (p b -q b , q a ); the second one with the vertices (p b -q b , q a ) and (0, p a ). The latter edges lie on distinct lines. The three above statements on Newton diagram follow from the inequality r a > r b , as in loc. cit. The germ of analytic curve {G = 0} ⊂ C 2 (t,τ ) at the origin is a union of two germs η 1 ∪ η 2 corresponding to the edges of the Newton diagram, as in loc. cit. Namely, the monomials (1 -r b )t p b and r b t p b -q b τ qa generating the first edge are asymptotically opposite (asymptotically "cancel out") along the germ η 1 , and all the other Taylor monomials of the function G are of higher order along η 1 , as in loc. cit. This implies that for every fixed small t there are exactly q a parameter values τ for which (t, τ ) ∈ η 1 , and they satisfy asymptotic equality (3.6). Similarly, the monomials r b t p b -q b τ qa and -cτ pa are asymptotically opposite along the germ η 2 , and for every fixed small t there are exactly p a -q a values τ such that (t, τ ) ∈ η 2 , and they satisfy asymptotic equality (3.7). Proposition 3.10 is proved. 2

G(t, τ ) = t p b (1 + o(1)) + r b t p b -q b (τ qa -t q b )(1 + o(1)) -cτ pa (1 + o(1)) = 0, ( 3
Corollary 3.11 Let a, b ⊂ CP 2 be irreducible germs of holomorphic curves at a point A ∈ C ∞ that are tangent to each other and transverse to C ∞ , let b be nonlinear. Let (x, y) be affine coordinates on C 2 with the x-axis being tangent to b at A. Let ξ 1 , . . . , ξ pa , ξ j = ξ j (t) be the points of intersection T t b ∩ a. Their x-coordinates have the following asymptotics, as t → A: Case 1): r a > r b . One has

x(ξ j ) = r b r b -1 x(t)(1 + o(1)) for 1 ≤ j ≤ q a , (3.12)
x(ξ j ) = o(x(t)) for j > q a . (

(Points satisfying (3.13) exist only when a is nonlinear.) Case 2): r a = r b . One has

x(ξ j ) = θ qa j x(t)(1 + o(1)), (3.14)
where θ j are the roots of the polynomial3 Q pa,qa,c (θ) = θ pa R pa,qa,c (θ -1 ) = (r -1)θ pa -rθ pa-qa + c;

(3.15) r = p a q a = p b q b > 1, c = c a c b .
Case 3): r a < r b . One has

x(t) = o(x(ξ j )).
The corollary follows from Proposition 3.10 by writing its asymptotics in the local coordinates (z, w) = ( 1x , y x ).

3.3 Intersections with the germs having the same projective Puiseux exponents. Proof of Theorem 3.6

Let A ∈ C ∞ , and let b ⊂ CP 2 be a nonlinear irreducible germ of analytic curve at A that is transverse to C ∞ . Let Γ ⊃ b be an arbitrary (T A b, A)-local multigerm containing b. Let (x, y) be affine coordinates on C 2 such that the x-axis is tangent to b at A. Proposition 3.12 Those points of the intersection T t b∩Γ whose x-coordinates are asymptotically equivalent to x(t) up to a nonzero multiplicative constant, as t → A, lie in the intersection of the line T t b with those irreducible germs a ⊂ Γ that are centered at A, tangent to b and for which r a ≥ r b , including the germ b. (Some of these germs a may be linear.) The asymptotics of their x-coordinates are given by either (3.12) if r a > r b , or (3.14) if r a = r b .

The proposition follows immediately from Corollaries 3.9 and 3.11 and the fact that the x-coordinates of the points of the intersection of the line T t b with those germs a in Γ that are not centered at A tend to finite limits, as t → A, while x(t) → ∞.

In the sequel we assume that the curve b has relative symmetry property with respect to the multigerm Γ. First we prove Theorem 3.6 in the next simplest special case in order to underline the main ideas, and then in the general case.

Special case: Γ \ b is a union of germs centered at A and transverse to b. Let us prove quadraticity of the germ b. To do this, we consider those intersection points of the tangent line T t b with Γ, whose x-coordinates have asymptotics νx(t)(1 + o(1)), ν = 0, as t → A. These are exactly the points of the intersection T t b ∩ b (Proposition 3.12). Their x-coordinates have asymptotics θ q j x(t)(1 + o(1)), where θ 1 , . . . , θ p are the roots of the polynomial W (θ) = Q p,q,1 (θ) = (r -1)θ p -rθ p-q + 1; here p = p b and q = q b are the degrees from the parametrization (3.1) of the germ b in the local chart (z, w) = ( 1x , y x ) (Corollary 3.11). The intersection points of the line T t b with the other, transverse germs of the curve Γ have x-coordinates with bigger asymptotics, by Corollary 3.9. This together with the relative symmetry property implies that the collection of x-coordinates of the points of intersection T t b ∩ b is invariant under the symmetry with respect to the point x(t). This implies that the collection of powers θ q j is invariant under the symmetry with respect to 1. Therefore,

p j=1 θ q j = p, (3.16)
by symmetry. On the other hand,

p j=1 θ q j = p r -1 . (3.17)
Indeed, the latter sum is independent of the free term of the polynomial W , being expressed via elementary symmetric polynomials of degrees at most q < p, which are independent of the free term. Therefore, it equals the sum of the q-th powers of nonzero roots of the polynomial (r -1)θ p -rθ p-q . All the latter q-th powers are equal to r r-1 , hence their sum equals qr r-1 = p r-1 . This proves (3.17). Formulas (3.16) and (3.17) together imply that p = p r-1 . Hence, r = 2.

General case. Let a 1 , . . . , a l ⊂ Γ be those irreducible germs that are centered at A, tangent to b and have the same projective Puiseux exponent: r a i = r b = r; b is one of them. Here l may be any natural number including 1. Let q a i , p a i , c a i be the corresponding degrees and coefficients from their parameterizations (3.1) in the local chart (z, w) = ( 1x , y x ). Let

c i = c a i c b , W i = Q pa i ,qa i ,c i (θ)
be the corresponding constants and polynomials from (3.15). Let θ ij , i = 1, . . . , l, j = 1, . . . , p a i denote the roots of the polynomials W i . Let Γ b ⊂ Γ denote the union of those germs in Γ that are centered at A, tangent to b and have Puiseux exponents greater than r = r b (some of them may be linear).

Let k 1 denote the number of those points ξ(t) of the intersection T t b ∩ Γ, for which x(ξ(t)) = o(x(t)), as t → A. These are exactly those points that either lie in Γ b and have x-coordinates o(x(t)), see (3.13), or lie in those germs in Γ that are not centered at A. Let k 2 denote the number of those points of the intersection T t b ∩ Γ b whose x-coordinates are asymptotic to

qa i ij = 2, r-2
r-1 (each of the latter powers being taken with the total multiplicity of the corresponding roots) is invariant under the symmetry of the line C with respect to 1.

Proof The points of intersection T t b ∩ Γ whose x-coordinates are o(x(t)) should be symmetric with respect to t to other intersection points with xcoordinates asymptotically equivalent to 2x(t) and vice versa. The latter should be points of intersection with the germs a i . This follows from Proposition 3.12 and the fact that they cannot be points of the intersection T t b∩Γ b . The latter statement follows from (3.12) and the inequality r r-1 = 2, which follows from the assumption that r = 2. This together with Corollary 3.11 and Proposition 3.12 implies that exactly k 1 powers θ qa i ij are equal to 2. Similarly, if the collection Γ b is non-empty, then the intersection T t b ∩ Γ b contains points whose x-coordinates are asymptotic to r r-1 x(t), by (3.12). Vice versa, the points of the intersection T t b ∩ Γ with the latter asymptotics lie in T t b ∩ Γ b . This follows from Corollary 3.11, Proposition 3.12 and the fact that they cannot be points of intersection with the germs a i : no number s = ( r r-1 )

1 qa i can be a root of a polynomial W i with c i = 0. Indeed,

W i (s) = (r -1)s pa i -rs pa i r r -1 -1 + c i = c i = 0.
The above points of the intersection T t b ∩ Γ b should be symmetric to the points of the intersection T t b ∩ Γ with x-coordinates asymptotically equivalent to (2 -r r-1 )x(t) = r-2 r-1 x(t). The latter points of intersection lie in the union of the germs a i , by Proposition 3.12 and due to r-2 r-1 = r r-1 . Therefore, exactly k 2 powers θ qa i ij are equal to r-2 r-1 . The points of the intersection T t b ∩ Γ having x-coordinates asymptotically equivalent to vx(t) with v = 0, 2, r r-1 , r-2 r-1 are symmetric with respect to t, by relative symmetry property. The collection of the corresponding asymptotic factors v, thus symmetric with respect to 1, coincides with the collection M of the powers θ

qa i ij = 2, r-2
r-1 , by the above arguments. The proposition is proved.

2 Proposition 3.14 Let r > 1. Consider a collection S r = {(p i , q i , c i )} i=1,...,N , q i , p i ∈ N, p i > q i , p i q i = r, c i ∈ C \ {0}, (3.18)
set W i (θ) = (r -1)θ p i -rθ p i -q i + c i . Let θ ij (j = 1, . . . , p i ) denote the roots of the polynomials W i . Let M denote the collection of those q i -th powers of roots θ ij that are different from 2 and r-2 r-1 : each power being taken with the total multiplicity of the corresponding roots. Let M be invariant under the symmetry of the line C with respect to 1. Let k 1 (k 2 ) denote the number of those pairs (i, j), for which θ q i ij equals 2 (respectively, r-2 r-1 ). Set

Π =

i p i = the cardinality of the collection of all the roots θ ij .

Then (r -2)Π = k 2 -k 1 (r -1). (3.19)
Proof The invariance of the collection M under the symmetry with respect to 1 implies that the sum of its elements equals the cardinality card(M ) = Π -k 1 -k 2 . On the other hand,

Π -k 1 -k 2 = x∈M x = i,j θ q i ij -2k 1 - r -2 r -1 k 2 , (3.20)
by definition. Let us calculate the latter right-hand side. One has

ij θ q i ij = i p i r -1 = Π r -1 , (3.21)
as in (3.17).

Substituting (3.21) to (3.20) yields Π r -1 -2k 1 - r -2 r -1 k 2 = Π -k 1 -k 2 ,
which is equivalent to (3.19). 2

Proposition 3.15 Let r > 1. Consider a collection S r as in (3.18), set

W i (θ) = (r -1)θ p i -rθ p i -q i + c i . Let θ ij (j = 1, . . . , p i ) denote the roots of the polynomials W i , set Π = N i=1 p i . Let k 1 (k 2
) denote the number of index pairs (i, j) for which θ q i ij equals 2 (respectively, r-2 r-1 ). Let equality (3.19) hold. Then r = 2.

Proof Let us represent r as an irreducible fraction r = p q , p, q ∈ N. One has p i = s i p, q i = s i q, s i = gcd(p i , q i ).

Suppose the contrary: equality (3.19) holds and r = 2. Case 1): r > 2. Hence, r -2 ≥ 1 q . One has

k 2 ≥ (r -2)Π ≥ 1 q Π > 1 p Π,
by (3.19). This implies that there exists a polynomial W i for which more than 1 p -th part of its roots have q i -th powers equal to r-2 r-1 . Thus, the number of the latter roots is no less than s i +1 = p i p +1. We will show that the above W i cannot exist. Let it exist, and let us fix it. None of its roots θ ij with θ q i ij = r-2 r-1 can be a multiple root. Indeed, the derivative of the polynomial W i (θ) equals θ p i -q i -1 (p i (r -1)θ q i -r(p i -q i )) = p i (r -1)θ p i -q i -1 (θ q i -1), since r(p i -q i ) = rq i (r -1) = p i (r -1). Therefore, the q i -th powers of the roots of the derivative are equal to 0, 1 = r-2 r-1 . Hence, W i has at least s i + 1 distinct roots θ = θ ij with θ q i ij = r-2 r-1 . By definition, the latter roots satisfy the equality

θ p i -q i ((r -1)θ q i -r) + c i = θ p i -q i ((r -2 -r) + c i = -2θ p i -q i + c i = 0 (3.23)
and have equal q i -th powers. Hence, their p i -th powers are also equal, by (3.23). Therefore, the ratio of any two above roots is simultaneously a q i -th and p i -th root of unity, and hence, an s i -th root of unity, since p i = s i p, q i = s i q and p, q are coprime. Therefore, the number of roots under question is no greater than s i . We get a contradiction. Case 2): 1 < r < 2. One has

k 1 ≥ Π 2 -r r -1 = Π 2q -p p -q ≥ 1 p -q Π > 1 p Π,
by (3.19). This implies that the there exists a polynomial W i that has at least s i + 1 roots whose q i -th powers are equal to 2, as in the previous case.

We then get a contradiction, as in the above discussion. The proposition is proved. 2 Lemma 3.16 Let r > 1, S r = {(p i , q i , c i )} i=1,...,N be a collection as in (3.18). Set W i (θ) = (r -1)θ p i -rθ p i -q i + c i . Let θ ij (j = 1, . . . , p i ) denote the roots of the polynomials W i . Let M denote the collection of those q i -th powers of roots θ ij that are different from 2 and r-2 r-1 : each power being taken with the total multiplicity of the corresponding roots. Let M be invariant under the symmetry of the line C with respect to 1. Then r = 2. Lemma 3.16 follows from Propositions 3.14 and 3.15. Lemma 3.16 together with Proposition 3.13 imply the statement of Theorem 3.6.

4 Invariants of singularities and Plücker formulas.

Proofs of Theorems 1.6 and 1.5

Here we prove Theorem 1.6 in Subsection 4.2. Then we prove Theorem 1.5 in Subsection 4.3. Afterwards in Subsection 4.4 we state and prove a purely algebraic-geometric Theorem 4.1 that is a direct consequence of Theorems 3.6 and 1.6 that seems to be of independent interest.

The proof of Theorem 1.6 essentially uses general Plücker and genus formulas for plane algebraic curves; the corresponding background material is presented in Subsection 4.1. The main observation is that the upper bound 2 to the projective Puiseux exponents of all transverse local branches of the curve and Plücker formulas yield that the singularity invariants of the considered curve must obey a relatively high lower bound. On the other hand, the contribution of the points in the infinite line C ∞ appears to be not sufficient to fit that lower bound unless the curve is a conic.

Invariants of plane curve singularities

For the reader's convenience, we recall here main definitions and formulas. Almost all this stuff is classically known (see [12, Chapter III], [20, §10], and the modern exposition in [START_REF] Greuel | Introduction to singularities and deformations[END_REF]Section I.3]).

Let (x 0 : x 1 : x 2 ) be homogeneous coordinates on CP 2 . Let α ⊂ CP 2 be a reduced, irreducible curve of degree d > 1, i.e., given by a homogeneous square-free, irreducible polynomial F (x 0 , x 1 , x 2 ) of degree d > 1. For any point A ∈ α and a sufficiently small closed ball V ⊂ CP 2 centered at A the intersection α ∩ V is topologically a bouquet of discs b i , 1 ≤ i ≤ r, given by the local branches of the germ (α, A).

(1) Multiplicity and dual multiplicity of a local branch. Given a local branch b i of the curve α at A ∈ α and its Puiseux parametrization (3.1), the number s(b i ) = q is called the multiplicity, and the number s * (b i ) = p -q the dual multiplicity of the branch b i . Note that s(b i ) is the intersection multiplicity of the branch b i with a transversal line, while s * (b i ) + s(b i ) is the intersection multiplicity with the tangent line T A b i . Observe also that the subquadraticity condition for b i is equivalent to the relation

s * (b i ) ≤ s(b i ) . (4.1)
(2) δ-invariant. Let f (x, y) = 0 be an equation of the germ (α, A) (just F = 0 rewritten in local coordinates x, y). Then α ε := {f (x, y) = ε} ∩ V , for 0 < |ε| 1, is a smooth surface with r holes (Milnor fiber). The δ-invariant of the germ (α, A) admits several equivalent definitions and topologically can be defined as the genus of the closed surface obtained by attaching a sphere with r holes to the surface α ε . The genus formula in the form of Hironaka [START_REF] Hironaka | Arithmetic genera and effective genera of algebraic curves[END_REF] reads

(d -1)(d -2) 2 = g(α) + A∈Sing(α) δ(α, A) ,
where g(α) is the geometric genus of α, i.e., the genus of the Riemann surface obtained by the resolution of singularities of α. In particular, we have

A∈Sing(α) δ(α, A) ≤ (d -1)(d -2) 2 . ( 4.2)
(3) Class of the singular point (κ-invariant). Given a germ (α, A) and local affine coordinates (x, y), suppose that the y-axis is not tangent to any of the local branches b i , 1 ≤ i ≤ r. Denote by α the polar curve of α defined by the equation ∂f ∂y = 0. The class of the germ (α, A) is defined by

κ(α, A) = (α • α ) A ,
the intersection multiplicity of α and α at A.

κ(α, A) = 2δ(α, A) + r i=1 (s(b i ) -1) . (4.3)
(4) Hessian of the singular (or inflection) point. The Hessian H α of the curve α is the curve given by the equation det ∂ 2 F ∂x i ∂x j 0≤i,j≤2 = 0. The Hessian of the germ (α, A) is

h(α, A) = (α • H α ) A ,
the intersection multiplicity of α and H α at A. It vanishes in all smooth points of α, where α quadratically intersects its tangent line. An expression for h(α, A) via the preceding invariants was found in [START_REF] Shustin | On invariants of singular points of algebraic curves[END_REF]Formula (2)]. It can be written as which means that the Newton diagram of α in these coordinates is the segment [(d, 0), (0, d/2)], i.e., the coefficients of all the monomials z i 1 w j 1 with (i, j) below this segment vanish. In the coordinates x = 1 z 1 , y = w 1 z 1 , this yields that the coefficients of all monomials x i y j with j < i vanish. The same consideration with the affine coordinates z 2 , w 2 in a neighborhood of the center B of the branch b 2 such that T B b 2 is the z 2 -axis and C ∞ is the w 2axis leads to the conclusion that the coefficients of all monomials x i y j with j > i vanish. This finally leaves the only Newton segment [(0, 0), (d/2, d/2)]. Note that a polynomial with such a Newton segment factors into the product of d 2 binomials of type xy -λ. Thus, d = 2 due to the irreducibility of α.

h(α, A) = 3κ(α, A) + r i=1 (s * (b i ) -s(b i )) . ( 4
               b∈Btr∪Btan (s(b) -1) = d -|B tr ∪ B tan | -b∈Btan s * (b) ≤ d -|B tr | -2|B tan |, b∈Btr s(b) = d -b∈Btan (s * (b) + s(b)) ≤ d -2|B tan |, b∈Btr (s(b) -2) + b∈Btan (s * (b) + s(b) -2) = d -2|B tr ∪ B tan |, (4.8
If |B tan | = 1 and |B tr | = 0, then we have an equality in (4.9); hence, all the above inequalities turn to be equalities, in particular, the second relation in (4.8), that is, s * (b) + s(b) = 2 for the unique branch of α centered on C ∞ , which finally means that d = 2.

Proof of Theorem 1.5

Let γ denote the complex projective Zariski closure of the curve C. Let α be its arbitrary nonlinear irreducible component. The curve α has neither singular, nor inflection points in C 2 (Theorem 2.2), and each its local branch transverse to the infinity line (if any) is quadratic, by Corollary 3.7. Therefore, α is a conic, by Theorem 1.6. Thus, the curve γ is a finite union of conics and lines, and C is a union of arcs of conics and lines. The latter union of arcs is finite: each potential end of an arc should be a singular point of the curve γ, and the number of singular points of an algebraic curve is finite. At least one conical arc is present, since C is a C 4 -smoothly immersed curve that does not lie in a line. The parameter interval (circle) of the curve C is thus split into a finite number of segments, each of them parameterizes an entire arc of conic (line) in C. Any two arcs parameterized by adjacent segments have contact of order at least 5, since C is a C 4 -smoothly immersed curve. Therefore, no conical arc can be adjacent to a linear arc, which implies that there are no linear arcs at all. No two arcs of distinct conics can be adjacent neither: otherwise, their intersection index would be greater than 4, by the above statement. This implies that C lies in just one conic and proves Theorem 1.5.

4.4 Appendix: a general corollary of Theorems 3.6 and 1.6

The following theorem is a direct consequence of Theorems 3.6 and 1.6. Theorem 4.1 Let α ⊂ CP 2 be an irreducible algebraic curve distinct from a line such that all its singular and inflection points (if any) lie in the infinity line C ∞ . Let for every point A ∈ α ∩ C ∞ each local branch β of the curve α at A that is transverse to the infinity line C ∞ (if any) have the local relative symmetry property with respect to some (T A β, A)-local multigerm Γ = Γ(β). Then α is a conic.

Proof Each local branch β as above is quadratic (Theorem 3.6). Hence, α is a conic, by Theorem 1.6.

2

Definition 3 . 3

 33 Let L ⊂ CP 2 be a line, and let A ∈ L. A (L, A)-local multigerm is a finite union of distinct irreducible germs of analytic curves b 1 , . . . , b N (called components) at base points A j ∈ L such that each germ at A j = A is different from the line L. (A germ at A can be arbitrary, in particular, it may coincide with the germ (L, A).) The (L, A)-localization of an algebraic curve in CP 2 is the corresponding (L, A)-local multigerm formed by all its local branches of the above type. Definition 3.4 Let C 2 ⊂ CP 2 be a given affine chart. Let A ∈ CP 2 , b ⊂ CP 2 be a nonlinear irreducible germ of analytic curve at A. The germ b has (local) relative symmetry property, if there exists a (T A b, A)-local multigerm Γ containing b such that for every t ∈ b∩C 2 close enough to A the intersection T t b ∩ Γ ∩ C 2 is symmetric with respect to the point t.

Definition 3 . 5 2 3. 2

 3522 The projective Puiseux exponent [13, p. 250, definition 2.9] of a nonlinear germ b is the ratior = r b = p b q b .The germ b is called quadratic, if r b = 2, and is called subquadratic, if r b ≤ 2. By definition, the projective Puiseux exponent of a germ of line equals infinity. Theorem 3.6 Let a nonlinear irreducible germ b of analytic curve in CP 2 at a point A ∈ C ∞ be transverse to C ∞ and have local relative symmetry property. Then it is quadratic. Theorem 3.6 is proved in Subsection 3.3. Corollary 3.7 Let a nonlinear irreducible germ b of analytic curve in CP 2 at a point A ∈ C ∞ be transverse to C ∞ and lie in the complex projective Zariski closure γ of a C 2 -smoothly immersed real curve generating a polynomially integrable multivalued outer billiard. Then b is quadratic. Proof Let C be the real curve under question, and let f be a polynomial integral of the corresponding outer billiard. Without loss of generality we consider that C is an immersed image of a connected curve, either an interval, or a circle (passing to a smaller curve) and f | γ ≡ 0, see Proposition 2.1. Let γ 0 denote the irreducible component of the curve γ containing the germ b. The component γ 0 is nonlinear and has the relative symmetry property with respect to the algebraic curve Γ = {f = 0} ⊃ γ: for every P ∈ γ ∩ C 2 the restriction of the integral f to the affine tangent line T P γ is invariant under the central symmetry with respect to P , as in the proof of Theorem 2.2. For every point u ∈ b close to A the intersection of the line T u b with the algebraic curve Γ coincides with the intersection of the line T u b and the (T A b, A)-localization of the curve Γ. This is implied by definition and the following facts: -for any neighborhood U of a given line L all the lines sufficiently close to L are contained in U ; -the point of intersection T u b ∩ T A b tends to A, as u → A. The local branch b of the curve γ 0 has local relative symmetry property with respect to the (T A b, A)-localization of the curve Γ, by the above discussion. This together with Theorem 3.6 implies that b is quadratic and proves the corollary. Asymptotics of intersection points Proposition 3.8 Let a, b be transverse irreducible germs of holomorphic curves at the origin in C 2 , let b be nonlinear. Let (z, w) be affine coordinates in a neighborhood of the origin in C 2 centered at 0 such that b is tangent to the z-axis at 0. Let t be the local parameter of the curve b as in (3.1):

) by [ 13 ,

 13 Proposition 2.10, p. 250]. Let Q t denote the intersection point of the line T t b with the w-axis. One has

2 Corollary 3 . 9 . 5 )

 2395 tends to the z-axis. In more detail, suppose the contrary: the above w-coordinates are not asymptotically equivalent, hence the differencew(ζ(t)) -w(Q t ) is asymptotically no less than cw(ζ(t)), c = 0. Then the ratio w(ζ(t))-w(Qt) z(ζ(t))would not tend to zero, since z(ζ(t)) = O(w(ζ(t))), as was mentioned above. Hence, the line Q t ζ(t) would not tend to the z-axis, -a contradiction. This together with (3.4) proves the second equality in (3.2). The proposition is proved. Let A ∈ C ∞ , and let (b, A) ⊂ CP 2 be a nonlinear irreducible germ of analytic curve that is transverse to C ∞ . Let (a, A) be another irreducible germ transverse to b, and let ξ 1 , . . . , ξ qa , ξ j = ξ j (t) be the points of intersection T t b ∩ a. Let (x, y) be affine coordinates in C 2 such that the x-axis is tangent to b at A. Then x(t) = o(x(ξ j (t))) for all j, as t → A.(3ProofTake the local coordinates (z, w) = ( 1 x , y x ) centered at A and apply the first formula in (3.2). 2 Proposition 3.10 (cf. [13, p. 268, Proposition 2.50] 2 ) Let a, b be irreducible germs of holomorphic curves at the origin in the plane C 2 with coordinates (z, w). Let a and b be tangent to the z-axis, and let b be nonlinear. Let c a and c b be the corresponding coefficients in (3.1). Then for every t small enough the intersection T t b ∩ a consists of p a points ξ 1 , . . . , ξ pa (or just one point ξ 1 , if a is the germ of the line T O b) whose z-coordinates have the following asymptotics, as t → 0. Case 1): r a > r b (including the linear case, when r a = ∞). One has

  .11) c = ca c b = 0, as in [13, p. 269, proof of Proposition 2.50]. The Newton diagram of the germ of analytic function

r r- 1

 1 x(t), see (3.12). Proposition 3.13 Let r = r b = 2. The collection of powers θ qa i ij of the above roots contains exactly k 1 powers equal to 2 and exactly k 2 powers equal to r-2 r-1 . The collection M of the other powers θ

. 4 ) 6

 46 In view of deg H α = 3(d -2), Bézout's theorem yields (a Plücker formula) Let α ⊂ CP 2 be a curve of degree d ≥ 2, satisfying the hypotheses of Theorem 1.6. We will show that d = 2.Observe thatδ(α, A) = κ(α, A) = h(α, A) = 0 for all points A ∈ α \ C ∞ .Denote by B tr , resp. B tan the set of the local branches of α centered on C ∞ and transversal, resp. tangent to C ∞ . Relation (4.1) holds for all the local branches b ∈ B tr , since they are subquadratic, by the condition of Theorem 1.6. Thus, their contributions to the sum i (s * (b i )-s(b i )) are non-positive. Therefore, from (4.4) and (4.5), we get 3d(d -2) ≤ 3 A∈α∩C∞ κ(α, A) + b∈Btan (s * (b) -s(b)) .

(4. 6 )

 6 Together with (4.2) and(4.3) this yields 3d(d -2) ≤ 6 A∈α∩C∞ δ(α, A) + 3 b∈Btr∪Btan (s(b) -1) + b∈Btan (s * (b) -s(b)) ≤ 3(d -1)(d -2) + 3 b∈Btr∪Btan (s(b) -1) + b∈Btan (s * (b) -s(b)) = 3(d -1)(d -(b) + s(b) -2) .(4.7) Developing d = (α • C ∞ ) into contributions of local branches b ∈ B tr ∪ B tan , we obtain

  ) and hence the sequence of relations (4.7) reduces to 2 ≥ |B tr | + 2|B tan | . (4.9) If B tan = ∅ and all the branches b ∈ B tr are centered at one point, then b∈Btr s(b) = d, and the intersection multiplicity of α with the tangent to one of the branches b ∈ B tr appears to be greater than d. This implies that the latter tangent line is contained in α. Thus, α splits off a line, contrary to the irreducibility assumption. If B tan = ∅, |B tr | = 2, and the two branches b 1 , b 2 ∈ B tr have distinct centers, we have an equality in (4.9); hence, equalities in all the above relations, in particular, in (4.6). Thus, in view of (4.4), (4.5) and the inequality s * (b i ) ≤ s(b i ) (subquadraticity), it means s * (b i ) = s(b i ), i = 1, 2. Intersecting α with the tangent lines to b 1 and b 2 , we obtain s(b i ) ≤ d 2 , i = 1, 2, while the intersection with C ∞ yields s(b 1 ) + s(b 2 ) = d. It follows that s(b 1 ) = s(b 2 ) = d 2 . Choosing affine coordinates in CP 2 \ C ∞ so that the coordinate axes are tangent to b 1 and b 2 (at infinity) respectively, we obtain that the Newton polygon of the defining polynomial of α is just the segment [(0, 0), (d/2, d/2)]. Indeed, in local affine coordinates z 1 , w 1 in a neighborhood of the center A of the branch b 1 such that the tangent line T A b 1 is the z 1 -axis and C ∞ is the w 1 -axis, we have b 1 given by z 1 = t d/2 , w 1 = c 1 t d (1 + o(1)), t ∈ (C, 0)

  It is well-known (see, for instance, [16, Propositions I.3.35 and I.3.38]) that

The case, when m = 1, was treated by S.L.Tabachnikov[START_REF] Tabachnikov | On algebraically integrable outer billiards[END_REF]. His arguments were extended to arbitrary m by M.Bialy and A.E.Mironov, who introduced the function F (their private communication, which repeats their arguments from [6, section 6])

The formulas from loc. cit. provide the inverse expressions, for the coordinate z(t) in terms of z(ξj(t)). They are equivalent to the formulas given here.

In the case, when b = a, one has c = 1, and the polynomial Qp a,qa ,1 has double root 1 corresponding to the tangency point t. It has roots θ with θ qa = 1, if and only if r = pa qa = 2.

Acknowledgements

We are grateful to Misha Bialy, Andrey Mironov and Sergei Tabachnikov for introducing us to polynomially integrable billiards, helpful discussions and providing the starting point for our work: Theorem 2.2. This work was partly done during the visits of the first author (A.Glutsyuk) to Sobolev Institute at Novosibirsk and to Tel Aviv University. He wishes to thank Andrey Mironov and Misha Bialy for their invitations and hospitality and both institutions for their hospitality and support. We are grateful to the referee for a very careful reading of the paper and very helpful remarks, which also led to major improvement of the first author's paper [14].

‡ Supported by part by RFBR grants 13-01-00969-a, 16-01-00748, 16-01-00766 and ANR grant ANR-13-JS01-0010. grant 176/15 from the Israeli Science Foundation.