Escape of mass in homogeneous dynamics in positive characteristic

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of 2-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue; that is, that for a large set of rays the measures fail to converge to the uniform probability measure on the space of 2-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of 2-lattices.


Introduction
Let F q be a finite field of order a positive power q of a prime p, and let K = F q (Y ) be the field of rational functions in one variable Y over F q . Let R ∞ = F q [Y ] be the ring of polynomials in Y over F q , let K ∞ = F q ((Y −1 )) be the field of formal Laurent series in Y −1 over F q and let X ∞ = PGL 2 (K ∞ )/ PGL 2 (R ∞ ) be the space of homothety classes of R ∞ -lattices in K ∞ × K ∞ (that is, of rank 2 free R ∞ -submodules spanning the vector plane K ∞ × K ∞ over K ∞ ). A point x ∈ X ∞ is called A ∞ -periodic if its orbit under the diagonal subgroup A ∞ of PGL 2 (K ∞ ) is compact. This orbit A ∞ x then carries a unique A ∞ -invariant probability measure, denoted by µ x . The aim of this paper is to study the asymptotic behavior of these measures µ x (and in particular to prove unexpected escape of mass phenomena) as x varies in arithmetically defined subsets of A ∞ -periodic points. We will give motivations for this problem in the second part of this introduction.
Recall that for every x 0 ∈ X ∞ and every prime polynomial ν in R ∞ , the Hecke tree T ν (x 0 ) with root x 0 is the connected component of x 0 in the graph with vertex set X ∞ , with an edge between the homothety classes of two R ∞ -lattices Λ and Λ when Λ ⊂ Λ and Λ /Λ is isomorphic to R ∞ /νR ∞ as an R ∞ -module. The boundary at infinity Ω of T ν (x 0 ) identifies with the projective line P 1 (K ν ) over the completion K ν of K associated to ν, and a point of Ω is called rational if it belongs to P 1 (K). (Note that the identification of Ω with P 1 (K ν ) is not canonical, but the notion of rationality is well defined.) For every ξ ∈ Ω, let (x ξ n ) n∈N be the vertices along the geodesic ray (called a Hecke ray) in T ν (x 0 ) from x 0 to ξ.
In what follows we fix an A ∞ -periodic point x 0 in X ∞ . Note that the vertices of the Hecke-tree T ν (x 0 ) then also have periodic A ∞ -orbits. Our aim is to understand the possible sets Θ ξ of weak-star accumulation points of the sequences of measures (µ x ξ n ) n∈N on X ∞ associated to the vertices of the Hecke ray with endpoint ξ, when ξ varies in Ω. For all ξ ∈ Ω and c > 0, we say that • ξ has c-escape of mass if there exists θ ∈ Θ ξ with θ(X ∞ ) ≤ 1 − c, • ξ has uniform c-escape of mass if for every θ ∈ Θ ξ we have θ(X ∞ ) ≤ 1 − c.
Here is a summary of our results.
Theorem 1 There exists c > 0 such that any rational ξ ∈ Ω has uniform c-escape of mass.
The following result also exhibits full espace of mass phenomena along Hecke rays.
The key approach to these results (proved in Subsection 4.1) is to use the geodesic flow on the quotient of the Bruhat-Tits tree of (PGL 2 , K ∞ ) (see for instance [Ser2] and Subsection 2.3) by the lattice PGL 2 (R ∞ ).
Theorem 1 proves an escape of mass phenomenon along only countably many Hecke rays. Using the remarkable fact that the above constant c is independent of the rational Hecke ray, we can strengthen this in the next result (see Subsection 4.2).
Theorem 3 There exists c > 0 such that the set of ξ ∈ Ω having c-escape of mass is uncountable.
As guided by the analogy with PGL 2 (R)/ PGL 2 (Z) (see below), we could still wonder if the part of the measure which does not go to infinity still equidistributes in X ∞ , that is, converges to a measure proportional to the homogeneous measure on X ∞ under PGL 2 (K ∞ ). The next result proves that this is also not always the case.
Theorem 4 There exists c > 0 such that for every A ∞ -periodic point x ∈ X ∞ , there exist ξ ∈ Ω and θ ∈ Θ ξ such that c µ x ≤ θ. In particular, θ is not absolutely continuous with respect to the homogeneous measure on X ∞ .
We give explicit constants c, c in the above statements. We will actually prove a stronger result, Theorem 21 in Subsection 4.4, which mixes the behaviors in Theorems 3 and 4. For this, the main tool (proved in Subsection 4.3) is an effective equidistribution result of sectors of Hecke spheres in positive characteristic, which we prove using the known exponential decay of matrix coefficients, see for instance [AtGP]. We refer for instance to the works of Dani-Margulis [DM], Clozel-Oh-Ullmo [COU], Clozel-Ullmo [CU], Eskin-Oh [EO], Benoist-Oh [BeO] for equidistribution results of Hecke spheresxs in zero characteristic.
As we shall see in the main body of the text, Theorems 1, 3 and 4 are valid upon replacing K by any global function field (see also Remark 9 for further extensions). In this more general case, there are several (albeit finitely many) ways to go to infinity in X ∞ , and we will give more precise results towards which cusp of X ∞ the escape of mass occurs.
Considering the usual analogy between function fields and number fields, the above results should be compared with the zero characteristic analogue, in which the behaviour is in sharp contrast. When R ∞ is replaced by Z, K ∞ by R, and ν by an integer prime, Aka and Shapira have proved in [AkS,Theo. 4.8] that the periodic measures µ x ξ n along (virtually any) ray in the corresponding Hecke tree equidistribute towards the homogeneous measure in the moduli space PGL 2 (R)/ PGL 2 (Z). This result was what motivated this work, which turned out to have a surprisingly different outcome.
The underlying phenomenon which changes drastically when passing from zero to positive characteristics is as follows. While in zero characteristics the size of the orbit A ∞ x ξ n is exponential in n, in positive characteristics, it is linear in n due to the presence of the Frobenius automorphism (see Theorem 13). When this is combined with the fact that rational rays diverge in a linear speed, we get the results regarding the escape of mass.
Although the rigidity displayed in zero characteristics completely breaks down, as demonstrated by the above results, we still believe that the following conjecture holds. It implies in particular, that the set of rays having uniform escape of mass (such as the rational rays) is a null set.
Conjecture 5 For almost any ξ ∈ Ω (with respect to the natural probability measure), the averages 1 N +1 N n=0 µ x ξ n converge to the homogeneous probability measure on X ∞ .
Conjecture 5 reflects our belief that the behaviour along rational rays is far from generic. In fact, after some computer experiments, we suggest the following.
Conjecture 6 For any rational ξ ∈ Ω, µ x ξ n converges to the zero measure.
This work raises many other natural questions which we plan on studying in subsequent works. A few examples are: Is the rationality of the ray characterized by an uniform (or full) escape of mass? Can we find irrational rays exhibiting an escape of mass in average? Do we have a criterion for the convergence (or convergence in average) towards (a multiple of) the homogeneous measure ? What is the Hausdorff dimension of the set of ξ for which Theorem 3 holds?
Although we are working with the dynamics of rank 1 torus, it is interesting to compare our results with the huge corpus of works in dynamics on noncompact spaces, in particular locally homogeneous ones or moduli spaces, precisely devoted to prove that there is no escape of mass to infinity for nice sequences of probability measures on these spaces. This is in particular the case in homogeneous dynamics -with real Lie groups, thereby in zero characteristic -(see for instance [EMa, BeQ]) or in Teichmüller dynamics (see for instance [EMi, Ham]).
Note that an escape of mass for the diagonal group is not a feature appearing only in positive characteristics. Over the reals, there are examples of escape of mass for the diagonal flow: for example, in [Sar] the author constructs a sequence of closed geodesics on the modular surface which converge to the zero measure (see also [Sha] for similar examples in higher dimensions). We stress though that these examples do not share the arithmetic relation between the measures along the sequence which is present in our results. Indeed, due to the results in [AkS], such an arithmetic relation cannot coexist with an escape of mass over the reals.
As another motivation for studying the limiting behaviour of µ x ξ n (also originating from the analogy with [AkS]), let us give a relation with the distribution properties of the periods of the continued fraction expansion of certain sequences of quadratic irrationals. We refer for instance to the surveys [Las, Schm] for background. We denote by be the set of quadratic irrationals over K in K ∞ . Assume for simplicity that the characteristic p is different from 2, and denote by f σ = f the Galois conjugate of f ∈ QI over K. Given an irrational f ∈ Y −1 O ∞ , we have f ∈ QI if and only if the continued fraction expansion is eventually periodic. We then denote by ν f the uniform probability on the periodic part of the orbit of f under It is then easy to prove that x f is A ∞ -periodic. Using the main results of [BrP], we may construct a natural map from (a full-measure subset of) X ∞ onto (a full-measure subset of) Y −1 O ∞ , sending the (normalized) homogeneous measure m ∞ on X ∞ to the (normalized) Haar measure on Y −1 O ∞ , A ∞ -orbits in X ∞ to Ψ-orbits in Y −1 O ∞ , and more precisely the A ∞invariant probability measure µ x f to the equiprobability ν f for every quadratic irrational f in Y −1 O ∞ . Hence the distribution properties of the periods of the continued fraction expansions of quadratic irrationals are related to the distribution properties of the A ∞orbits in X ∞ .

Global function fields
We refer for instance to [Ros,Ser1] for the content of this subsection. Let F q be a finite field with q elements, where q is a positive power of a prime p. Let K be a global function field over F q , that is, the function field of a geometrically connected smooth projective curve C over F q , or equivalently an extension of F q of transcendance degree 1, in which F q is algebraically closed. The set P of primes of K is the set of closed points of C, or equivalently the set of discrete valuations of K, trivial on F × q , with value group exactly Z. We fix an element in P that we denote by ∞, and we denote by P f the set P − {∞}.
For every ω ∈ P, we denote by R ω the affine algebra of the affine curve C − {ω} (which is a Dedekind ring), by v ω the discrete valuation of K associated to ω (with the usual convention that v ω (0) = +∞), by K ω the associated completion of K (and again by v ω the extension of v ω to K ω ), by O ω its local ring, by π ω a uniformizer of O ω , by k ω its residual field (that we identify with its canonical lift in O ω ), and by deg(ω) the degree of k ω over F q . We assume, as we may using for instance the Riemann-Roch theorem, that π ν belongs to R ∞ if ν ∈ P f . Note that R ∞ ⊂ O ν if ν ∈ P f (since an element in R ∞ has no pole at the closed point ν = ∞ of C), and that R ∞ [π −1 ν ] ∩ O ν = R ∞ . We normalize the absolute value |·| ω associated to v ω by |x| ω = |k ω | −vω(x) = q − deg ω vω(x) for every x ∈ K ω . In particular, the product formula For every finite extension K ω of K ω , we denote again by v ω the unique extension of v ω to a valuation on K ω , and by e( K ω , its ramification index (see for instance [Ser1,§2]).
For instance, if C is the projective line P 1 and if ∞ = [1 : 0] is its usual point at infinity, and the uniformizers π ν for ν ∈ P f may be taken to be the monic prime polynomials in R ∞ , with deg ν the degree of the polynomial π ν . This is the example considered in the introduction.

Generalisation to rank-one semi-simple groups
The aim of this subsection is to explain to which K ∞ -rank-one groups the tools introduced in this paper are applying besides PGL 2 . But for the readability, we will restrict to this last case at the end of this subsection, giving the group-theoretic notation we are going to use. We refer for instance to [Tit1,Tit2] for the already known content of this subsection.
Let G be a connected semi-simple linear algebraic group defined over K, with K ∞ -rank one. We fix an embedding G → GL N for some N ∈ N. The example considered in the introduction is G = PGL 2 (which is adjoint and absolutely simple).
For every ω ∈ P and every algebraic subgroup H of G defined over K ω (for instance if H is defined over K), we set H ω = H(K ω ), which is a non-Archimedian Lie group.
For every ω ∈ P, we define Γ ω = G(R ω ), which is a lattice in the locally compact group G ω . For instance, when G = PGL 2 and C = P 1 , the lattice Γ ∞ is called Nagao's lattice [Nag] (or Weil's modular group [Wei]).
For every ω ∈ P, we denote by X ω the totally disconnected locally compact space Γ ω \G ω (contrarily to the introduction, we consider the left quotient, as it makes the connection with Bruhat-Tits theory easier). As we want to study phenomena of escape of mass at infinity for measures on X ∞ , we require that X ∞ is not compact. For instance, when G = PGL 2 , the space X ∞ is non compact, and identifies by Given ν ∈ P f , let S = {∞, ν} and letΓ S be the S-arithmetic group G(R ∞ [π ν −1 ]), which embeds diagonally in the locally compact group G S = G ∞ × G ν as a lattice, and let X S = Γ S \G S . We identify G ∞ and G ν , hence any subgroup of them, with their images in G S by the maps x → (x, e) and y → (e, y).
Note that when ν ∈ P f , the K ν -rank of G may be 1 (as in the case G = PGL 2 ) or not. For instance, let D be a (finite dimensional) central simple algebra over K which is ramified at ∞ (that is, D ∞ = D ⊗ K K ∞ ) is a division algebra). Then the algebraic group G with G(L) = PGL 2 (D ⊗ K L) for every K-algebra L is an (adjoint absolutely quasi-simple) connected semi-simple linear algebraic group defined over K, with K ∞ -rank one. For all ν ∈ P f , the group G has K ν -rank 1 if and only if D ramifies at ν (that is, when D ν = D ⊗ K K ν is a division algebra).
The next two results are not necessary for the main results of the paper, but they will be used to explain the restrictions on the considered algebraic groups. The first one follows from a well-known argument of weak approximation.
Lemma 7 Let ν ∈ P f , if the K ν -rank of G is 1, then there exist tori A in G defined over K, which splits over both K ∞ and K ν (hence is a maximal K ∞ -split and K ν -split torus).
Proof. By [PR,Theo. 2] applied to the semisimple connected algebraic group G defined over the infinite field K, there exists m ∈ N − {0} such that the closure of the image of the diagonal embedding of G(K) in G ∞ × G ν contains the subgroup of G ∞ × G ν generated by m-th powers. Let γ ∞ and γ ν be nontrivial elements in G(K) which split over K ∞ and K ν respectively. There hence exists an element in G(K) arbitrarily close to both γ m ∞ and γ m ν , which therefore splits simultaneously over K ∞ and K ν .
Proposition 8 Let H be an adjoint, absolutely quasi-simple, connected, semi-simple algebraic group over a local field F of F -rank one. Let T be a maximal F -split torus, Z its centralizer, P a minimal parabolic subgroup of H over F , and U its unipotent radical. If H is isomorphic over F to the algebraic group L → PGL 2 (D ⊗ F L) for every F -algebra L, where D is a central division algebra over F , then Z(F ) acts transitively on U (F ) − {0} by conjugation. If H is isomorphic over F to the algebraic group L → PU 1,1 (D ⊗ F L) for every F -algebra L, where D is a quaternion division algebra over F , then Z(F ) acts non transitively with finitely many orbits on U (F ) − {0} by conjugation. Otherwise, Z(F ) acts with infinitely many orbits on U (F ) − {0} by conjugation.
In particular, by the classification theorem [Tit1], if furthermore F = K ν for some ν ∈ P and H is defined and isotropic over K, then Z(F ) acts transitively on U (F ) − {0} by conjugation if and only if H is isomorphic over K to PGL 2 (D) where D is a central division algebra over K, and Z(F ) acts non transitively with finitely many orbits on U (F ) − {0} by conjugation if and only if H is isomorphic over K to PU 1,1 (D) where D is a quaternion division algebra over K.
If U is non abelien (or equivalently if the (relative) root system of H is not reduced), it is easy to see that the action of Z(F ) on U (F ) − {0} by conjugation has infinitely many orbits. Conversely, assume that U is abelien. When F = C (then H = PGL 2 (C) ) or F = R (then H = PO(1, n)), the action of Z(F ) on U (F )−{0} by conjugation is transitive. Hence assume that F is non Archimedian. By the classification theorem [Tit2], up to isomorphism, H is either PGL 2 (D) for a central division algebra D over F , or PU 1,1 (D) for a quaternion division algebra D over F and the Hermitian form h(z 1 , z 2 ) = z 1 z 2 + z 2 z 1 .
In the first of the above two cases, we may take The transitivity of the action by conjugation of Z on U − {0} follows hence from the transitivity of the action of D × × D × on D − {0} by (a, d) · b = abd −1 , which is immediate.
In the second case, we denote by z → z the canonical involution in the quaternion division algebra D over F , by N : x → xx and Tr : x → x + x its (reduced) norm and trace, and by (1, i, j, k) a standard basis of D over F . Recall that F × /(F × ) 2 is finite and non trivial. Indeed, this group is isomorphic to Tr(x) = 0} be the K-vector space of purely imaginary elements of D, endowed with the action of the orthogonal group O(N |Im D ) of the restriction to Im D of the norm. Since F × /(F × ) 2 is finite and N (F × ) = (F × ) 2 , there exists a finite subset A of F × such that every line in Im D contains a vector whose norm lies in A. By Witt's theorem, the group O(N |Im D ) hence acts with finitely many orbits on the lines of Im D.
The group SL 2 (D) acts by g · M = t g M g on the 6-dimensional F -vector space , and note that the restriction Q |M ⊥ 0 is non degenerate. We consider the basis of M ⊥ 0 , and we sometimes write matrices by blocs in the decomposition (e 1 , (e 2 , e 3 , e 4 ), e 5 ).
, whose unipotent radical is, by an easy computation, For every λ ∈ F , the action of , the action of T on each line in U has finitely many orbits (and at least two). The fact that the action of Z(F ) on U (F ) − {0} by conjugation has finitely many orbits (and at least two) hence follows from the fact that the action of O(N |Im D ) on the lines of Im D has finitely many orbits.
From now on in this paper, we fix ν ∈ P f and we denote by G = PGL 2 the (adjoint semi-simple absolutely simple) projective linear algebraic group over K in dimension 2, so that . Whenever necessary, we embed PGL 2 in GL 3 by the adjoint representation on the vector space of traceless 2-by-2 matrices.
Let A be the diagonal subgroup of G, that is, the algebraic subgroup of G consisting in the elements represented by diagonal matrices, which is a (split) maximal torus of G defined over K.
• again by v ω the map from the abelian group Remark 9 An appropriate version of this paper (including loss of mass phenomena of the homogeneous probability measures on the periodic orbits of the points along appropriate rays of the Hecke tree of any given periodic point of X ∞ ) is valid when we replace G by the linear algebraic group over K defined where D is a (finite dimensional) central division algebra over K which ramifies at the places ∞ and ν, and we endow the algebraic group G with a R ∞ -structure such that G( [Rei] for any information on orders), • or by G(L) = PU 1,1 (D ⊗ K L) for every K-algebra L, where D is a quaternion algebra over K (and the underlying Hermitian form is (z 1 , z 2 ) → z 1 z 2 + z 2 z 1 ), allowing, thanks to the transitivity properties described in Proposition 8, to prove a modified version of Theorem 13, when we replace Γ ∞ by a congruence subgroup and when we replace A by any torus over K in G which splits over both K ∞ and K ν (which exists by Lemma 7). But for the sake of simplicity, we stick to the above choice of (ν, G, A, Γ ∞ ).

Bruhat-Tits trees
Let (K, ν, G, A) and the associated notation be as in Subsection 2.2 before Remark 9.
Trees. Let T be a locally finite tree. Its set of vertices V T is endowed with the maximal distance for which two adjacent distinct vertices are at distance 1. A geodesic ray or line in T is an isometric map from N or Z to its set of vertices. The set of geodesic lines of T , endowed with the compact-open topology, is denoted by G T . An end of T is an equivalence class of geodesic rays, when two geodesic rays are equivalent if the intersection of their images is the image of a geodesic ray. The set of ends of T , endowed with the (compact, totally disconnected) quotient topology of the compact-open topology, is denoted by ∂ ∞ T , and called the boundary at infinity of T .
The translation length of an isometry γ of T is It is invariant under conjugation of γ in the isometry group of T . We will say that γ is loxodromic if T (γ) > 0, in which case there exists a unique image of a geodesic line in T on which γ translates a distance T (γ), called the translation axis of γ.
The geodesic flow (with discrete times) (φ m ) m∈Z on T is the right action (G T ×Z) → G T of Z on G T by translations at the source, defined by for all m ∈ Z and : Z → V T in G T . Given a group Γ of automorphisms of T , the geodesic flow on T induces a right action of Z on Γ\G T , also called the geodesic flow of Γ\T , and again denoted by (φ m ) m∈Z .
The tree of PGL 2 over local fields. For ω ∈ S = {∞, ν}, let T ω be the Bruhat-Tits tree of (G, K ω ), see for instance [Tit2]. We use its description given in [Ser2]. Recall We identify as usual the projective line P 1 (K ω ) with K ω ∪{∞} using the map K ω (x, y) → xy −1 . There exists one and only one homeomorphism between the boundary at infinity ∂ ∞ T ω of T ω and P 1 (K ω ) such that the (continuous) extension to ∂ ∞ T ω of the isometric action of G ω on T ω corresponds to the projective action of G ω on P 1 (K ω ). From now on, we identify ∂ ∞ T ω and P 1 (K ω ) by this homeomorphism.
The group G ω hence acts simply transitively on the set of ordered triples of distinct points in ∂ ∞ T ω . In particular, the group G ω acts transitively on the space G T ω of geodesic lines in T ω . The stabilizer under this action of the geodesic line Besides, by [Ser2,page 108], the translation length on Using the group morphism v ω : A ω → Z, the action by translations on the right of We denote by π ∞ : the canonical projection (see the diagram at the beginning of Section 2). The previous equation proves that π ∞ is equivariant with respect to the morphism v ∞ : A ∞ → Z, where A ∞ acts by translation on the right on X ∞ and Z by the (quotient) geodesic flow on Γ ∞ \G T ∞ : for all x ∈ X ∞ and a ∈ A, The principal bundle π ∞ : acts transitively on the geodesic rays in T ν starting from * ν . Thus Γ ∞ preserves and acts transitively on the sphere in T ν of any given radius centered at * ν . For every g ∈ G ν , there hence exists γ ∈ Γ ∞ and n ∈ N such that γ In particular, which is transitive and free on the fibers of π ∞ . Hence π ∞ : X S → X ∞ is a principal bundle under the group G(O ν ), which gives an identification between X ∞ = Γ ∞ \G ∞ and X S /G(O ν ) = Γ S \G S /G(O ν ) (see the diagram at the beginning of Section 2).
Ends of the modular graph at the place ∞ and heights.
The quotient graph Γ ∞ \T ∞ will be called the modular graph at ∞ of K. By for instance [Ser2], the set of cusps Γ ∞ \P 1 (K) is finite, and Γ ∞ \T ∞ is the disjoint union of a finite connected subgraph containing Γ ∞ * ∞ and of maximal open geodesic rays h where h z (called a cuspidal ray) is the image by the canonical projection T ∞ → Γ ∞ \T ∞ of a geodesic ray whose point at infinity in P 1 (K) ⊂ ∂ ∞ T ∞ is equal to z. Conversely, any geodesic ray whose point at infinity lies in P 1 (K) ⊂ ∂ ∞ T ∞ contains a subray that maps injectively by the canonical projection [Fre]) of Γ ∞ \T ∞ by its finite set of ends E ∞ . This set of ends is indeed finite, in bijection with Γ ∞ \P 1 (K) by the map which associates to z ∈ Γ ∞ \P 1 (K) the end towards which the cuspidal ray h z converges. See for instance [Ser2] for a geometric interpretation of E ∞ in terms of the curve C.
Let X ∞ = X ∞ E ∞ and let p ∞ : X ∞ → Γ ∞ \T ∞ be the map equal to the identity map on E ∞ and to the canonical projection on X ∞ (see the diagram at the beginning of Section 2). Since p ∞ is a proper map, this defined a compactification of X ∞ , by endowing X ∞ with the compact metrisable topology generated by the open subsets of U and the sets p ∞ −1 (U ) with U an open neighborhood of a point in E ∞ . We will say that E ∞ is the set of cusps of X ∞ , and we will indicate towards which cusp of X ∞ the escape of mass occurs.
For every x ∈ X ∞ , define the height of x in X ∞ by Lemma 10 For all g ∈ G ∞ and x ∈ X ∞ , we have By the triangle inequality and since the projection map The second assertion follows if p ∞ (x) and p ∞ (xg ) simultaneously belong or do not belong to (the image of) h z . If for instance p ∞ (x) belongs to h z and p ∞ (xg ) does not belong to h z , then and the result holds as above.
Example: Assume that C is the projective line over F q and that ∞ is its usual point at infinity. Then the (image of the) geodesic ray in T ∞ starting from * ∞ with point at infinity ∞ ∈ P 1 (K ∞ ), which is The quotient graph of finite groups Γ ∞ \\T ∞ , whose underlying graph is the geodesic ray Γ ∞ \T ∞ , is called the modular ray.
the modular ray Γ ∞ \\T ∞ (which has only one end) is given by the following figure.
The full-down property in the modular graph (see for instance [Ser2,Lub]). If ρ is a geodesic ray in T ∞ whose image is a cuspidal ray in Γ ∞ \T ∞ , the stabilizers of the vertices of ρ different from the origin of ρ are strictly increasing along the ray. Hence the image in Γ ∞ \T ∞ of a geodesic ray in T ∞ satisfies the following full-down property: if it starts to go down along the image of a cuspidal ray h z for some z ∈ E ∞ , then it needs to go all the way down to h z (0). As explained in [Ser2,Pau], this full-down property has the following consequence: the image by the canonical map T ∞ → Γ ∞ \T ∞ of a geodesic ray ρ in T ∞ starting from * ∞ either is an infinite sequence a 0 b 0 a 1 b 1 a 2 b 2 . . . of concatenations of paths a i (possibly reduced to points) in the finite graph Γ ∞ \T ∞ − z∈E∞ h z (]0, +∞[) and back and forth paths b i (of even lengths at least 2) from the origin h z i (0) of the cuspidal ray h z i to itself inside this ray, if ρ ends in an irrational point at infinity (that is, in P 1 (K ∞ ) − P 1 (K)), or starts by such a finite sequence and then follows some cuspidal ray to infinity, otherwise.
2.4 A ∞ -periodic orbits in X ∞ Let (K, ν, G, A) and the associated notation be as in Subsection 2.2 before Remark 9. Let us give a description of the compact orbits for the action by translations on the right of the subgroup A ∞ on X ∞ = Γ ∞ \G ∞ .
Proposition 11 For every g ∈ G ∞ , the following assertions are equivalent, where x = Γ ∞ g ∈ X ∞ : (1) there exists a unique A ∞ -invariant probability measure on the orbit xA ∞ ; (2) the subgroup A ∞ ∩ g −1 Γ ∞ g is a (uniform) lattice in A ∞ ; (3) the orbit of π ∞ (x) under the geodesic flow (φ n ) n∈Z on Γ ∞ \G T ∞ is periodic; (4) there exists γ 0 ∈ Γ ∞ and t 0 ∈ K × ∞ with v ∞ (t 0 ) positive and minimal such that If one of these conditions is satisfied, we say that x is A ∞ -periodic, and the unique A ∞ -invariant probability measure on xA ∞ is denoted by µ x .
The elements γ 0 ∈ Γ ∞ and t 0 ∈ K × ∞ are said to be associated with g. Note that they depend on the choice of the representative g of x: if γ 0 is associated with g, then γ −1 γ 0 γ is associated with γg for every γ ∈ Γ ∞ . Furthermore, γ 0 is primitive (not a proper power of an element of Γ ∞ ) and loxodromic on T ∞ . The period of π ∞ (x) under the geodesic flow (φ n ) n∈Z is the translation length of γ 0 on T ∞ , which is equal to v ∞ (t 0 ), and depends only on x.

13
The  (3) and (4). Let us now prove the additional properties of (γ 0 , t 0 ) and discuss its uniqueness. Assume that n in the above proof is minimal. Then γ 0 is primitive and loxodromic, with translation axis the image of , translation length n, which is the period of Γ ∞ under the geodesic flow.
positive and minimal. Then n = n and γ 0 = φ n ( ). Hence γ −1 0 γ 0 belongs to the pointwise stabilizer in Γ ∞ of the image of , which is the finite group gA(O ∞ )g −1 ∩ Γ ∞ . Therefore there exists

Hecke trees
Let (K, ν, G, A) and the associated notation be as in Subsection 2.2 before Remark 9.
The set X ∞ of homothety classes of R ∞ -lattices in K ∞ × K ∞ is the set of vertices of a graph, whose non-oriented edges are the pairs {x, x } of vertices such that there exists representatives Λ of x and Λ of x such that Λ ⊂ Λ and Λ /Λ is isomorphic to R ∞ /π ν R ∞ . The action of G ∞ on X ∞ extends to an (isometric) action by graph automorphisms on this graph.
For every x ∈ X ∞ , the connected component of x in this graph is a (|k ν | + 1)-regular tree, called the (ν)-Hecke tree of x, and denoted by T ν (x). We have T ν (x)g = T ν (xg) for all x ∈ X ∞ and g ∈ G ∞ . A (ν-)Hecke ray from x is a geodesic ray in the Hecke tree T ν (x) starting from x.
The following description of the ν-Hecke trees in X ∞ is well known, and is given, besides in order to fix the notation, only for the sake of completeness.
Lemma 12 Let g ∈ G ∞ and x = Γ ∞ g its image in X ∞ . The map from G ν to X ∞ defined by g → π ∞ (Γ S (g, g )) induces an isometric map hec g from the vertex set V T ν = G v /G(O ν ) of the Bruhat-Tits tree T ν onto the vertex set V T ν (x) of the Hecke tree T ν (x), sending * ν to x. For every γ 0 ∈ Γ ∞ , the map hec g conjugates the action of γ 0 on T ν to the right action of g −1 γ 0 g ∈ Γ ∞ on V T ν (x): for every y ∈ V T ν , we have hec g (γ 0 y) = hec g (y) g −1 γ 0 g .
For all h ∈ G ∞ such that Γ ∞ h = x, we have hec g = hec h if and only if g = h; furthermore, the following diagram commutes: Note that hec g depends on g and not only on x. We will denote again by hec g the (continuous) extension ∂ ∞ T ν → ∂ ∞ T ν (x) of hec g to the boundaries at infinity of the Bruhat-Tits and Hecke trees.
Proof. Since the action by translations on the right of G(O ν ) on X S preserves the fibers of the bundle map π ∞ : X S → X ∞ , the map g → π ∞ (Γ S (g, g )) does induce a map hec g : where γ ∈ Γ ∞ and n ∈ N. By Equation (3), any element in G ν may be written γ a n ν g for some γ ∈ Γ ∞ , n ∈ N and g ∈ G(O ν ). Hence, the elements in hec g (V T ν ) are the points π ∞ (Γ S (g, γ a n ν g )) = Γ ∞ a −n ν γ −1 g where g ∈ G(O ν ), γ ∈ Γ ∞ and n ∈ N. Therefore hec g (V T ν ) = V T ν (x).
If y, y ∈ V T ν are joined by an edge in T ν , then again by density of Γ ∞ in G(O ν ), there exists an element in Γ ∞ mapping the edge between y and y into the geodesic ray with vertices (a n ν * ν ) n∈N . Up to exchanging y and y , there hence exists n ∈ N and γ ∈ Γ ∞ such that γ −1 y = a n ν * ν and γ −1 y = a n+1 ν * ν . In particular, hec g (y) = Γ ∞ a −n ν γ −1 g is joined by an edge to hec g (y ) = Γ ∞ a −n−1 ν γ −1 g in the Hecke tree T ν (x). Hence hec g induces a surjective graph morphism between the trees T ν and T ν (x). Since both trees are regular of degree |k ν | + 1, the map hec g is an isomorphism of trees.
Equation (5) follows by writing y ∈ V T ν = G ν /G(O ν ) as y = g G(O ν ) for some g ∈ Γ S (see the line following Equation (3)), and by using the following equalities: Let h be another element in G ∞ such that Γ ∞ h = x. Since G ν = Γ S G(O ν ) and by the definition of π ∞ , we have hec g = hec h if and only if Γ ∞ γ −1 g = Γ ∞ γ −1 h for every γ ∈ Γ S , that is γ −1 (gh −1 )γ ∈ Γ ∞ for every γ ∈ Γ S . Writing gh −1 = a b c d and using γ = e, we may take a, b, c, d ∈ R ∞ . Since the order of vanishing at a point of C − {∞} of an element of R ∞ is nonnegative and v ν (π ν ) = 1, we have v ∞ (π ν ) = 0 by the product formula. Taking γ = π n ν 0 0 1 gives π n ν c, π −n ν b ∈ R ∞ for every n ∈ Z, that is c = b = 0. Taking γ = 1 π n ν 0 1 gives π n ν (a − d) ∈ R ∞ for every n ∈ Z, that is a = d. Hence hec g = hec h if and only if gh −1 is the identity element in Γ ∞ = G(R ∞ ).
The other claims are left to the reader.
3 Dynamics of the modular group at the infinite place on the Bruhat-Tits tree at a finite place Let (K, ν, G, A) and the associated notation be as in Subsection 2.2 before Remark 9.
In this Section, we study the dynamics of Γ ∞ on the Bruhat-Tits tree T ν of (G, K ν ).
Hence Γ ∞ does act on T ν , and for every n ∈ N, every γ 0 ∈ Γ ∞ preserves the sphere S ν (n) = S Tν ( * ν , n) of center * ν and radius n in T ν . Since S ν (n) is finite, every orbit in S ν (n) of the cyclic group γ 0 Z generated by γ 0 is periodic. The following linear growth property of these periodic orbits is a remarkable feature of the positive characteristic.
Theorem 13 Let γ 0 be an element in Γ ∞ which is loxodromic on T ∞ . Let K ν = K ν (γ 0 ) be the splitting field of γ 0 over K ν , with local ring O ν , uniformizer π ν and residual field k ν . Let e ν = e ν (γ 0 ) be the ramification index e( K ν , K ν ) of K ν over K ν . Let d ν = d ν (γ 0 ) be the smallest positive integer such that the image of γ 0 dν in G( k ν ) (by reduction modulo π ν O ν ) is the identity. Let r ν = r ν (γ 0 ) be the biggest positive integer such that the image of is not the identity. Then there exists a constant κ ν = κ ν (γ 0 ) ∈ N such that for every big enough n ∈ N, the maximal cardinality m n = m n (γ 0 ) of an orbit of This result implies that the sequence (m n ) n∈N has linear growth: for every n ∈ N big enough, we have and that if γ 0 is diagonalisable over K ν , then for every k ∈ N big enough m rν p k +κν ≤ d ν p k .
Proof. We start the proof by the following lemma on the growth of the valuations of the powers of the elements of O ν with their constant terms removed, which concentrates the positive characteristic feature.
Lemma 14 Let a ∈ k × ν , λ ∈ a + π ν O ν and n ∈ N. Define m n (λ) = min{k ∈ N − {0} : λ k ∈ a k + π n ν O ν } and r λ = v ν (λ − a) > 0. Then for every n > r λ , In particular, m n (λ) < p r λ n for every n > r λ and m r λ p k (λ) = p k for every k ∈ N − {0}. Proof. Up to replacing λ by λ a , we may assume that a = 1. To simplify the notation, let r = r λ . For every k ∈ N−{0}, consider the expansion of k in base p given by k = s i=0 a i p i where s ∈ N and a i ∈ {0, . . . , p − 1}. Let v p (k) = inf{i ∈ N : ∀ j < i, a j = 0 } be the p-adic valuation of k. Then, using the Frobenius automorphism, and the fact that a i is invertible in the characteristic subfield F p , hence in O ν , if and only if a i is non zero, we have Hence for every n ∈ N, we have λ k ∈ 1 + π n ν O ν if and only if rp vp(k) ≥ n. Therefore, for every n > r, if rp m−1 < n ≤ rp m (that is, if m = log p n r ), we have the equalities m n (λ) = min{k ∈ N − {0} : v p (k) = m} = p m . The result follows. Now, let γ 0 ∈ Γ ∞ be loxodromic on T ∞ . Note that the constant d ν is well defined since R ∞ ⊂ O ν ⊂ O ν . As we have seen in Section 2.3, there exist λ ± in a finite extension of K such that the element γ 0 is conjugated to λ + 0 0 λ − and K ν = K ν ( λ + λ − ). Note that λ − and λ + are distinct since γ 0 is not the identity element.
Let T ν be the Bruhat-Tits tree of (G, K ν ), and * ν = e G( O ν ) its standard base point in V T ν = G( K ν )/G( O ν ). The value group of (the unique extension of) the valuation v ν on K × ν contains the value group Z of the valuation v ν on K × ν with index e ν . By the correspondance of the action on the right of A( K ω ) on G( K ω )/A( O ω ) and the action of the geodesic flow on the geodesic lines in T ν , the sphere S ν (n) of center * ν and radius n in T ν is naturally contained in the sphere S Tν ( * ν , e ν n) of center * ν and radius e ν n in T ν , for every n ∈ N. Therefore, up to replacing K ν by K ν , we may assume that γ 0 is diagonalisable over K ν , and we prove that the cardinality of every orbit of γ 0 Z in S ν (n) is at most d ν p log p n+κν rν for every n ∈ N, for some κ ν ∈ N.
Note that the coefficients λ ± have absolute value 1 in K ν . Indeed, λ + 0 0 λ − may be choosen to be conjugated to a representative of γ 0 in GL 2 (R ∞ ). Hence λ ± satisfy an equation P (λ ± ) = 0 with P a monic quadratic polynomial with coefficients in R ∞ ⊂ O ν . Therefore |λ ± | ν 2 ≤ max{|λ ± | ν , 1}, so that |λ ± | ν ≤ 1, and equality holds by replacing γ 0 by its inverse. Hence λ ± ∈ a ± + π ν O ν with a ± ∈ k × ν . By the finiteness of k × ν , there exists a smallest d ν ∈ N − {0} such that a − dν = a + dν . Note that d ν coincides with the notation introduced in the statement of Theorem 13. Let Since γ 0 is loxodromic on T ∞ , no power of γ 0 is the identity, hence r ν > 0. Note that r ν coincides with the notation introduced in the statement of Theorem 13. Up to replacing γ 0 by γ 0 dν , to modify λ ± by a common multiple by an element of k × ν , and to proving that m n (γ 0 ) ≤ p log p n+κν rν for some κ ν ∈ N and for n big enough, we may assume that the constant terms in k × ν of λ ± are equal to 1, so that d ν = 1.
Since λ − = λ + , the centralizer Z Gν (γ 0 ) of γ 0 in G ν is the abelian group h A ν h −1 . Note that h is well defined modulo multiplication on the right by an element of A ν . Let 0 : n → a n ν * ν be the geodesic line in T ν from 0 ∈ ∂ ∞ T ν to ∞ ∈ ∂ ∞ T ν , through * ν at time n = 0, which is pointwise fixed by A(O ν ). The group A ν preserves 0 (Z) and acts transitively on it. Note that the projective action of A(O ν ) on P 1 (K ν ) fixes 0 and ∞, and acts transitively on Up to multiplying h on the right by an element of A ν , we may assume that the closest point to * ν on (the image of) is h * ν = (0). Let s ν = s ν (γ 0 ) ∈ N be the distance between * ν and h * ν in T ν (see the picture below).
Let pr : V T ν → (Z) be the closest point map on the geodesic line . For all n, k ∈ N, define (see the above picture) For all n, k ∈ N with 0 ≤ k < s ν , let E n, k be the set of x ∈ S ν (n) such that the length of the common segment [ * ν , h * ν ] ∩ [ * ν , x] is equal to k . Then we have a partition Since γ 0 fixes * ν , h(0) and h(∞), it pointwise fixes (Z) ∪ [ * ν , h * ν ]. Hence the above partition of S ν (n) is invariant under γ 0 .
Note that E n, k is exactly the set of points at distance n − |k| − s ν from ha k ν * ν = (k) on a geodesic ray from ha k ν * ν to a point in h(π −k ν O × ν ) ⊂ P 1 (K ν ). Hence for any two points in E n, k (with n, k fixed), there exists an element in the centralizer of γ 0 mapping one to the other. In particular, the cardinality c n, k = Card(γ Z 0 y) is independent of y ∈ E n, k . Since ha k−1 ν h −1 centralizes γ 0 and ha k−1 ν h −1 E n−|k|+1, 1 ⊂ E n, k , we have c n, k = c n−|k|+1, 1 . For every n ∈ N, we have c n , 1 ≤ c n +1, 1 , since the closest point map E n +1, 1 → E n , 1 is onto and equivariant under γ 0 .
Note that h(0) and h(∞) do not belong to P 1 (K), since γ 0 , being loxodromic on T ∞ , fixes no point of P 1 (K). The positive subray of 0 hence has no subray whose image is entirely contained in the image of . Therefore 0 (N) ∩ (Z) is either empty or the set of vertices of a compact interval [ (0), (k 0 )] for some k 0 ∈ Z.
Therefore, by the proof of Lemma 14 applied with λ = λ + λ − , since the constant r λ of Lemma 14 is equal to r ν by Equation (8), we have, if n is big enough, This concludes the proof of Theorem 13.

Escape of mass along Hecke rays of A ∞ -periodic points
Let (K, ν, G, A) and the associated notation be as in Subsection 2.2 before Remark 9. We fix from now on an A ∞ -periodic point x 0 in X ∞ = Γ ∞ \G ∞ , as well as a representative g 0 of x 0 in Γ ∞ , so that x 0 = Γ ∞ g 0 . In this section, we prove our main results on the asymptotic behavior of the A ∞ -invariant probability measures µ x supported on the A ∞ -orbits in X ∞ of the vertices x of the ν-Hecke tree T ν (x 0 ) of x 0 , as x tends to infinity in this tree along rays. We will recall below a proof that every vertex of Let P( X ∞ ) be the space of probability measures on the compactification X ∞ = X ∞ ∪ E ∞ by its finite set of cusps E ∞ = Γ ∞ \P 1 (K) (see Subsection 2.3). Let ξ ∈ ∂ ∞ T ν (x 0 ) be an end of the ν-Hecke tree of x 0 . Let Θ ξ be the subset of P( X ∞ ) consisting of the weak-star accumulation points of the sequence (µ x ξ n ) n∈N of A ∞ -invariant probability measures on the vertices (x ξ n ) n∈N along the geodesic ray in T ν (x 0 ) from x 0 to ξ. For all c > 0 and z ∈ E ∞ , we say that • ξ has c-escape of mass if there exists θ ∈ Θ ξ with θ(E ∞ ) ≥ c.
• ξ has uniform c-escape of mass towards the cusp z if for every θ ∈ Θ ξ we have θ({z}) ≥ c.

Uniform escape of mass along rational Hecke rays
We start this subsection by defining the rational Hecke rays in the ν-Hecke tree T ν (x 0 ) of x 0 , and we will then prove Theorem 15, a uniform escape of mass phenomenon for the A ∞ -invariant probability measures µ x , as x tends to infinity along these rays.
The group G(K) acts transitively on P 1 (K), but its subgroups Γ ∞ = G(R ∞ ) and The sets E ∞ = Γ ∞ \P 1 (K) (with order at most the class number of R ∞ ) and Γ S \P 1 (K) are finite and both canonical maps Γ ∞ \P 1 (K) → Γ S \P 1 (K) → G(K)\P 1 (K) may be non injective. Note that for instance when C is the projective line over F q and ∞ its usual point at infinity, then R ∞ is principal, and Γ ∞ does act transitively on P 1 (K).
Since Γ ∞ preserves P 1 (K) and by the commutativity of the diagram (6), the image hec g 0 (P 1 (K)) ⊂ ∂ ∞ T ν (x 0 ) by hec g 0 of the set P 1 (K) of rational points of ∂ ∞ T ν = P 1 (K ν ) does not depend on the choice of the representative g 0 of x 0 , nor does the image by hec g 0 of the orbit of ∞ by any subgroup of G(K) containing Γ ∞ , as for instance hec g 0 (Γ S ∞).
A Hecke ray in T ν (x 0 ), as well as its point at infinity, is said to be rational if its point at infinity belongs to hec g 0 (P 1 (K)), and S-rational if its point at infinity belongs to hec g 0 (Γ S ∞). In particular when Γ ∞ acts transitively on P 1 (K) (that is, when the graph Γ ∞ \T ∞ has only one end, as for instance when C is the projective line over F q and ∞ its usual point at infinity), these two notions coincides. But there are examples of functions fields when not all rational ends of T ν (x 0 ) are S-rational (the two inclusions Γ ∞ ∞ ⊂ Γ S ∞ ⊂ P 1 (K) may be strict).
If ξ is a rational end of T ν (x 0 ), the cusp of X ∞ associated to ξ is z ξ = Γ ∞ γ∞ ∈ E ∞ , where γ ∈ G(K) is such that ξ = hec g 0 (γ∞). Note that z ξ does not depend on the choices of g 0 or γ. If ξ is S-rational, we say that z ξ is an S-cusp of X ∞ .
Theorem 15 There exists c = c(x 0 ) > 0 such that every rational end ξ of the Hecke tree of x 0 has uniform c-escape of mass, and if furthermore ξ is S-rational, then ξ has uniform c-escape of mass towards the cusp of X ∞ associated to ξ.
In particular, since the orbits of γ 0 on V T ν are finite, every x ∈ V T ν (x 0 ) is also A ∞periodic and the A ∞ -invariant probability measure µ x on the compact orbit xA ∞ is well defined. Furthermore, with the notation of Theorem 13, for every n ≥ n ξ , the orbit under the geodesic flow of π ∞ (x n ) = π ∞ (hec g 0 (γ γy n+r ξ )) is periodic, with period λ n bounded as follows: Let d be the distance in the graph Γ ∞ \T ∞ . Recall that p ∞ : X ∞ → Γ ∞ \T ∞ is the map Γ ∞ g → Γ ∞ g * ∞ (see the diagram at the beginning of Section 2). Using • Lemma 10 with κ = κ ξ = d Tν ( * ν , (γ γ) −1 g 0 * ν ) for the first inequality, • the definition of the height (see Equation (4)) and Equation (10) with the notation β n = ( b γ,n+r ξ γ ) −1 b γ,n+r ξ γ for the second equality, we have, for all n ∈ N, Recall that β n belongs to G(K), hence preserves the set of geodesic rays in T ∞ ending in P 1 (K) ⊂ ∂ ∞ T ∞ , and takes finitely many values as n varies. Let Recall that any geodesic ray in T ∞ ending in P 1 (K) has a subray that isometrically injects into Γ ∞ \T ∞ . Hence using Equation (12) and the triangle inequality, there exist constants n ξ ≥ n ξ and κ ξ , κ ξ ≥ 0 such that for every integer n ≥ n ξ , This argument in fact proves that ht ∞, zn (x n ) ≥ n |v ∞ (π ν )| − κ ξ for n big enough, where z n is the cusp defined above. For every n ∈ N, let µ n = (π ∞ ) * µ xn , which is the equiprobability on the finite orbit of π ∞ (x n ) under the geodesic flow on Γ ∞ \G T ∞ . Recall that the pushforwards of measures by proper continuous maps preserve the total mass, and are weak-star continuous. The map π ∞ is a fibration with compact fiber, hence a proper map. Therefore ξ has uniform c-escape of mass (repectively uniform c-escape of mass towards its associated cusp z ξ = Γ ∞ γ γ∞) if and only if for every weak-star accumulation point θ of (µ n ) n∈N in the space of probability measures on Γ ∞ \G T ∞ , we have θ (E ∞ ) ≥ c (respectively θ ({z ξ }) ≥ c).
Let o : Γ ∞ \G T ∞ → Γ ∞ \V T ∞ be the origin map Γ ∞ → Γ ∞ (0), which is a proper map. For all N ∈ N, let which are open subsets of Γ ∞ \G T ∞ , which accumulate as N → +∞ exactly to E ∞ (ξ) ⊂ Γ ∞ \G T ∞ . By the full-down property (see Subsection 2.3), the orbit under the geodesic flow of π ∞ (x n ) passes at a distance from Γ ∞ * ∞ which is bounded by the diameter N 0 of the finite graph Γ ∞ \T ∞ − z∈E∞ h z (]0 + ∞[). Recall that this orbit is periodic, of period denoted by λ n . Hence if N ≥ N 0 and if ht(x n ) ≥ N , the origins of φ i (π ∞ (x n )) for 0 ≤ i ≤ λ n needs to range twice over all points at distance between N and ht(x n ) on a geodesic ray in Γ ∞ \T ∞ between Γ ∞ * ∞ and o(ρ ∞ (x n )). Hence if n is big enough, by the comment following Equation (13) and by Equation (11), we have By the linear growth property of (m n (γ 0 )) n∈N (see Equation (7) and the notation of Theorem 13), the right hand side of Equation (14) as a limit as n → +∞ at least Hence for every weak-star accumulation point θ of (µ n ) n∈N , we have θ (E ∞ (ξ)) ≥ c. This proves the result.
This assumption is for instance satisfied if −1 is not a square modulo p (as for p = 3), if p = q and if γ 0 = Y 1 1 0 , since ∆ = Y 2 + 4. By the previous arguments, for every rational end ξ ∈ Ω, there exists an element θ ∈ Θ ξ which vanishes on X ∞ . This proves Theorem 2 in the introduction. The above proof also gives a speed of escape of mass when LOM(γ 0 ) = 1: for every compact subset C of X ∞ , we have µ xn k (C) = O( 1 n k ) when n k = r ν (γ 0 ) p k + κ ν (γ 0 ).
Let us give one more estimation on the constant LOM(γ 0 ) when p = 2 and v ν (tr γ 0 ) > 0. We then have square and 2 otherwise). The constant terms a ± = ± −4 det γ 0 ∈ k ν × of λ ± are opposite (and non zero), hence d = 2. By Equation (8) Hence LOM(γ 0 ) ≤ 1 2 , with equality if and only if tr γ 0 is a constant multiple of a power of π ν , as for instance when π ν = Y and γ 0 = Y 1 1 0 . For these elements where equality holds, at least half the mass escapes to infinity along subsequences of every rational Hecke ray.

Escape of mass along uncountably many Hecke rays
In the previous subsection, we proved escape of mass phenomena along countably many Hecke rays, the rational ones. In this subsection, we use the uniformity of the escape of mass in Theorem 15 in order to prove that an escape of mass (towards prescribed cusps of X ∞ ) actually occurs along uncountably many Hecke rays. We first introduce some notation that we will use from now on in this paper. We denote by Ω = ∂ ∞ T ν (x 0 ) the boundary at infinity of the Hecke tree T ν (x 0 ) of x 0 . For every ξ ∈ Ω, we denote by [x 0 , ξ[ the geodesic ray in T ν (x 0 ) starting from x 0 and converging to ξ. We denote by (x ξ n ) n∈N the sequence of vertices of [x 0 , ξ[, in this order along this ray. In particular, x ξ 0 = x 0 and d(x ξ k , x ξ n ) = |k − n|.
Let x ∈ V T ν (x 0 ). We define the sector of x by and, for every n ∈ N, the sector-sphere of x of radius n by The depth of the cone C x or of the sector Ω x of x is defined to be the distance in the Hecke tree T ν (x 0 ) from x to x 0 . The sector-sphere S n x is nonempty if and only if n is at least this depth. For every ξ ∈ Ω, the sequences (C x ξ n ) n∈N and (Ω x ξ n ) n∈N are strictly decreasing, with Ω x 0 = Ω, Note that if two cones (or sectors) intersect nontrivially, then one of them is contained in the other. Also, sectors are nonempty compact-open sets in Ω and in particular contain infinitely many rational ends, and even infinitely many S-rational ends.
Theorem 17 There exists c = c(x 0 ) > 0 such that, for every S-cusp z ∈ E ∞ of X ∞ , the set of ξ ∈ Ω having c-escape of mass towards the cusp z is uncountable.
In particular, the set of ξ ∈ Ω having c-escape of mass is uncountable. Theorem 3 in the Introduction follows immediately, being the case when C is the projective line, in which case X ∞ has only one cusp.
Proof. Let c = c(x 0 ) > 0 be the constant introduced in Theorem 15. For every S-cusp z ∈ E ∞ , we fix a fundamental system (V n ) n∈N of open neighborhoods of z in X ∞ = X ∞ ∪ E ∞ , so that {z} = n∈N V n . For all n ∈ N, let Σ n = {0, 1} n be the set of words of length n in 0 and 1. Let Σ = n∈N Σ n be the set of finite words in 0 and 1.
We are going to define a map ψ : Σ → V T ν (x 0 ) with the following properties: For all n ∈ N and α ∈ Σ n , (1) if β is an initial subword of α, then Ω ψ(α) ⊂ Ω ψ(β) , (2) if β is an initial subword of α with β = α, then the intersection Ω ψ(α0) ∩ Ω ψ(α1) is empty, (3) the depth of the sector Ω ψ(α) is at least n, Assume for the moment that such a map ψ is constructed. Let Σ ∞ = {0, 1} N , which is uncountable. For every w ∈ Σ ∞ , let w n be the initial subword of length n of w. Note that by Properties (1) and (3), for every w ∈ Σ ∞ , the sequence of sectors (Ω ψ(wn) ) n∈N is strictly nested, and its intersection contains a single point, denoted by ξ w . Furthermore, for every n ∈ N, we have w n ∈ [x 0 , ξ w [ . Note that by Property (2), the map w → ξ w from Σ ∞ to Ω is injective. By Property (4), for every w ∈ Σ ∞ , if θ w is a weak-star accumulation point of (µ ψ(wn) ) n∈N in the space P( X ∞ ) of probability measures on the compact space X ∞ , then θ w ({z}) ≥ c. Hence ξ w has c-escape of mass towards the cusp z. This proves Theorem 17.
By density, there exist distinct points ξ 0 and ξ 1 in Ω ψ(α) which are rational and whose associated cusps z ξ 0 and z ξ 1 of X ∞ respectively are both equal to z. By Theorem 15, ξ 0 and ξ 1 both have c-escape of mass towards the cusp z.
Let j ∈ {0, 1}. We claim that there exists n j ≥ m 0 such that µ n+2 , which contradicts the fact that ξ j has c-escape of mass towards the cusp z.

Effective equidistribution of sector-spheres
The aim of this section is to prove an effective statement regarding the equidistribution in X ∞ of the sector-spheres of the vertices of the Hecke tree of x 0 , Theorem 18, by using the effective decay of matrix coefficients for the action of G S on L 2 (X S ). This sectorial effective equidistribution result will be the main tool used in Subsection 4.4 in order to prove Theorem 4 and its improvements. We first introduce some notation.
We denote by |E| the cardinality of any finite set E and by ∆ x the unit Dirac mass at any point x of any measurable space. For all x ∈ V T ν (x 0 ) and n ∈ N with n ≥ k where k = d Tν (x 0 ) (x 0 , x) is the depth of the sector C x , let η n, x be the uniform probability measure on the (finite nonempty) sector-sphere S n x : that we consider as a probability measure on the locally compact space X ∞ with support S n x . Since the ν-Hecke tree of x 0 (as is the Bruhat-Tits tree T ν ) is |P 1 (k ν )|-regular, note that |S n x | = |k ν | n−k if x = x 0 and n ≥ k, and that |S n x | = (|k ν | + 1)|k ν | n−1 if x = x 0 and n > 0.
For every place ω ∈ P, we define W ω = G(O ω ), which is a maximal compact-open subgroup of G ω , and W S = W ∞ ×W ν ⊂ G ∞ ×G ν = G S , which is a maximal compact-open subgroup of G S . We denote by m ∞ (respectively m S ) the Haar measure on G ∞ (respectively G S ), normalized so that m ∞ (W ∞ ) = 1 (respectively m S (W S ) = 1). We again denote by m ∞ (respectively m S ) the measure on X ∞ (respectively X S ) such that the covering map G ∞ → X ∞ = Γ ∞ \G ∞ (respectively G S → X S = Γ S \G S ) locally preserves the measures. Note that this measure on X ∞ (respectively X S ) is nonzero and finite, but is not necessarily a probability measure, the above normalisation of the Haar measures will turn out to be more convenient. For every k ∈ [1, +∞], we define L k (X ∞ ) = L k (X ∞ , m ∞ ) (respectively L k (X S ) = L k (X S , m S ) ).
The group G = G ∞ (respectively G = G S ) acts (on the left) on the complex vector space of maps ψ from X = X ∞ (respectively X = X S ) to C, by right translation on the source: For every g ∈ G, if R g : X → X is the right translation x → xg, then gψ = ψ • R g : x → ψ(xg).
A map ψ from X to C is locally constant if there exists a compact-open subgroup U of W = W ∞ (respectively W = W S ) which leaves ψ invariant: ∀ g ∈ U, gψ = ψ , or equivalently, if ψ is constant on each orbit of U under the right action of G on X. Note that ψ is continuous, since the orbits of U are compact-open subsets. We define as the dimension of the complex vector space generated by the images of ψ under the elements of W , which is finite, and even satisfies d f ≤ [W : U ]. We define the lc-norm of every bounded locally constant map ψ : X → C by ψ lc = d ψ ψ ∞ .
Though the lc-norm does not satisfy the triangle inequality, we have λψ lc = |λ| ψ lc for every λ ∈ C. We denote by lc(X) the vector space of bounded locally constant maps ψ from X to C.
Finally, given a set A and maps f, g : A → [0, +∞[ , we will write f g if there exists a constant c > 0 such that f (a) ≤ c g(a) for all a ∈ A. If f and g depend on a parameter p, we write f p g if there exists a constant c > 0, possibly depending on the parameter p, such that f (a) ≤ c g(a) for all a ∈ A.
The following result strenghtens the well-known result of equidistribution of full Hecke spheres (see for instance the works of Dani-Margulis [DM], Clozel-Oh-Ullmo [COU], Clozel-Ullmo [CU], Eskin-Oh [EO], Benoist-Oh [BeO] in characteristic 0), to an equidistribution result of sector-spheres, which is furthermore effective. Taking x = x 0 gives as a particular case an effective equidistribution result of the full Hecke spheres.
Theorem 18 There exists δ > 0 such that for every x ∈ V T ν (x 0 ), we have for all n x 1 and ψ ∈ lc(X ∞ ).
Proof. Let us fix x ∈ V T ν (x 0 ) and ξ = ξ x ∈ Ω x , so that x = x ξ k for some fixed k = k x ∈ N (see the picture at the beginning of Subsection 4.2).
Step 1: Thickening the sector-spheres. Note that the sector-spheres are measure zero subsets of X ∞ . In order to be able to apply (effective) mixing arguments, we have to replace them by (regular) bump fonctions around them. In this step, we will define nice compact-open neighborhoods of the sector-spheres, whose characteristic functions will be our bump fonctions. By the construction of the sector-spheres, it is more natural to lift the sector-spheres in X S and to work in the bundle X S over X ∞ .
We will hence use a lot the W ν -bundle map π ∞ (see Subsection 2.3) from X S = Γ S \G S to X ∞ = Γ ∞ \G ∞ , defined by Γ S (g, h) → Γ ∞ g whenever h ∈ W ν . Recall (see Subsection 2.5) that the map hec g 0 from the Bruhat-Tits tree T ν to the Hecke tree T ν (x 0 ), defined on V T ν = G ν /W ν by hW ν → π ∞ (Γ S (g 0 , h)) is an isomorphism of trees, and we identify ∂ ∞ T ν = P 1 (K ν ) and Ω by (the extension to the boundary at infinity of) this map. We endow T ν (x 0 ) with the (left) action of G ν making hec g 0 equivariant. Since W ν = G(O ν ) acts transitively on Ω = P 1 (O ν ), we also fix w = w x ∈ W ν such that w∞ = ξ, where ∞ = [1 : 0]. For all n ∈ N, we denote by B n ν the stabiliser in W ν of the point x ∞ n at distance n from x 0 on the geodesic ray [x 0 , +∞[ in the Hecke tree T ν (x 0 ). The group B ν = B k ν acts transitively on the sector-spheres S n x ∞ k of x ∞ k for all n ∈ N. As we have already seen, for all n ∈ N, we have x ∞ n = hec g 0 (a n ν * ν ) = π ∞ (Γ S (g 0 , a n ν )) . Note that x ξ n = wx ∞ n for all n ∈ N. In particular, x = wx ∞ k , hence wB ν w −1 is the stabilizer in W ν of x. It acts transitively on the sector-spheres S n x of x for all n ∈ N, with stabilizer of x ξ n equal to wB n ν w −1 . Therefore, for all n ∈ N, S n x = wB ν w −1 x ξ n = wB ν x ∞ n = π ∞ (Γ S (g 0 , wB ν a n ν )) .
Now that we have this nice description of the sector-spheres, let us define nice neighborhoods of them.
Lemma 19 There exist σ 1 , σ 2 > 0 and a nondecreasing family (B ∞ ) >0 of compact-open subgroups of W ∞ , which is a fundamental system of neighborhoods of the identity element in W ∞ , and which satisfies and ∀ a ∈ A ∞ , a −1 B ∞ a ⊂ B e σ 2 |v∞(a)| ∞ .
Proof. For every n ∈ N, let Z n be the kernel of the reduction modulo π n+1 For all a, b, c, d ∈ K ∞ and t ∈ K × ∞ , we have Hence, using the isomorphism α ∞ : K × ∞ → A ∞ defined just above Remark 9, we have a −1 Z n a ⊂ Z n−|v∞(a)| for all a ∈ A ∞ and n ≥ |v ∞ (a)| in N. Equation (18) (which will only be used in Subsection 4.4) follows with σ 2 = log [Z 1 : Z 0 ].
For every > 0, we finally define the following compact-open subset of X S U = Γ S (g 0 B ∞ , wB ν ) , so that, for all n ∈ N, the image π ∞ (U a n ν ) of its translate by a n ν is a (small when is small) neighborhood of the sector-sphere S n x in X ∞ , by Equation (16).
Step 2: Using the decay of matrix coefficients. In this step, we use the following theorem about effective decay of matrix coefficients for the action of G S on L 2 (X S ) (see for instance [AtGP]). For every g = (g ∞ , g ν ) ∈ G S = G ∞ × G ν , we denote by |g| S the maximum of the norms of the adjoint representations of g ∞ , g ν (for the operator norm on the 3 × 3 matrices with entries in K ∞ , K ν ).
Theorem 20 There exists δ 1 > 0 such that for all locally constant maps ϕ, ψ ∈ L 2 (X S ) and for every g ∈ G S . Now, let us fix ψ ∈ lc(X ∞ ). We denote by ψ = ψ • π ∞ its lift to X S , which is constant on each right W ν -orbit, hence is locally constant (since invariant under U × W ν if ψ is invariant under U ). Note that ψ ∈ L 2 (X S ) since m S is finite and ψ is bounded. By the normalization of the Haar measures, we have m S ( ψ) = m ∞ (ψ) and m S (X S ) = m ∞ (X ∞ ) .
For every > 0, let ϕ = 1 m S (U ) 1 U be the normalized characteristic function of U , so that m S (ϕ ) = 1. The map ϕ : X S → C is locally constant, since it is invariant under the right action of the compact-open subgroup B ∞ × B ν of W S . We have