Application of the convergence of the spatio-temporal processes for visits to small sets
aa r X i v : . [ m a t h . D S ] J u l APPLICATION OF THE CONVERGENCE OF THESPATIO-TEMPORAL PROCESSES FOR VISITS TO SMALLSETS
FRANC¸ OISE P`ENE AND BENOˆIT SAUSSOL
Abstract.
The goal of this article is to point out the importance ofspatio-temporal processes in different questions of quantitative recur-rence. We focus on applications to the study of the number of visits toa small set before the first visit to another set (question arising from aprevious work by Kifer and Rapaport), the study of high records, thestudy of line processes, the study of the time spent by a flow in a smallset. We illustrate these applications by results on billiards or geodesicflows. This paper contains in particular new result of convergence indistribution of the spatio temporal processes associated to visits by theSinai billiard flow to a small neighbourhood of orbitrary points in thebilliard domain.
Contents
1. Introduction 12. Convergence results for transformations and special flows 33. Number of visits to a small set before the first visit to a secondsmall set 44. Number of high records 65. Line process of random geodesics 86. Time spent by a flow in a small set 11Appendix A. Visits of the Sinai billiard flow to a finite union ofballs for the position 14References 221.
Introduction
Let (Ω , F , µ, T ) or (Ω , F , µ, Y = ( Y t ) t ≥ ) be a probability preserving dy-namical system in discrete or continuous times. Let ( A ε ) ε be a family ofmeasurable subsets of Ω with µ ( A ε ) →
0+ as ε →
0. Given a family of mea-surable normalization functions H ε : A ε → V where V is a locally compactmetric space endowed with its Borel σ -algebra V , we study the family ofspatio-temporal point processes ( N ε ) ε on [0 , + ∞ ) × V given by N ε ( x ) := N ( T, A ε , h ε , H ε ) := X n ≥ T n ( x ) ∈ A ε δ ( nh ε ,H ε ( T n ( x ))) for a map T (1) Date : August 3, 2020.2000
Mathematics Subject Classification.
Primary: 37B20. or N ε ( x ) := N ( Y, A ε , h ε , H ε ) = X t> Y t enters A ε δ ( th ε ,H ε ( Y t ( x ))) for a flow Y . (2)We are interested in results of convergence in distribution of ( N ε ) ε> to apoint process P as ε → • Study of the visits in a small neighborhood of an hyperbolic peri-odic point of a transformation (see [21, Section 5], with applicationto Anosov maps).Such visits occurs by clusters (once a point visits such a neighbour-hood, it stays close to the periodic point during an unbounded timebefore living this area). The idea we used to study these clusterswas to consider a process N ε corresponding to the last (or first)position of the clusters. • Convergence of a normalized Birkhoff sum processes n − α ⌊ nt ⌋− X k =0 f ◦ T k t ≥ n ≥ to an α -stable process. In [25] Tyran-Kami´nska provided criteriaensuring such a result. One of the conditions is the convergence of N /n = N ( T, {| f | > γn α } , /n, n − α f ( · ))(for every γ >
0) to some Poisson point process. The general resultsof [21] combined with the criteria of [25] have been used in [13] toprove convergence to a L´evy process for the Birkhoff sum processof H¨older observable of billiards in dispersing domains with cusps.We won’t detail again the above applications. Our goal here is to emphasizeon further ones.After recalling in Section 2 below the general results of convergence ofspatio-temporal point processes to Poisson point processes established in[21], we present in the remaining sections four other important applicationsof such convergence results: • The number of visits to (or of the time spent in) a small set beforethe first visit to a second small set (motivated by Kifer and Ra-paport [16]), with application to the Sinai billiard flow with finitehorizon, • The evolution of the number of records larger than some threshold,with an application to billiards with corners and cusps of orderlarger than 2,
PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 3 • The Line process of random geodesics (motivated by Athreya, Lal-ley, Sapir and Wroten [2]), • The time spent by a flow in a small set, with application to theSinai billiard flow with finite horizon.Appendix A contains a new theorem of convergence of point processes forthe Sinai billiard flow and for neighborhoods of arbitrary positions in thebilliard domain, which is used in the examples that illustrate the applicationsabove. Finally we also present an application to the closest approach by thebilliard flow.2.
Convergence results for transformations and special flows
We set E := [0 , + ∞ ) × V and we endow it with its Borel σ -algebra E = B ([0 , + ∞ )) ⊗ V . We also consider the family of measures ( m ε ) ε on( V, V ) defined by m ε := µ ( H − ε ( · ) | A ε ) (3)and W a family stable by finite unions and intersections of relatively compactopen subsets of V , that generates the σ -algebra V . Let λ be the Lebesguemeasure on [0 , ∞ ).We will approximate the point process defined by (1) or (2) by a Poissonpoint process on E . Given η a σ -finite measure on ( E, E ), recall that aprocess N is a Poisson point process on E of intensity η if(i) N is a point process (i.e. N = P i δ x i with x i E -valued randomvariables),(ii) For every pairwise disjoint Borel sets B , ..., B n ⊂ E , the randomvariables N ( B ) , ..., N ( B n ) are independent Poisson random vari-ables with respective parameters η ( B ) , ..., η ( B n ).Let M p ( E ) be the space of all point measures defined on E , endowed withthe topology of vague convergence; it is metrizable as a complete separablemetric space. A family of point processes ( N ε ) ε converges in distributionto N if for any bounded continuous function f : M p ( E ) → R the followingconvergence holds true E ( f ( N ε )) → E ( f ( N )) , as ε → . (4)For a collection A of measurable subsets of Ω, we define the followingquantity: ∆( A ) := sup A ∈A ,B ∈ σ ( ∪ ∞ n =1 T − n A ) | µ ( A ∩ B ) − µ ( A ) µ ( B ) | . (5)We set λ for the Lebesgue measure on [0 , ∞ ). Theorem 2.1. (Convergence result for transformations [21, Theorem 2.1] )We assume that (i) for any finite subset W of W we have ∆( H − ε W ) = o ( µ ( A ε )) , (ii) there exists a measure m on ( V, V ) such that for every F ∈ W , m ( ∂F ) = 0 and lim ε → µ ( H − ε ( F ) | A ε ) converges to m ( F ) .Then the family of point processes ( N ε ) ε converges strongly in distribution,as ε → , to a Poisson point process P of intensity λ × m . i.e. with respect to any probability measure absolutely continuous w.r.t. µ FRANC¸ OISE P`ENE AND BENOˆIT SAUSSOL
In particular, for every relatively compact open B ⊂ E such that ( λ × m )( ∂B ) = 0 , ( N ε ( B )) ε converges in distribution, as ε → , to a Poissonrandom variable with parameter ( λ × m )( B ) . Theorem 2.2. (Convergence result for special flows [21, Theorem 2.3] ) As-sume (Ω , µ, Y = ( Y t ) t ) can be represented as a special flow over a probabilitypreserving dynamical system ( M, ν, F ) with roof function τ : M → (0 , + ∞ ) with M ⊂ Ω and set Π : Ω → M for the projection such that Π( Y s ( x )) = x for all x ∈ M and all s ∈ [0 , τ ( x )) .Assume moreover that Y enters at most once in A ε between two consecutivevisits to M and that there exists a family of measurable normalization func-tions G ε : M → V such that the family of point processes ( N ( F, Π( A ε ) , h ε , G ε )) ε converges in distribution, as ε → and with respect to some probability mea-sure ˜ ν ≪ ν , to a Poisson point process of intensity λ × m , where m is somemeasure on ( V, V ) , then the family of point processes ( N ( Y, A ε , h ε / E ν [ τ ] , G ε ◦ Π)) ε converges in distribution, as ε → (with respect to any probabilitymeasure absolutely continuous with respect to µ ), to a Poisson process P ofintensity λ × m . Number of visits to a small set before the first visit to asecond small set
Suppose B ε and B ε are two disjoint sets. We define the spatio-temporalprocess N ε with A ε = B ε ∪ B ε , H ε ( x ) = ℓ if x ∈ B ℓε , ℓ = 0 ,
1, that is on[0 , + ∞ ) × { , } N ε ( x ) = ∞ X n =1 1 X ℓ =0 δ ( nµ ( A ε ) ,ℓ ) B ℓε ( T n x ) (6)in the case of a transformation T or N ε ( x ) = X t> X ℓ =0 δ ( th ε ,ℓ ) Y t enters B ℓε (7)in the case of a flow Y . In [16] Kifer and Rapaport studied the distribution ofa (multiple) event T n x ∈ B ε until a (multiple) hazard T n ( x ) ∈ B ε . We stickhere to single event and hazard and define, in the case of a transformation T , M ε ( x ) := τ B ε ( x ) X n =1 B ε ( T n x ) , (8)where we set τ B ( x ) := inf { n ≥ T n ( x ) ∈ B } or, in the case of a flow Y : M ε ( x ) := X t ∈ (0 ,τ B ε ( x )) Y t enters B ε , (9)where we set τ B ( x ) := inf { t > Y t ( x ) ∈ B } . The process M ε counts thenumber of entrances of the flow in the 1-set before its first visit to the 0-set.In the case of a flow, it is also natural to consider the following process M ′ ε PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 5 measuring the time spent by the flow in the 1-set before its first visit to the0-set: M ′ ε ( x ) := Z τ B ε ( x )0 B ε ◦ Y s ( x ) ds . (10)In view of the study of this last process, we will consider the following processmeasuring the time spent by the flow in each set: L ε := X j =0 X t : Y t enters B jε δ th ε ,j,a ε D Bjε ◦ Y t ε> with D A := τ Ω \ A . Theorem 3.1.
Let p ∈ (0 , and P be a probability measure on Ω . Assume,in the case of a flow, that lim ε → P ( B ε ∪ B ε ) =0.If the spatio-temporal process N ε defined as in (6) or (7) converges,with respect to P , to a PPP of intensity λ × B ( p ) where B ( p ) denotes theBernoulli measure with parameter p (for a transformation we expect p =lim ε → µ ( B ε ) /µ ( A ε ) ), then the process ( M ε ) ε has asymptotically geometricdistribution, more precisely it converges in distribution to M with P ( M = k ) = p k (1 − p ) for any k ≥ ; in particular the asymptotic value for thecommitor function is lim ε → P ( τ B ε < τ B ε ) = lim ε → P ( M ε = 0) = 1 − p. In the case of a flow, if ( a ε τ Ω \ B ε ) ε converges in probability P to 0 and if ( L ε ) ε> supported on [0 , + ∞ ) × { , } × ¯ R + converges in distribution withrespect to P to a PPP L with intensity λ × P j =0 p j ( δ j × m ′ j ) where the m ′ j are probability measures, then ( a ε M ′ ε ) ε converges to P M i =1 X i where ( X i ) i isa sequence of i.i.d. random variables with distribution m ′ and independentof M where M is as above.Proof. We first observe that the mapping J : ξ ∈ M p ([0 , + ∞ ) × { , } ) ξ ([0 , τ ] × { } )is continuous where τ = sup { t ≥ ξ ([0 , t ] × { } ) = 0 } is continuous at a.e.realization ξ of χ := P P P ( λ ×B ( p )). Indeed, ξ ( ·×{ } ) and ξ ( ·×{ } ) are therealization of two homogeneous independent Poisson process hence τ is a.s.not an atom of ξ ( · × { } ). Observe that, in the case of a transformation, M ε = J ( N ε ) and in the case of a flow P ( M ε = J ( N ε )) = P ( Y ∈ B ε ∪ B ε ) →
0. Therefore, by the continuous mapping theorem, M ε converges indistribution to G := J ( χ ).We now compute the law of G . The first hazard τ has an exponentialdistribution with parameter 1 − p , while χ ( · ) := χ ( · × { } ) is a Poissonpoint process with intensity pλ , and the two are independent. Therefore,for any k ∈ NP ( G = k ) = P ( χ ([0 , τ ]) = k )= Z ∞ e − pt ( pt ) k k ! (1 − p ) e − (1 − p ) t dt = (1 − p ) p k . FRANC¸ OISE P`ENE AND BENOˆIT SAUSSOL
This ends the proof of the first points of the Theorem. Let us now prove thelast one. We use the fact that the mapping J : ξ ∈ M p ([0 , + ∞ ) × { , } × ¯ R + ) R [0 ,τ ] ×{ }× [0 ,K ] z dξ ( t, j, z ) is continuous at a.e. realization ξ of χ and conclude as above by the continuous mapping theorem and the Slutzkylemma since a ε M ′ ε = { Y B ε } (cid:0) J ( L ε ) + a ε τ Ω \ B ε (cid:1) . (cid:3) Example 3.2.
Consider the billiard flow ( Y t ) t associated to a Sinai billiardwith finite horizon in a domain Q ⊂ T (see Appendix for details). Let P beany probability measure on Ω := Q × S absolutely continuous with respectto Lebesgue. We fix two distinct point positions q , q ∈ Q and two positivereal numbers r , r > . Set B iε := B ( q i , r i ε ) × S and d i = 2 − q i ∈ ∂Q .Then ( M ε ) ε converges in distribution with respect to P to M with P ( M = k ) = p k (1 − p ) for any k ≥ and with p = d r d r + d r .Moreover ( ε − M ′ ε ) ε converges in distribution with respect to P to r P M i =1 Y i where ( Y i ) i is a sequence of i.i.d. random variables with density y y √ − y [0 , ( y ) independent of M , with M as above.Proof. Recall that the billiard flow Y preserves the normalized Lebesguemeasure µ on Q × S . In view of applying Theorem 3.1, observe first thatlim ε → P ( B ε ∪ B ε ) = 0 and E [ ε − τ Ω \ B ε ] ≤ r P ( B ε ), thus ( ετ Ω \ B ε ) ε con-verges in probability P to 0.As a direct consequence of Theorem A.1, the family of spatio-temporalprocesses ( N ε ) ε> given by (7), with h ε = ( d r + d r ) πεArea ( Q ) , converges in distri-bution to a PPP of intensity λ × B ( d r d r + d r ) and so the first conclusionsof Theorem 3.1 holds true with p = d r d r + d r . This ends the proof of theconvergence ( M ε ) ε .Due to Theorem 6.2, ( L ε ) ε with a ε = ε and h ε as previously converges in dis-tribution to a PPP with intensity λ × P j =0 p j ( δ j × m ′ j ) where p j := d j r j d r + d r and where m ′ j has density y y r j q r j − y [0 , r j ] ( y ). Thus the last conclu-sion of Theorem 3.1 holds also true with these notations. We conclude bytaking Y i = X i / (2 r ). (cid:3) Number of high records
We define the high records point process by R f ( u, ℓ ) = ∞ X k =1 δ ku { f ◦ T k > max( ℓ,f,...,f ◦ T k − ) } . The successive times of records of an observable along an orbit are obviouslytractable from the time and values of the observations along this orbit. Thefollowing proposition states that this is still the case for the correspondingasymptotic distributions. This has already been noticed in [11], in particularin the context of Extremal events. Our result is similar to the proof of [11,Theorem 3.1] from [11, Theorem 5.1].
Proposition 4.1.
Let (Ω , F , µ, T ) be a probability preserving dynamical sys-tem and f : Ω → [0 , + ∞ ) be a measurable function. Assume the family PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 7 (cid:0) N ε = N ( T, { f > ε − } , h ε , / ( εf )) (cid:1) ε> of point processes on [0 , + ∞ ) × [0 , converges in distribution with respect to P to a Poisson point process ofintensity λ × m with m a probability measure on [0 , without any atom.Then (cid:0) R f ( h ε , ε − ) (cid:1) ε> converges in distribution, as ε → to a Point pro-cess R = P ∞ ℓ =1 Z ℓ δ T ℓ where T ℓ = P ℓi =1 X i , the X i are independent standardexponential random variable and the Z ℓ are independent random variable ofBernoulli distribution with respective parameters ℓ − , and the two sequencesare independent.Proof. Define the mapping F : ξ = X i δ ( t i ,v i ) ∈ M p ([0 , ∞ ) × [0 , X i ∈ I ( ξ ) δ t i , where I ( ξ ) are the records of ξ , defined by those i such that for any j onehas t j < t i = ⇒ v j > v i . The map F is continuous at each ξ such that the t i ’s, and the v i ’s, are distincts. This is the case for a.e. realization ξ of aPoisson process of intensity λ × m . Therefore by the continuous mappingtheorem R f ( h ε , ε − ) = F ( N ε ) converges to χ = F ( P P P ( λ × m )).We are left to compute the distribution. Observe that P P P ( λ × m )is distributed as P ∞ ℓ =1 δ ( T ℓ ,W ℓ ) with ( T ℓ ) as in the statement and the W ℓ are i.i.d. with distribution m , the two sequences being independent. Let Z ℓ = 1 { W ℓ is a record } . By [23, Proposition 4.3] the Z ℓ are independent, haveprobability 1 /ℓ , and when Z ℓ = 1 we keep the point T ℓ . (cid:3) In particular, for every t > ε − before the time th − ε corresponds to R f ( h ε , ε − )([0 , t ]) and the conclu-sion of Proposition 4.1 implies that it converges to P N t ℓ =1 Z ℓ where Z ℓ are asin Proposition 4.1 and where ( N s ) s is a standard Poisson Process indepen-dent of ( Z ℓ ) ℓ . Example 4.2.
Consider a dispersive billiard with corner and cusps of max-imal order β ∗ > as in [13] . Consider the induced system (Ω , µ, T ) cor-responding to the successive reflection times outside a neighbourhood U ofcusps and write R ( x ) for the number of reflections in U starting from x . Set α = β ∗ β ∗ − ∈ (1 , .Setting A ε := { R ◦ T − > ε − } , it has been proved in [13, Lemma 4.5] thatthere exists an explicit c > such that µ ( A ε ) ∼ c ε α as ε → .The assumptions of Proposition 4.1 hold true with f = R ◦ T − and h ε = µ ( A ε ) ∼ c ε α . So the same assumptions hold true with h ε = c ε α .Furthermore the number R n of records of R higher than n /α before the n -th reflection outside cusps converges to P Nℓ =1 Z ℓ where Z ℓ are as in Propo-sition 4.1 and where N is a Poisson random variable of parameter c andindependent of ( Z ℓ ) ℓ . FRANC¸ OISE P`ENE AND BENOˆIT SAUSSOL
Proof.
It follows from the proof of [13, Lemma 4.8] that the family of pointprocesses ( N ( T, A ε , µ ( A ε ) , εR ◦ T − )) ε on [0 , + ∞ ) × [1 , + ∞ ] converges indistribution to a PPP with intensity of density ( t, y ) αy − α − y> withrespect to the Lebesgue measure.Therefore the assumptions of Proposition 4.1 hold true with f = R ◦ T − and h ε = µ ( A ε ) ∼ c ε α . So the same assumptions hold true with h ε = c ε α .This ends the proof of the first part.For the second we apply Proposition 4.1 with ε = n − α . (cid:3) Line process of random geodesics
We study the line process generated by a geodesic as in [2] and recovertheir main result. Let N be a compact Riemannian surface of negativecurvature. The geodesic flow ( Y t ) t on the unit tangent bundle Ω = T N preserves the Liouville measure µ . Let π N : T N → N be the canonicalprojection ( q, v ) q . We denote by D ( q, ε ) the ball in N of radius ε . Wenow state the main theorem, postponing the details and precise definitionsthereafter. Theorem 5.1.
Fix q ∈ N . For any a > , the intersection of the neighbor-hood D ( q , ε ) with the geodesic segment π N ( { Y t ( x ) , ≤ t ≤ aε − } ) , where x is taken at random on (Ω , µ ) , converges in distribution, after normalization,as ε → , to a Homogeneous Poisson line process in the unit disk of intensity a/Area ( N ) . A Poisson line process in the unit disk D of the plane, of intensity κ ∈ (0 , ∞ ), is a probabilistic process which draw lines in the disk. Each line L is parametrized by ( r, θ ) ∈ [ − , × [0 , π ] where L = { ( x, y ) ∈ D : r = x cos θ + y sin θ } , and the parameters ( r, θ ) are produced by a Poisson point process of intensity κπ drdθ on [ − , × [0 , π ]. Equivalently, changing the parametrization to O • • θrs ϕ L Figure 1.
Parametrization of the line L by ( r, θ ) or ( s, ϕ ).( s, ϕ ) where s ∈ ∂D =: S is one point of intersection of the line with theunit circle and ϕ is the angle between the line L (directed into the disk) [13, Lemma 4.8] states that this convergence is true in the set of point processes on[0 , + ∞ ) × [1 , + ∞ ), but its proof can be adapted in a straighforward way to obtain ourpurpose by considering not only intervals of the form ( c, c ′ ) but also intervals of the form( c, + ∞ ]. PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 9 and the normal at s pointing inside the disk (see Figure 1), gives a Poissonpoint process of intensity κ cos ϕ π dsdϕ (the jacobian is cos ϕ and each line hastwo representations in this parametrization). The intensity κ in the theoremis equal to a/Area ( N ), therefore the intensity in this parametrization willbe a πArea ( N ) cos ϕdsdϕ = aV ol ( T N ) cos ϕdsdϕ . The convergence of a pointprocess in this parametrization implies it in the original one (by continuityof the change of parameter; see [23, Proposition 3.18]).The exponential map exp q is a local diffeomorphism on a neighborhood U ⊂ T q N of 0. Thus its inverse is well defined on D ( q , ε ) for ε small enoughso that B (0 , ε ) ⊂ U . We identify T q N with R . Set V = S × [ − π , π ]. For q ∈ D ( q , ε ) we let s ε ( q ) = ε − exp − q ( q ) and for q ∈ ∂D ( q , ε ) and v ∈ T q N we denote by φ q ( v ) the angle between the normal at q pointing inside thedisk and v (see Figure 2). q • •• qφ q ( v ) v γ Figure 2.
A geodesic arc γ entering the ball D ( q , ε ).The intersection I aε ( x ) := π N ( Y [0 ,aε − ] ( x )) ∩ D ( q , ε ) consists of finitelymany geodesic arcs γ i := π N ( Y [ t i ,t i + ℓ i ] ( x )), where ℓ i is the length of thearc; we drop the dependence on x and ε for simplicity. The arcs γ i arefully crossing the ball, except possibly for the two extremities (at t = 0 or t = aε − ) which could give an incomplete arc. The later happens with avanishing probability as ε →
0, therefore we will ignore this eventuality. Thearc γ i enters the ball at the position q i with direction v i where ( q i , v i ) := Y t i ( x ).When ε →
0, the geodesic arcs γ i which compose the intersection I aε be-come more and more straight. This justifies the definition of the convergencein distribution of I aε as the convergence in distribution of the point process X i δ ( s ε ( q i ) ,φ q ( v i )) . (11)Loosely speaking, we identify the images s ε ( γ i ) with the chord of the unitdisk D originated in s ε ( q i ) and direction v i .We now proceed with the proof of the theorem. Let A ε ⊂ T N be theset of points ( q, v ) such that q ∈ ∂D ( q , ε ) and v is pointing inside the ball.We define on A ε H ε ( q, v ) = ( s ε ( q ) , φ q ( v )) ∈ V. (12)The theorem is a byproduct of the following result for the geodesic flow. Proposition 5.2.
The process of entrances in the ball for the position forthe geodesic flow N ( Y, A ε , ε/Area ( N ) , H ε ) on [0 , + ∞ [ × V converges to aPoisson point process with intensity π cos ϕdtdsdϕ .Proof of Theorem 5.1. The counting process L aε ( · ) := N ( Y, A ε , ε/Area ( N ) , H ε )([0 , a/Area ( N )] × · ) (13)produces a point ( s, ϕ ) each time that the geodesic flow Y t enters in D ( q , ε )for some t such that 2 εt/Area ( N ) ≤ a/Area ( N ), that is t ≤ aε − . ByProposition 5.2 and the continuous mapping theorem the point process L aε converges to a Poisson point process of intensity aArea ( N ) 14 π cos ϕdsdϕ . Bythe above discussion, in particular (11), this completes the proof of thetheorem. (cid:3) We emphasize that this proof only uses the convergence stated in Propo-sition 5.2, therefore it applies for more general ’geodesic-like flows’, for in-stance the argument applies immediately to billiards systems, using Theo-rem A.1 in place of Proposition 5.2.
Proof of Proposition 5.2.
The first step is to construct a Markov section forthe geodesic flow, subordinated to a finite family of disks D i ⊂ T N . Fixsome δ > X i ) i of size δ , in particular diam X i < δ and T N = ∪ i Y [ − δ, ( X i ). One canchoose the disks D i ⊃ X i in such a way that D i ⊂ { ( q, v ) : q ∈ Q i , | ∠ ( n q , v ) | > π − δ } where Q i are C curve in N and n q is the normal vector to Q i at q (with q n q continuous). Without loss of generality we assume that q
6∈ ∪ i Q i .The flow ( Y t ) is represented by a special flow over the Poincar´e section M := ∪ i X i , with a C roof function τ . Let Π be the projection onto M along the flow in backward time. The flow ( T N, ( Y t ) , µ ) projects downto a system ( M, F, ν ), conjugated to a subshift of finite type with a Gibbsmeasure of a H¨older potential. In order to apply Theorem 2.2 we need tocheck that the set A ε := Π A ε and H ε ( x ) := H ε ( Y s ( x )) where s > Y s ( x ) ∈ A ε fulfills the hypotheses of Theorem 2.1.For that we will apply [21, Proposition 3.2]. The Poincar´e map F has ahyperbolic structure with an exponential rate, thus it satisfies the setting of[21, Proposition 3.2] with any polynomial rate α , in particular α = 4 works.Here the boundary is meant in the induced topology on M . It suffices toprove that for some p ε = o ( ν ( A ε )) one has (i) ν ( τ A ε ≤ p ε ) = o (1) and (ii) ν (( ∂A ε ) [ p − αε ] ) = o ( ν ( A ε )), the two other assumptions being trivially satisfiedin our situation.Measure of A ε : The Liouville measure µ is the product of the normalizedsurface on N times the Haar measure on T N . Its projection ν to thePoincar´e section satisfies dν = c ν cos ϕdrdϕ for some normalizing constant c ν = (cid:16)P i R X i cos ϕdrdϕ (cid:17) − , where r is the curvilinear abscissa on Q i and ϕ the angle between the velocity and the normal to Q i . Moreover we have dµ = ( R M τ dν ) − dν × dt | M τ where M τ = { ( x, t ) : x ∈ M, ≤ t < τ ( x ) } . PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 11
The geodesic flow preserves the measure cos ϕdrdϕ from A ε ⊂ M to A ε ,therefore ν ( A ε ) = c ν Z A ε cos ϕdrdϕ = c ν Z A ε cos ϕdrdϕ = c ν Z ∂D ( q ,ε ) dr Z π/ − π/ cos ϕdϕ ∼ c ν πε. Short returns: For any q ∈ D ( q , ε ), let R ε ( q ) be the set of v ∈ T q N suchthat the geodesic segment γ [0 ,ε − / ] ( q, v ) enters again D ( q , ε ) after leaving D ( q , ε ). The result of [2, Lemma 5.3] ensures the existence of K > q ∈ D ( q , ε ) Leb ( R ε ( q )) ≤ Kε − / = K √ ε. Therefore, setting ˆ A ε = { ( q, v ) ∈ A ε : v ∈ R ε ( q ) } we get that the bidimen-sional Lebesgue measure of ˆ A ε is O ( ε / ). A fortiori since the projection Πpreserves the measure cos ϕdrdϕ we get ν (Π ˆ A ε ) = c ν Z Π ˆ A ε cos ϕdrdϕ = c ν Z ˆ A ε cos ϕdrdϕ = O ( ε / ) . Let p ε = ⌊ (max τ ) − ε − / ⌋ and notice that A ε ∩ { τ A ε ≤ p ε } ⊂ Π ˆ A ε . By theprevious estimates we get ν ( A ε ∩ { τ A ε ≤ p ε } ) = O ( ε / ) . Hence ν ( τ A ε ≤ p ε | A ε ) = o (1) . This is the assumption (i).We now prove (ii). The boundary of A ε in the induced topology of M isincluded in the set of Π( q, v ) where v is tangent to the boundary of ∂D ( q , ε ).This defines for each i such that X i ∩ A ε is nonempty at most two C curvesin D i of finite length (by transversality), therefore its ε -neighborhood hasa measure O ( ε ).Finally, the measure dm ε = ( H ε ) ∗ ν ( ·| A ε ) is equal to the measure dm := π cos ϕdsdϕ , since the measure cos ϕdrdϕ is preserved by the inverse of theprojection Π from A ε to A ε and H ε has constant jacobian ε in these coor-dinates. By Theorem 2.1 the point process N ( F, A ε , ν ( A ε ) , H ε ) convergesto a Poisson point process of intensity λ × m . Applying Theorem 2.2 with h ε = c ν πε and h ′ ε = h ε /E ν ( τ ) we get that N ( Y, A ε , h ′ ε , H ε ) converges to aPoisson point process of intensity λ × m . In addition, Z M τ dν = c ν Z M τ cos ϕdrdϕ = c ν Z M τ cos ϕdtdrdϕ = c ν V ol ( T N ) . Thus, since
V ol ( T N ) = 2 πArea ( N ) we get that h ′ ε = εArea ( N ) , proving theproposition. (cid:3) Time spent by a flow in a small set
Given a flow Y = ( Y t ) t defined on Ω and a set A ⊂ Ω, a very naturalquestion is to study the time spent by the flow in the set A , that is the localtime L T ( A ) given by following quantity : L T ( A ) := λ ( { t ∈ [0 , T ] : Y t ∈ A } ) . This quantity measures the time spent by the flow Y in the set A betweentime 0 and time T (the symbol L refers to the local time). We also write D A := inf { t > Y t A } for the duration of the present visit to the set A . Proposition 6.1.
Let J ≥ and Y = ( Y t ) t ≥ be a flow defined on (Ω , F , P ) .Assume that ( N ε = N ( Y, A ε , h ε , H ε )) ε> converges in distribution (with re-spect to P ) to a PPP N of intensity λ × m with H ε ( A ε ) ⊂ V = { , ..., J }× W where m = P Jj =1 ( p j δ j × m j ) , with P Jj =1 p j = 1 and where m j are prob-ability measure on some separable metric space W . Suppose in additionthat, for some a ε and each x entering in A ε , a ε D A ε ( x ) = D ε ( H ε ( x )) with lim ε → D ε ( j, w ) =: D j ( w ) uniformly in w ∈ W , where D j : W → ¯ R + iscontinuous.Then L ε = X t : Y t ( x ) enters A ε δ th ε ,H (1) ε ( Y t ( x )) ,a ε D Aε ◦ Y t ( x ) ε> converges in distribution with respect to P to a PPP L on [0 , + ∞ ) ×{ , ..., J }× ¯ R + with intensity λ × P Jj =1 ( p j δ j × ( D j ) ∗ ( m j )) .If moreover a ε D A ε P → , then, for every T > , (( a ε L ( j ) ⌊ t/h ε ⌋ ( A ε )) t ∈ [0 ,T ] ,j =1 ,...,J ) ε> converges in distribution to (cid:18)P N ( j ) t k =1 X ( j ) k (cid:19) t ∈ [0 ,T ] ,j =1 ,...,J as ε → , where ( N ( j ) t ) t> are independent Poisson process with parameter p j and where ( X ( j ) k ) k ≥ are independent sequences of independent identically distributedrandom variables with distribution P Jj =1 p j ( D j ) ∗ ( m j ) independent of ( N ( j ) t ) t> Proof.
Observe that, for every ǫ ≥ L ε = ( ψ ε ) ∗ ( N ε ) with ψ ε : ( t, j, w ) ( t, j, D ε ( j, w )) if ε > ψ : ( t, j, w ) ( t, j, D j ( w )). Using [23,Proposition 3.13] we prove the first statement.Assume now that a ε D A ε P →
0. Then a ε L ( j ) t/h ε = a ε min( t/h ε , D A ε ) + Z [0 ,t ] ×{ j }× ¯ R + z d L ε ( s, i, z ) ! t ∈ [0 ,T ] ,j =1 ,...,J which converges to (cid:16) a ε L ( j ) t/h ε = R [0 ,t ] ×{ j }× ¯ R + z d L ( s, i, z ) (cid:17) t ∈ [0 ,T ] ,j =1 ,...,J . (cid:3) We apply the previous result to the dispersive billiard flow in a Sinai bil-liard with finite horizon.
Theorem 6.2 (Time spent by the billiard flow in a shrinking ball for theposition) . Consider the billiard flow associated to a Sinai billiard with finitehorizon in a domain Q ⊂ T (see Appendix for details). Recall that thisflow preserves the normalized Lebesgue measure on Q × S . Let J be apositive integer. Let q , ..., q J ∈ Q be a J pairwise distinct fixed position inthe billiard domain and r , ..., r J be J positive real numbers. We set d j = 2 PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 13 if q j ∂Q and d j = 1 if q j ∈ ∂Q and d := P Jj =1 d j r j and also L ε = J X j =1 X t : Y t ( x ) enters B ( q j ,r j ε ) × S δ dπεtArea ( Q ) ,j,ε − D B ( qj,rjε ) × S ◦ Y t ( x ) ε> and L ( j ) t/ε := Z tε { Y s ( · ) ∈ B ( q j ,r j ε ) × S } ds . Then, ( L ε ) ε> converges strongly in distribution to a PPP L with intensity λ × P Jj =1 d j r j d ( δ j × m ′ j ) where m ′ j is the distribution of r j X with X a randomvariable of density y y . arccos ′ ( y ) [0 , ( y ) = y √ − y [0 , ( y ) .Moreover, for every T > , (( ε − L ( j ) t/ε ) t ∈ [0 ,T ] ,j =1 ,...,J ) ε> converges stronglyin distribution to (cid:18) r j P N ( j ) t k =1 X ( j ) k (cid:19) t ∈ [0 ,T ] ,j =1 ,...,J as ε → , where ( N ( j ) t ) t> are independent Poisson process with parameter d j Area ( Q ) d π , where ( X k ) k ≥ is a sequence of independent identically distributed random variables withdensity x x √ − x [0 , ( x ) independent of ( N t ) t> .Proof of Theorem 6.2. Due to Theorem A.1, we know that the family ofprocesses J X j =1 X t : ( Y s ( y )) s enters B ( q j ,ε ) × S at time t δ (cid:18) dπεtArea ( Q ) , Π Q ( Yt ( y )) − qjε , Π V ( Y t ( y )) (cid:19) converges in distribution (when y is distributed with respect to any prob-ability measure absolutely continuous with respect to the Lebesgue mea-sure on M ) as ε → λ × ˜ m where ˜ m is the probability measure on { , ..., J } × S × S with density( j, p, ~u ) P Jj =1 r j d j d d j π h ( − p ) , ~u i + {h p,~n qj i≥ } with d := P Jj =1 d j r j .We will apply Proposition 6.1 with A ε := S Jj =1 B ( q j , εr j ) × S and H ε ( q, ~v ) = (cid:16) j, −→ q j qr j ε , ~v (cid:17) if q ∈ ∂B ( q j , r j ε ).Let x = ( q, ~v ) entering in B ( q j , ε ) × S . If the billiard flow crosses B ( q J , ε ) × S before any collision off ∂Q , then ε − D B ( q j ,r j ε ) × S ( q, ~v ) = 2 ε − \ ( −→ qq j , ~v ) = D ( H ε ( x )) , with D ( j, p, ~u ) = 2 r j \ ( − p, ~u ). This is always the case if q j ∂Q . But, if q j ∈ ∂Q , it can also happen that the billiard flow collides ∂Q at a point q ′ ∈ B ( q j , ε ) before exiting B ( q j , ε ) × S . Then the point q ′ is at distancein O ( ε ) of the tangent line to ∂Q at q j , and the tangent line of ∂Q at q ′ makes an angle in O ( ε ) with the tangent line of ∂Q at q j . In this case ε − D B ( q j ,r j ε ) × S ( q, ~v ) = 2 ε − \ ( −→ qq j , ~v ) + O ( ε ) = D ( H ε ( x )) + O ( ε ) , uniformly in x = ( q, ~v ) and ε . In any case, we set a ε = ε − and D ε = D + O ( ε ).Applying now Proposition 6.1, we infer that ( L ε ) ε> converges strongly in distribution to a PPP L with intensity λ × P Jj =1 d j r j P Jj ′ =1 d j ′ r j ′ ( δ j × ( D j ) ∗ ( m j )),with D j ( p, ~u ) = 2 r j \ ( − p, ~u ) and m j the probability measure on S × S withdensity ( p, ~u ) d j π h ( − p ) , ~u i + {h p,~n qj i≥ } . It remains to identify the distribution ( D j ) ∗ ( m j ). By the transfer formula,we obtain Z ∞ h ( D j ( p, ~u )) dm j ( p, ~u ) = 12 d j π Z S × S h ( r j h− p, ~u i ) h ( − p ) , ~u i + {h p,~n qj i≥ } dp d~u = 12 Z π − π h (2 r j cos ϕ ) cos ϕ dϕ = Z π h (2 r j cos ϕ ) cos ϕ dϕ = Z h ( r j y )(arccos( · / ′ ( y ) y dy . Thus we have proved that the probability distribution ( D j ) ∗ ˜ m j is the distri-bution of r j X with X a random variable of density y y . arccos ′ ( y ) [0 , ( y ) = y √ − y [0 , ( y ).We can apply the last point of Proposition 6.1 since ε − D A ε ≤ j r j A ε P → P absolutely continuous with respect to theLebesgue measure on Q × S . (cid:3) Appendix A. Visits of the Sinai billiard flow to a finite unionof balls for the position
In this appendix we are interested in spatio temporal processes for theSinai billiard flow with finite horizon.Let us start by recalling the model and introducing notations. We considera finite family { O i , i = 1 , ..., I } of convex open sets of the two-dimensionaltorus T = R / Z . We consider the billiard domain Q = T \ S Ii =1 O i and callthe O i obstacles. We assume that these obstacles have C -smooth boundarywith non null curvature and that their closures are pairwise disjoint. Weconsider a point particle moving in Q in the following way: the point particlegoes straight at unit speed in Q and obeys to the classical Descartes reflexionlaw when it collides an obstacle. We then define the billiard flow ( Y t ) t ∈ R as follows. Y t ( q, ~v ) = ( q t , v t ) is the couple position-velocity of the pointparticle at time t if the particle has position q and velocity ~v at time 0.To avoid any confusion, we consider that the billiard flow is defined on thequotient ( Q × S ) / R , with R is the equivalence relation corresponding tothe identification of pre-collisional and post-collisional vectors at a reflectiontime: ( q, ~v ) R ( q ′ , ~v ′ ) ⇔ ( q, ~v ) = ( q ′ , ~v ′ ) or ~v ′ = ~v − h ~n q , ~v i ~n q , where ~n q is the unit normal vector to ∂Q at q directed inward Q if q ∈ ∂Q ,with convention ~n q = 0 if q ∂Q . This flow preserves the normalized PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 15
Lebesgue measure µ on Q × S .We assume moreover that every billiard trajectory meets ∂Q (finite horizonassumption).Let us write Π Q : Q × S → Q and Π V : Q × S → S for the canonicalprojections given respectively by Π Q ( q, ~v ) = q and Π V ( q, ~v ) = ~v . Theorem A.1 (Visits of the billiard flow to a finite union of shrinkingballs for the position) . Let q , ..., q J ∈ Q be pairwise distinct positions inthe billiard domain and r , ..., r j be positive real numbers. We set d j = 2 if q j ∂Q and d j = 1 if q j ∈ ∂Q and d = P Jj =1 d j r j .Then, the family of processes J X j =1 X t : ( Y s ( y )) s enters B ( q j ,εr j ) × S at time t δ (cid:18) dπεtArea ( Q ) ,j, Π Q ( Yt ( y )) − qjrjε , Π V ( Y t ( y )) (cid:19) converges in distribution (when y is distributed with respect to any proba-bility measure absolutely continuous with respect to the Lebesgue measureon M ) as ε → to a Poisson Point Process with intensity λ × ˜ m where ˜ m is the probability measure on V := { , ..., J } × S × S with density ( j, p, ~u ) r j dπ h ( − p ) , ~u i + {h p,~n qj i≥ } . Observe that if q j ∈ ∂Q , the set of p ∈ S satisfying h p, ~n q j i ≥ S when q j is in the interior of Q .This result has already been proved in [21, Theorem 4.4] for J = 1 andLebesgue-almost every position q . The extension to a finite number ofpoints is relatively easy. The most difficult part is to treat all the possiblespositions in the billiard domain.Along the paper we provided various applications of this theorem to dif-ferent questions. We present here a result on the closest approaches to agiven point in the billiard table by the orbit of the billiard flow. Example A.2.
Consider the billiard flow associated to a Sinai billiard withfinite horizon in a domain Q ⊂ T . Consider a fixed position q ∈ Q . Set d = 2 − q ∈ ∂Q . During each visit of the flow to B ( q , ε ) , the closest distanceto q is given by L ( q, ~v ) := ε | sin ∠ ( −→ qq , ~v ) | where ( q, ~v ) is the entry point.Then the family of closest approach point process (cid:0) C ε := N ( Y, B ( q , ε ) × S , dε/Area ( Q ) , ε − L (cid:1) ε> on [0 , + ∞ ) × [0 , converges in distribution (with respect to any probabilitymeasure absolutely continuous with respect to the Lebesgue measure on Q × S ) to a PPP with intensity 1.Proof. Due to Theorem A.1, the family of spatio-temporal processes( N ε := N ( Y, B ( q , ε ) × S , dε/Area ( Q ) , H ε ) ε> with H ε ( q, ~v ) = ( ε − −→ q q, ~v ) converges in distribution (with respect to anyprobability measure absolutely continuous with respect to Lebesgue on Q × S ) to a PPP of intensity λ × ˜ m where ˜ m is the probability measure on S × S with density ( p, ~u ) dπ h ( − p ) , ~u i + {h p,~n q i≥ } (where ~n q is the unit normal vector to ∂Q at q directed inward Q if q ∈ ∂Q , ~n q = 0 otherwise).Observe that C ε = ˜ G ( N ε ) , with ˜ G ( t, p, ~u ) = ( t, G ( p, ~u )) where G ( p, ~u ) = ( t, | sin ∠ ( − p, ~u ) | ). Thus ( C ε ) ε converges strongly in distribution to the PPP with intensity λ × G ∗ ( ˜ m ) andit remains to identify ˜ m = G ∗ ( ˜ m ). Due to the transfer formula, we obtain Z ∞ h ( G ( p, ~u )) d ˜ m ( p, ~u ) = 12 dπ Z S × S h ( | sin ∠ ( − p, ~u ) | )(cos ∠ ( − p, ~u )) + {h p,~n qj i≥ } dp d~u = 12 Z π − π h ( | sin ϕ | ) cos ϕ dϕ = Z π h (sin ϕ ) cos ϕ dϕ = Z h ( y ) dy . (cid:3) Proof of Theorem A.1.
Due to [28, Theorem 1], it is enough to prove theresult for the convergence in distribution with respect to µ . Assume ε > min j = j ′ q j q j ′ . We use the representation of the billiard flow as a special flowover the discrete time billiard system ( M, ν, F ) corresponding to collisiontimes and with τ the length of the free flight before next collision.Set e A ε = S Jj =1 e A ( j ) ε , where e A ( j ) ε is the set of the configuration entering in A ( j ) ε := ( Q ∩ B ( q j , ε )) × S , i.e. e A ( j ) ε is the set of ( q, ~v ) ∈ ( Q ∩ ∂B ( q j , ε ]) × S s.t. h ~qq , ~v i >
0. Set also A ε := S Jj =1 A ( j ) ε .Set h ′ ε := dπε/Area ( Q ) and H ε ( q ′ , ~v ) = ( j, −−→ q j q ′ r j ε , ~v ) if q ′ ∈ ∂B ( q j , r j ε ). Here M is the set of reflected unit vectors based on ∂Q , ν is the probability mea-sure with density proportional to ( q, ~v )
7→ h ~n ( q ) , ~v i , where ~n ( q ) is the unitvector normal to ∂Q at q directed towards Q and F : M → M is the trans-formation mapping a configuration at a collision time to the configurationcorresponding to the next collision time.The normalizing function G ε is given by G ε ( x ) = H ε ( Y τ ( Y ) e Aε ( x ) ( x )) with τ ( Y ) e A ε ( y ) := inf { t > Y t ( y ) ∈ e A ε } .As in the setting of Theorem 2.2, we write Π for the projection on M , that isΠ( q ′ , ~v ) = ( q, ~v ) is the post-collisional vector at the previous collision time.We take here h ε := ν (Π( e A ε )).As for [21, Theorem 4.4], we will apply [21, Proposition 3.2] after checkingits assumptions. We define e A ( j ) ε := { ( q, ~v ) ∈ ∂B ( q j , ε ) × S : h−→ qq j , ~v i ≥ } .(i) Measure of the set. We have to adapt slightly the first item of theproof of [21, Theorem 4.4] which deals with the asymptotic be-haviour of ν ( B ε ) with B ε := Π( e A ε ). Observe that B ε = S Jj =1 B ( j ) ε with B ( j ) ε := Π( e A ( j ) ε ), i.e. B ( j ) ε is the set of configurations ( q, ~v ) ∈ M such that the billiard trajectory ( Y t ( q )) t ≥ will enter B ( q j , εr j ) be-fore touching ∂Q . As seen in [19, Lemma 5.1],if q j ∈ Q \ ∂Q, ν ( B ( j ) ε ) = | Q ∩ ∂B ( q j , r j ε ) || ∂Q | = 2 πr j ε | ∂Q | . PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 17
With exactly the same proof, we obtain thatif q j ∈ ∂Q, ν ( B ( j ) ε ) = | Q ∩ ∂B ( q j , r j ε ) || ∂Q | ∼ πr j ε | ∂Q | . Moreover, for every distinct j, j ′ , B ( j ) ε ∩ B ( j ′ ) ε is contained in Π( B ( x j,j ′ , K j,j ′ ε ) ∪ B ( x j ′ ,j , K j,j ′ ε )) where x j,j ′ = (cid:16) q j , −−→ q j q ′ j q j q ′ j (cid:17) and K j,j ′ = max (cid:16) , q j q j ′ (cid:17) .So, due to [19, Lemma 5.1], ν ( B ( j ) ε ∩ B ( j ′ ) ε ) = O ( ε ) = o ( ε ). Hencewe conclude that ν ( B ε ) ∼ J X j =1 ν ( B ( j ) ε ) ∼ dπε | ∂Q | , as ε → N ( Y, A ε , h ′ ε , H ε ) = J X j =1 N ( Y, A ( j ) ε , h ′ ε , H ε ) ≥ N ( Y, A ′ ε , h ′ ε , H ε ) , where A ′ ε = S Jj =1 Π − (Π( e A ( j ) ε )) \ S j ′ = j Π − (Π( e A ( j ′ ) ε )) and that, forall T > E µ (cid:2)(cid:0) N ( Y, A ε , h ′ ε , H ε ) − N ( Y, A ′ ε , h ′ ε , G ε ) (cid:1) ([0 , T ] × V ) (cid:3) ≤ T max τ h ε (min τ ) X j,j ′ : j = j ′ ν (cid:16) e A ( j ) ε ∩ e A ( j ′ ) ε (cid:17) = o (1) , where we used the representation of Y as a special flow over ( M, ν, F )due to the fact, proved in the previous item, that for any distinctlabels j, j ′ , ν (cid:16) e A ( j ) ε ∩ e A ( j ′ ) ε (cid:17) = o ( ε ). Thus it is enough to prove theconvergence in distribution of N ( Y, A ′ ε , h ′ ε , H ε ) with respect to µ .(iii) The same argument ensures that, with respect to ν , the conver-gence in distribution of N ( F, B ε , h ε , G ε ) to P is equivalent to theconvergence in distribution of N ( F, B ′ ε , h ε , G ε ), with B ′ ε := Π( A ′ ε ).(iv) Note that ν (( ∂B ε ) [ ε δ ] ) = o ( ν ( B ε )), for every δ > σ > ν ( τ B ε ≤ ε − σ | B ε ) = o (1), where τ B is here the first time k ≥ F k ( · ) ∈ B .(vi) Now let us prove that ( ν ( G − ε ( · ) | B ε )) ε> converges to ˜ m as ε → µ on { , ..., J } × S × S with density( j, p, ~u ) r j h ( − p ) , ~u i + .Observe first that ˜ m = ˜ µ ( ·| A ) with A := S Jj =1 A ( j ) and A ( j ) := (cid:8) ( p, ~u ) ∈ S × S : h ( − p ) , ~u i ≥ , h p, ~n q j i ≥ (cid:9) and second that ν ( G − ε ( · ) | B ε ) = e µ ( ·| G ε ( B ε )). But˜ µ ( A \ G ε ( B ε )) ≤ J X j =1 ˜ µ H ε Y τ ( Y ) e A ( j ) ε ( · ) [ j ′ = j ( B ( j ) ε ∩ B ( j ′ ) ε ) ≤ J X j =1 τ | ∂Q | r j εν [ j = j ′ ( B ( j ) ε ∩ B ( j ′ ) ε ) = o ( ν ( B ε ))and G ε ( B ε ) \ A corresponds to points ( p, ~u ) ∈ S × S with q j ∈ ∂Q with 0 < h p, ~u i ≤ O ( ε ), thus˜ µ ( G ε ( B ε ) \ A ) = O ( ε ) . This ends the proof of the convergence in distribution of the familyof measures ( ν ( G − ε ( · ) | B ε )) ε> to ˜ m as ε → W we use [21, Proposition 3.4].Thus, due to [21, Proposition 3.2], we conclude the convergence of dis-tribution with respect to ν of ( N ( F, B ε , h ε , G ε )) ε> and so, due to (ii),of ( N ( F, B ′ ε , h ε , G ε )) ε> to a PPP P with intensity λ × ˜ m . Applyingnow Theorem 2.2, we deduce the strong convergence in distribution of( N ( F, A ′ ε , h ε / E ν [ τ ] , H ε )) ε> to P and so, due to (iii), the convergence indistribution with respect to µ of ( N ( F, A ε , h ε / E ν [ τ ] , H ε )) ε> to P . Now weconclude by [28, Theorem 1] and by noticing that h ε E ν [ τ ] = dπε | ∂Q | E ν [ τ ] = dπεArea ( Q ) = h ′ ε . (cid:3) Lemma A.3. ∀ σ ∈ (0 , , ν ( τ B ε ≤ ε − σ | B ε ) = o (1) (14) Proof.
This point corresponds to the the second item of the proof of [21,Theorem 4.4], which for Lebesgue-almost every point came from [19, Lemma6.4]. To prove (14), we write ν ( τ B ε ≤ ε − σ | B ε ) ≤ ⌊ ε − σ ⌋ X k =1 ν ( F − n ( B ε ) | B ε ) . (15)Thus our goal to bound ν ( F − n ( B ε ) | B ε ).Step 1: Useful notations.We parametrize M by S Ii =1 { i } × ( R / | ∂O i | Z ) × (cid:2) − π ; π (cid:3) . A reflected vector( q, ~v ) ∈ M is represented by ( i, r, ϕ ) if q ∈ ∂ Γ i as curvilinear absciss r ∂O i and if ϕ is the angular measure in [ − π/ , π/
2] of ( ~n ( q ) , ~v ) where ~n ( q ) is thenormal vector to ∂Q at q .For any C -curve γ in M , we write ℓ ( γ ) for the euclidean length in the ( r, ϕ )coordinates of γ . If moreover γ is given in coordinates by ϕ = φ ( r ), thenwe also write p ( γ ) := R γ cos( φ ( r )) dr . We define the time until the nextreflection in the future by τ ( q, ~v ) := min { s > q + s~v ∈ ∂Q } . PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 19
It will be useful to define S := { ϕ = ± π/ } . Recall that, for every k ≥ F k defines a C -diffeomorphism from M \ S − k to M \ S k with S − k := S km =0 F − m ( S ) and S k := S km =0 F m ( S ).Step 2: Geometric study of B ε and of F ( B ε ).Moreover the boundary of each connected component of B ε (resp. F ( B ε ))is made with a bounded number of C curves of the following forms: • curves of S , corresponding, in ( r, ϕ )-coordinates, to { ϕ = ± π } . • C curves of F − ( S ) (resp. F ( S )), which have the form ϕ = φ ( r ) with φ a C decreasing (resp. increasing) function satisfyingmin κ ≤ | φ ′ ( r ) | ≤ max κ + τ , where κ ( q ) is the curvature of ∂Q at q ∈ ∂Q and where τ is the free flight length before the nextcollision time. • if q ∂Q : C curves, corresponding to the set of points x =( q, ~v ) ∈ M (resp. F ( x )) such that [Π Q ( x ) , Π Q ( F ( x ))] is tangentto ∂B ( q , ε ). These curves have the form ϕ = φ ε ( r ) with φ ε a de-creasing (resp. increasing) function satisfying min κ ≤ | φ ′ ε ( r ) | ≤ max κ + d ( q ,∂Q ) − ε ≤ max κ + τ ), with τ := d ( q , ∂Q ) as soon as ε < τ . • if q ∈ ∂Q : C curves, corresponding to the set of points x =( q, ~v ) ∈ M (resp. F ( x )) such that [Π Q ( x ) , Π Q ( F ( x ))] is tangentto ∂B ( q , ε ) or such that Π Q ( F ( x )) is an extremity of B ( q , ε ) ∩ Q and [Π Q ( x ) , Π Q ( F ( x ))] contains no other point of B ( q , ε ). Thesecurves have the form ϕ = φ ε ( r ) with φ ε a decreasing (resp. increas-ing) function satisfying min κ ≤ | φ ′ ε ( r ) | .The points x = ( q, ~v ) ∈ M , with d ( q, q ) ≪ B ( q , ε ) × S are contained in a union R ε of two rectangles of width O ( ε / ) for the position (around q ) andof width O ( ε ) for the velocity direction (around the tangent vectorsto ∂Q at q ).In B ε \ R ε (resp. F ( B ε ) \ ( R ε ∪ Π − Q ( B ( q , ε )))) we also have | φ ′ ε ( r ) | ≤ max κ + τ with τ := min τ as soon as ε < τ .We say that a curve γ of M satisfies assumption (C) if it is givenby ϕ = φ ( r ) with φ C -smooth, increasing and such that min κ ≤ φ ′ ≤ max κ + τ . We recall the following facts. • There exist C , C > λ > γ satisfyingAssumption (C) and every integer m such that γ ∩ S − m = ∅ , F m γ is a C -smooth curve satisfying assumption (C) and C p ( F m γ ) ≥ λ m p ( γ ) and ℓ ( γ ) ≤ C p p ( F γ ). • There exist C > λ > λ / such that, for every integer m ,the number of connected components of M \ S − m is less than C λ m .Moreover S − m is made of curves ϕ = φ ( r ) with φ C -smooth andstrictly decreasing. • If γ ⊂ M \ S − is given by ϕ = φ ( r ) or r = r ( ϕ ) with φ or r increasing and C smooth, then F γ is C , is given by ϕ = φ ( r )with min κ ≤ φ ′ ≤ max κ + τ . Moreover R F γ dϕ ≥ R γ dϕ . We observe that there exist K ′ > ε > ε ∈ (0 , ε ), F ( B ε ) \ R ε is made of a bounded number of connected components V ( i ) ε each of which is a strip of width at most K ′ ε of the following form in( r, ϕ )-coordinates: • { ( r, ϕ ) : r ∈ J, φ ( i )1 ( r ) ≤ ϕ ≤ φ ( i )2 ( r ) } (with J an interval) andis delimited by two continuous piecewise C curves γ j given by ϕ = φ j ( r ) satisfying assumption (C) and k φ ( i )1 − φ ( i )2 k ∞ ≤ K ′ ε . • or possibly, if q ∈ ∂Q , { ( r, ϕ ) : r ( i )1 ,ε ≤ r ≤ r ( i )2 ,ε } with | r ( i )1 ,ε − r ( i )2 ,ε | ≤ K ′ ε .In particular, with the previous notations, any connected component V ( i ) ε of F ( B ε ) \ R ε has the form S u ∈ [0 , e γ ( i ) u , where e γ ( i ) u corresponds to the graph { ψ ( i ) ( u, r ) = ( r, uφ ( i )1 ( r ) + (1 − u ) φ ( i )2 ( r )) : r ∈ J i } (or possibly { ψ ( i ) ( u, ϕ ) =( ur ( i )1 ,ε + (1 − u ) r ( i )2 ,ε , ϕ ) , ϕ ∈ J i } if q ∈ ∂Q ). Thus ∀ E ∈ B ( M ) , ν ( E ∩ F ( B ε \ R ε )) ≤ Leb ( E ∩ F ( B ε \ R ε ))2 | Q |≤ X i | ∂Q | Z J i × [0 , ψ ( i ) ( u,s ) ∈ E (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u ψ ( i ) ( u, s ) (cid:12)(cid:12)(cid:12)(cid:12) dsdu ≤ K ′ ε | ∂Q | sup [0 , ℓ ( E ∩ e γ u ) . (16)Step 3: Scarcity of very quick returns.Let us prove the existence of K > ∀ s ≥ , ∀ ε < τ , ν ( F − s − ( B ε ) | B ε ) ≤ K ( λ /λ ) s ε . (17)Let u ∈ (0 , ε ). We define γ be a connected component of e γ u ∩ F ( B ε ) ∩ F − s ( B ε ). The curve γ satisfies Assumption (C) or is vertical. In any case,any connected component of F ( γ ) satisfies Assumption (C) and ℓ ( γ ) ≤ C p p ( F ( γ )) (indeed, if γ is vertical, then ℓ ( γ ) ≤ τ p ( F ( γ )). It follows ℓ ( γ ) ≤ C p p ( F ( γ )) ≤ C q C λ − s p ( F s γ ) ≤ C ′ q C λ − s K ′ ε using first the fact that F ( γ ) is an increasing curve contained in M \ S − s and second the fact that F s γ is is an increasing curve satisfying Condition(C) and contained in B ε . Since F ( e γ u ) \ S s contains at most C λ s connectedcomponents, using (16), we obtain ν ( F − s − ( B ε ) ∩ B ε \ R ε ) = ν ( F − s ( B ε ) ∩ F ( B ε \ R ε )) ≤ K ′ ε | ∂Q | sup [0 , C λ s C ′ p C λ s ε . We conclude by using the fact that ν ( B ε ) = πε | ∂Q | and that ν ( R ε ) = O ( ε ).Step 4: Scarcity of intermediate quick returns.We prove now that for any a >
0, there exists s a > ε − sa X n = − a log ε ν ( B ε ∩ F − n B ε ) = o ( ν ( B ε )) . (18) PPLICATIONS OF SPATIO-TEMPORAL RARE EVENTS PROCESSES 21
Since ν ( B ε ) ≈ ε and ν ( R ε ) = O ( ε ), up to adding the condition s a < / ν ( B ε ∩ F − n B ε ) replaced by ν (( B ε \ R ε ) ∩ F − n ( B ε )).If q ∈ ∂Q and if e γ u is vertical, we replace it in the argument below by theconnected components of F ( e γ u ) and will conclude by noticing that; for anymeasurable set A , ℓ ( e γ u ∩ F − ( A )) ≤ C ′′ ℓ ( F ( e γ u ∩ A )).We denote the k th homogeneity strip by H k for k = 0 and set H = ∪ | k | 3. Let k ε = ε − s and H ε = ∪ | k |≤ k ε H k . For any u ∈ [0 , e γ k,u = e γ u ∩ H k . Each e γ k,u is aweakly homogeneous unstable curve.We cut each curve e γ k,u into small pieces e γ k,u,i such that each F j e γ k,u,i , j = 0 , . . . , n is contained in a homogeneity strip and a connected componentof M \ S . For x ∈ e γ k,u,i we denote by r n ( x ) the distance (in F n e γ u ) of F n ( x )to the boundary of F n e γ k,u,i .Recall that the growth lemma [6, Theorem 5.52] ensures the existence of θ ∈ (0 , c > γ one has ℓ ( γ ∩ { r n < δ } ) ≤ cθ n δ + cδℓ ( γ ) . (19)Therefore, ℓ ( e γ u ∩ F − n ( B ε ) \ H ε ) ≤ X | k |≤ k ε ℓ ( ∩{ r n ≥ ε − s } ∩ F − n ( B ε )) + ℓ ( e γ u,k ∩ { r n < ε − s } ) . The first term inside the above sum is bounded by the sum P i ℓ ( e γ u,k,i ∩ F − n ( B ε )) over those i ’s such that F n ( e γ u,k,i ) is of size larger than ε − s . Inparticular ℓ ( e γ u,k,i ) ≥ ε − s . On the other hand, by transversality ℓ ( F n ( e γ u,k,i ) ∩ B ε ) ≤ cε. By distortion (See Lemma 5.27 in [6]) we obtain ℓ ( e γ u,k,i ∩ F − n ( B ε )) ≤ cε s ℓ ( e γ u,k,i ) . Summing up over these i gives the first term inside the sum is bounded by ℓ ( e γ u,k ∩ { r n ≥ ε − s } ∩ F − n ( B ε )) ≤ cε s ℓ ( e γ u,k,i ) . Thus ℓ ( e γ u,k ∩ { r n < ε − s } ) ≤ cθ n ε − s + cε − s ℓ ( e γ u,k ) . A final summation over k gives ℓ ( e γ u ∩ F − n ( B ε ) \ H ε ) ≤ c ( ε s + ε − s ) ℓ ( e γ u ) + ck ε θ n ε − s . This combined with (16) leads to ν ( F ( B ε \ R ε ) ∩ F − n ( B ε )) ≤ ν ( F ( B ε \ R ε ) ∩ H ε ) + O ( ε s ) = O ( ε s ν ( B ε )) . where we use the fact that B ε \ H ε is contained in a uniformly boundedunion of rectangles of horizontal width O ( ε ) and contained in the k − ε = ε s -neighbourhood of S . We take s a < min( s, ). see [6] for notations and definitions. Step 5: End of the proof of (14).Choose a = 1 / (4 log( λ /λ / ). Observe that, due to (17), we have − a log ε X s =1 µ ( F − s A ε | A ε ) ≤ K λ /λ − λ /λ ) − a log ε ε / ≤ K λ /λ − ε / . This combined with (18) leads to ε − sa X n =1 ν ( F − n B ε | B ε ) = o (1) . (20)Let σ > 1. In view of (15), it remains to control ν ( F − n B ε | B ε ) for theintermediate integers n such that ε − s a ≤ n ≤ ε − σ . We approximate the set B ε by the union e B ε of connected components of M \ ( S − k ( ε ) ∪ S k ( ε ) ) thatintersects B ε , with k ( ε ) = ⌊| log ε | ⌋ . There exists e C > e θ ∈ (0 , 1) suchthat, for all positive integer k , the diameter of each connected componentof M \ ( S − k ∪ S k ) is less than e C e θ k .Thus B ε ⊂ e B ε and ν ( e B ε \ B ε ) ≤ ν (cid:16) ( ∂B ε ) [ e C e θ k ( ε ) ] (cid:17) = O ( ε e θ k ( ε ) ). But, due to[20, Lemma 4.1], we also have ∀ m > , ∀ n ≥ k ( ε ) , ν (cid:16) e B ε ∩ F − n e B ε (cid:17) = ν ( e B ε ) + O ( n − m ν ( e B ε )) . Since k ( ε ) = o ( ε − s a ) and thus ∀ m > , ε − σ X n = ε − sa ν ( F − n B ε | B ε ) ≤ O (cid:16) ε − σ + ε s a ( m − − σ + e θ k ( ε ) (cid:17) = o (1) , as ε → 0, since σ < e θ ∈ (0 , k ( ε ) → + ∞ and by taking m > σs a .This combined with (20) and (15) ends the proof of (14). (cid:3) References [1] J. F. Alves and D. Azevedo, Statistical properties of diffeomorphims with weak in-variant manifolds. 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