Extremes and extremal indices for level set observables on hyperbolic systems

Consider an ergodic measure preserving dynamical system (T, X, μ), and an observable ϕ:X→R . For the time series X n (x) = ϕ(T n (x)), we establish limit laws for the maximum process M n = max k⩽n X k in the case where ϕ is an observable maximized on a line segment, and (T, X, μ) is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on hyperbolic toral automorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.


Introduction
Suppose we have a time-series (X n ) of real-valued random variables defined on a probability space (X, μ) and let M n := max{X 1 , . . . , X n } be the sequence of successive maxima of (X i ). There is a well-developed theory for these maximum values in the setting of (X n ) i.i.d [12,25]. If we consider a dynamical system (T, X, μ) such that T : X → X and an observable φ : X → R, we can define a stochastic process by for x ∈ X. In the case of modeling deterministic physical phenomenon, T is usually taken as an ergodic, measure-preserving transformation, μ a probability measure and φ is a function with some regularity, for example (locally) Hölder [36]. In extreme value literature, it is typically assumed that φ is a function of the distance d(x, p) to a distinguished point p for some metric d so that φ(x) = f(d(x, p)) for x ∈ X, and f is a monotone decreasing function f : (0, ∞) → R. In this instance sup x∈X φ(x) = lim x→p φ(x), and hence the set {φ(x) u} corresponds to a neighborhood about p. We shall refer to the set of all points x ∈ X for which φ(x) achieves its maximum (with sup x∈X φ(x) = ∞ allowed) as the extremal set S. For convenience (and almost by convention) the observation φ(x) = −log d(x, p) is often used, but scaling relations translate extreme value results for one functional form to another quite easily provided the extremal set of φ is unchanged. If the observable φ(x) = −log d(x, p) is changed to another function of d(x, p), then S remains equal to {p}. However, if the underlying extremal set S is changed, e.g. going from a point to a curve, then the proofs of extreme value results and the results themselves do not translate and new approaches are required. Indeed, even if the extremal set changes from one point to another, then the extreme value laws may change (e.g. p periodic versus p non-periodic give different distributional extreme value laws) [10,16,20,34].
Since the value of the function φ • T n (x) is larger the closer T n (x) is to the extremal set S, there is a close relation between extreme value statistics for the time series X n = φ • T n (x) and return-time statistics to nested sets about S [5, 8, 11, 18-20, 26, 29, 33]. We focus on extreme value theory in this paper but it would be possible, though computationally very difficult, to derive return time distributions which are simple Poisson (in the cases in which the extremal index is θ = 1) and compound Poisson (in the cases in which the extremal index θ < 1). The parameters in the compound Poisson distribution would in particular be difficult to compute but this would constitute an interesting investigation. We expect this work could be carried out using basically the same toolkit from extreme value theory. These parameters are calculated for functions maximized at periodic orbits in the setting of a hyperbolic toral automorphism [10] and in [4] for functions maximized at periodic orbits in Sinai dispersing billiard systems. We discuss the concept of extremal index below, it is a number 0 θ 1 which roughly quantifies the clustering of exceedances. We will say θ = 1 is a trivial extremal index and θ < 1 a nontrivial extremal index. For results along these lines see [10,20,24].
Recent literature has focused on the case where the extremal set S is a single point {p}. In this paper we address some scenarios of interest where the observable is maximized on sets other than unique points in phase space, and in turn describe how the extreme value law depends on the geometry of S. We also describe a dynamical mechanism giving rise to a nontrivial extremal index which is not due to periodicity. The recent preprint [28] provides a different and axiomatic approach to determining the limit laws (especially simple and compound Poisson distributions) for entry times into neighborhoods of sets of measure zero in dynamical systems. They present similar results to this paper on coupled map lattices and consider other dynamical and statistical examples, including some systems with polynomial decay of correlations. We address here cases that are not easily captured by axiomatic approaches. This happens for example, if the extremal set S fails certain transversality assumptions relative to the local (or global) stable and unstable manifolds of the system. We discuss these situations further in sections 2.1 and 4.

Background on extremes for dynamical systems
Suppose (X n ) is a stationary process with probability distribution function F X (u) := μ(X u). We define an extreme value law (EVL) in the following way. Given τ ∈ R, let u n (τ ) be a sequence satisfying nμ(X 0 > u n (τ )) → τ , (1.1) as n → ∞. We say that (X n ) satisfies an extreme value law if μ(M n u n (τ )) → e −θτ (1.2) for some θ ∈ (0, 1]. Here, θ is called the extremal index and 1 θ roughly measures the average number of exceedances in a time window given that one exceedance has occurred. When (X n ) is i.i.d. and has a regularly varying tail it can be shown that this limit exists and θ = 1.
In the dependent setting for stationary (X n ) the existence of an EVL has been shown provided dependence conditions D(u n ) (mixing condition) and D (u n ) (recurrence condition) or similar conditions hold for the system [19,35]. Freitas et al [17], based on Collet's work, in turn gave a condition D 2 (u n ) which has the full force of D(u n ) in that together with D (u n ) it implies the existence of an EVL and is easier to check in the dynamical setting. We describe more precisely these three conditions below.
There are, however, no general techniques for proving conditions D 2 (u n ) and D (u n ) and checking the latter is usually hard. D (u n ) is a short returns condition that is not implied by an exponential decay of correlations. However D 2 (u n ) often follows from a suitable rate of decay of correlations. Collet [8] used the rate of decay of correlation of Hölder observations to establish D(u n ) for certain one-dimensional non-uniformly expanding maps. Condition D 2 (u n ) is easier to establish in the dynamical setting by estimating the rate of decay of correlations of Hölder continuous observables or those of bounded variation and in practice is easier to verify.
For completeness we now state conditions D(u n ), D 2 (u n ) and D (u n ). If {X n } is a stochastic process define M j,l := max{X j , X j+1 , . . . , X j+l }.
We will often write M 0,n as M n . We write F i 1 ,...,i n (u) for the joint distribution F i 1 ,...,i n (u) = μ(X i 1 u, X i 2 u, . . . , X i n u). Condition D(u n ) [35] We say condition D(u n ) holds for the sequence X 0 , X 1 , . . . if for any integers where γ(n, t) is non-increasing in t for each n and nγ(n, t n ) → 0 as n → ∞ for some sequence t n = o(n), t n → ∞. Condition D 2 (u n ) [17] We say condition D 2 (u n ) holds for the sequence X 0 , X 1 , . . . if for any integers l, t and n where γ(n, t) is non-increasing in t for each n and nγ(n, t n ) → 0 as n → ∞ for some sequence t n = o(n), t n → ∞. Condition D (u n ) [35] We say condition D (u n ) holds for the sequence X 0 , Condition D (u n ) controls the measure of the set of points of (X 0 > u n ) which return to the set relatively quickly, and is a condition that rules out 'short returns'. It is not a consequence of exponential decay of correlations and usually dynamical and geometric arguments are needed to verify condition D (u n ) in specific cases.
In the dynamical case if the time series of observations X n = φ • T n satisfy D(u n ) (or D 2 (u n )) and D (u n ) (or some variation thereof ) then an EVL holds. In these results, we have extremal index θ = 1 for observables of the form φ(x) = f(d(x, p)), maximized at generic p ∈ X provided p is non-periodic [10,16,20,24,30,34]. For periodic p, EVLs have been derived for these systems with index θ < 1 [4,10,16,20,34,36].
For statistical estimation and fitting schemes such as block maxima or peak over thresholds methods [12], it is desirable to get a limit along linear sequences of the form u n (y) = y/a n + b n . Here the emphasis is changed and the sequence u n (y) is now required to be linear in y. For example suppose φ(x) = −log x is an observable on the doubling map of the interval [0, 1], Tx = (2x) mod 1, which preserves Lebesgue measure μ. The condition nμ(φ > u n (y)) = y implies u n (y) = log n − log y. Furthermore we know that nμ(φ > u n (y)) = y implies μ(M n u n (y)) → e −y . This is a nonlinear scaling. If we change variables to Y = −log y we obtain In general if we restrict to linear scalings y ∈ R, we obtain a limit nμ(X 0 > y a n + b n ) → h(y) and hence μ(a n (M n − b n ) y) → e −h(y) = G(y), (n → ∞).
These three forms can be combined into a unified generalized extreme value (GEV) distribution (up to scale and location u → u−α σ ): The case ξ = 0 corresponds to the Gumbel distribution, ξ > 0 corresponds to a Fréchet distribution, while ξ < 0 corresponds to a Weibull distribution. Numerical fitting schemes for the GEV distribution are renormalized under place and scale transformations so that the extremal index (EI) is 1 [9, theorem 5.2]. Although it is theoretically possible to recover the EI by considering it as a function of these transformations, estimates in this way would have an undetectable level of error. Techniques to directly compute the EI, referred to as blocks and runs estimators, have been proposed [36, section 3.4]. Both methods utilize the definition of the EI (outlined above) by numerically estimating the ratio of the number of exceedances in a cluster to the total number of exceedances. Where these differ is in their definitions of a cluster; the blocks estimator splits the data into fixed blocks of size k n so that a cluster is defined by the number of exceedances inside each fixed block while the runs estimator introduces a run length of q n so that any two exceedances separated by a time gap of less than q n belongs to the same cluster. The problem with using these estimators in practical applications is their heavy dependence on choice the of k n and q n , respectively.
In recent literature, a more systematic approach is taken by first considering the point process describing exceedances which, under certain regularity conditions, follows some Poisson process in the limit [36]. The extremal index can then be seen as the expected value of this process and obtained through likelihood estimation. The issue of dependence in the likelihood approach is addressed in [41], and the resulting estimate of the extremal index is derived as the maximum likelihood estimator of expected value for systems satisfying condition D 2 (u n ). The setup is as follows.
Let q be a fixed quantile and N be the number of recurrences above the chosen quantile. Define T i for i = 1, . . . , N − 1 as the length of time between each consecutive recurrence, In other words, N c is the number of clusters found by counting the set of recurrences separated by a time gap of at least length 1. Then Süveges estimator [41] is given bŷ In a similar way to [13], we use equation (1.5) to numerically estimate the extremal index of the coupled map and Anosov systems found in section 3.7. For dynamical systems, the corresponding problem of finding scaling constants a n , b n depends on both the regularity of μ and that of the observable φ(x) = f(d(x, p)) in the vicinity of the point p.
For more general dynamical systems, these scaling relations depend on how the invariant measure scales on sets that shrink to p. This problem has been addressed in the case where μ admits a smooth or regularly varying density function h. However, for general measures (such as Sinai Ruelle Bowen measures) and general observables, estimating μ(X > y/a n + b n ) becomes more delicate, see [21,33]. However, an extreme law can still be obtained along some nonlinear sequence u n (y), with bounds on the growth of u n (y), see [27].
Furthermore, for deterministic dynamical systems the extremal index parameter θ may be nontrivial due to periodicity. For the doubling map discussed above, if p is a periodic point then θ = 1 − 1 2 q where q is the period of the periodic point (see [20,34]). In this article, we consider cases where φ is maximized on a more general extremal set S. For general S we cannot rely on previous methods adapted to observables of the form φ = f(d(x, p)).

Physical and energy-like observables
In the study of extreme events in dynamical systems, having in mind applications to weather and climate modeling, the notion of a physical observable was introduced and described in [33,37,40]. By physical observables we mean those of form and v a specified vector in R d . The former observable has planar level sets, while the latter has ellipsoidal level sets. In weather applications, these observables correspond to measuring (respectively) the momentum and kinetic energy of the system. The level geometries of φ introduced additional technicalities in establishing extreme laws relative to the cases where the level sets are metric balls. These issues are discussed in detail in [33], where S had a complicated geometry but its intersection with the attractor of the system was still a single point. In this article, we mainly consider energy-like observables for which the extremal set S is achieved on a line segment. We also discuss other extremal sets in section 4.

Organization of the paper
In section 2 we describe our main results on: hyperbolic automorphisms of the two-torus, Sinai dispersing billiard maps, and coupled uniformly expanding maps. We calculate the extreme value distribution, the extremal index and in some cases describe briefly the Poisson return time process. In particular we describe a method for obtaining a nontrivial extremal index which is not due to periodic behavior but rather self-intersection of a set of non-periodic points under the dynamics. Beyond existing approaches, we have to develop arguments that deal with both the geometry of S, and the recurrence properties of the dynamical systems under consideration. In our examples the underlying invariant measures have regular densities with respect to Lebesgue measure. This enables us to obtain analytic results on the GEV parameters and the extremal index. We also compare our results to numerical schemes, see section 3.7. We conclude with a discussion 4 on how the methods we have developed might be applied to general observables whose extremal sets have more complicated geometries.

Hyperbolic automorphisms of the two-torus
We consider hyperbolic toral automorphisms of the two-dimensional torus T 2 induced by a matrix with integer entries, det(T ) = ±1 and no eigenvalues on the unit circle. We will assume that both eigenvalues are positive in what follows to simplify the discussion and proofs. Such maps preserve Haar measure μ on T 2 . A well-known example is the Arnold cat map 2 1 1 1 .
We consider T 2 as the unit square with usual identifications with universal cover R 2 . T preserves the Haar measure μ on T 2 and has exponential decay of correlations for Lipschitz functions, in the sense that there exists Λ ∈ (0, 1) such that where C is a constant independent of φ, ψ and . Lip is the Lipschitz norm [1]. For a set D, we define d H (x, D) = inf{d(x, y) : y ∈ D} (for Hausdorff distance) , where d is the distance in ambient (usually Euclidean) metric. D denotes the closure of D and we define } is an neighborhood of D. As outlined in section 1.2, the observables we consider take the form φ(x) = f(d H (x, L)) where x = (x 1 , x 2 ) ∈ T 2 and L ⊂ T is a line segment with direction vectorL and finite length l(L). The function f : [0, ∞) → R is a smooth monotone decreasing function. We will take f(u) = −log(u). To fix notations, we also need to later consider -tubes around S. Thus if S is a line, or curve, and is small, then S is a thin tube.
The matrix dT has two unit eigenvectors v + and v − corresponding to the respective eigenvalues λ + = λ > 1, and λ − = λ −1 < 1. We can writeL = αv + + βv − for some coefficients α, β and so DT nL = αλ n + v + + βλ n − v − . If we let v (n) denote a unit vector in the direction of DT nL and α = 0, β = 0 then v (n) aligns with the direction v + as n → ∞ If L is aligned with the unstable direction, we may lift L toL on a fundamental domain of the cover R 2 of T 2 and writeL =p 1 + tv + , t ∈ [0, l(L)],p 1 ∈ R 2 . Thus L = π(p 1 + tv + ), t ∈ [0, l(L)] where π : R 2 → T 2 is the usual projection π : R 2 → R 2 /Z 2 . We write the endpoint ofL asp 2 , i.e.p 2 =p 1 + l(L)v + . We will also identify the vectors πp 1 and πp 2 with the corresponding points p 1 and p 2 in T 2 . Similarly if L is aligned with the stable direction, we may lift L toL on a fundamental domain of the cover R 2 of T 2 and write L = π(p 1 + tv − ), t ∈ [0, l(L)], p 1 ∈ R 2 . Again we write the endpoint ofL asp 2 , i.e.p 2 =p 1 + l(L)v − . We will also identify the vectorsp 1 andp 2 with the corresponding points they project to under π, written p 1 and p 2 .
where the extremal index θ is determined by these cases. If: (a) L is not aligned with the stable v − or unstable v + direction then θ = 1; (b) L is aligned with the unstable direction v + and π(p 1 + tv + ), −∞ < t < ∞ contains no periodic points then θ = 1; (c) If L is aligned with the stable direction v − and π(p 1 + tv − ), −∞ < t < ∞ contains no periodic points then θ = 1; (d) L is aligned with the stable v − or unstable v + direction and L contains a periodic point of prime period q then θ = 1 − λ −q ; (e) L is aligned with the unstable direction v + , L contains no periodic points but π(p 1 + tv + ), −∞ < t < ∞ contains a periodic point ζ of prime period q; then L ∩ T q L = ∅ implies and all values of θ in this range can be realized depending on the length and placement of L; we obtain corresponding limit laws.

Remark 2.4.
Since all periodic points of T have rational coefficients ( p 1 q 1 , p 2 q 2 ) and v + , v − have irrational slopes it follows that if π(p 1 + tv + ), −∞ < t < ∞ contains a periodic point it contains at most one, and similarly for π( Using exponential decay of correlations of the map, we show that for small -tubes L around L, we have (for all j sufficiently large) μ(T j L ∩ L ) Cμ(L ) 2 . This enables us to easily verify the form of the D(u n ) condition of Leadbetter et al [35] that we use.
The argument in the case that L is aligned with v + turns out to be the most subtle. We need a detailed analysis of how the forward images T j L wrap around the torus. It is clear that these forward images are dense, but we need quantitative information on how quickly these images become uniformly distributed. Such considerations are not necessary in the case where S is a single point, e.g. as discussed in [10], and furthermore this scenario is not easily captured by axiomatic approaches, as discussed in [5,28]. The close alignment of S with the unstable manifold appears non-generic in this hyperbolic toral automorphism example. However, for general observables one could imagine level set geometries failing transversality conditions generically, e.g. if {φ > u n } has a non-trivial boundary, which perhaps coils or accumulates upon itself. These scenarios would have to be treated on a case by case basis.

Sinai dispersing billiards maps
We now consider another setting in which it is natural to have a smooth observable maximized on a line segment. Suppose Γ = {Γ i , i = 1 : k} is a family of pairwise disjoint, simply connected C 3 curves with strictly positive curvature on the two-dimensional torus T 2 . The billiard flow B t is the dynamical system generated by the motion of a point particle in which moves with constant unit velocity inside Q until it hits Γ, then it undergoes an elastic collision where angle of incidence equals angle of reflection. If each Γ i is a circle and the system is lifted periodically to R 2 then this system is called a periodic Lorentz gas and was a model in the pioneering work of Lorentz on electron motion in conductors.
It is often easier to consider the billiard map T : ∂Q → ∂Q, derive statistical properties for it and then deduce corresponding properties for the flow. In this paper we will focus on limit laws for the billiard map. Let r be the natural one-dimensional coordinate of Γ given by arc-length and let n(r) be the outward normal to Γ at the point r. For each r ∈ Γ the tangent space at r consists of unit vectors v such that (n(r), v) 0. We identify such a unit vector v with an angle ϑ ∈ [−π/2, π/2]. The phase space M is then parametrized by M := ∂Q = Γ × [−π/2, π/2], and M consists of the points (r, ϑ). T : M → M is the Poincaré map that gives the position and angle T(r, ϑ) = (r 1 , ϑ 1 ) after a point (r, ϑ) flows under B t and collides again with M, according to the rule angle of incidence equals angle of reflection. The billiard map preserves a measure dμ = c M cos ϑdrdϑ equivalent to two-dimensional Lebesgue measure dm = drdϑ with density ρ(r, θ) = c M cos ϑ.
For this class of billiards the stable and unstable foliations lie in strict cones C u and C s in that the graphs ϑ(r) of local unstable manifolds have uniform bounds on the slopes of their tangent vectors which lie in the cone C u , s 0 dϑ dr s 1 , and similarly tangents to local stable manifolds lie in a cone C s , −t 1 dϑ dr −t 0 , for some strictly positive constants s 0 , t 0 , s 1 , t 1 . We will assume a line segment L with direction vectorL is uniformly transverse to C s and C u . More precisely, we will consider functions maximized on line segments , which are transverse to the stable and unstable cone of directions. For example the line segment r = r 0 , which is a position on the table rather than the point (r 0 , ϑ 0 ) (which is in phase space). We note that [39] studied distributional and almost sure return time limit laws to the position r = r 0 . In our setting the precise extreme law (Weibull, Fréchet or Gumbell) depends upon the observable we choose but results may be transformed from one observable to another in a standard way. We will take the function φ(r, ϑ) = 1 − d H (x, L) which, because it is bounded, will lead to a Weibull distribution. We assume the finite horizon condition, namely that the time of flight of the billiard flow between collisions is bounded above and also away from zero. Under the finite horizon condition Young [43] proved that the billiard map has exponential decay of correlations for Hölder observables. A good reference for background results for this section are the papers [2,3,7,43] and the book [6].
Let L be a line segment transverse to the stable and unstable cones and φ(r, ϑ) = 1 − d H (x, L). Let y > 0. We define a sequence u n (y) = y/a n + b n by the requirement nμ{φ > u n (y)} = y. Apart from complication arising from the invariant measure having a cosine term, a n scales like 1 n . The set {φ > u n } is a rectangle U n with center L roughly of width Cy n for some constant C. Note that we assume L is not aligned in either the unstable or the stable direction, so the following result is expected from the hyperbolic toral automorphism case.
Then μ(M n u n (y)) → e −y as n → ∞. In particular the extreme index θ = 1.
Remark 2.6. We now make some remarks on what we conjecture in the case that a line segment or curve L is contained in, i.e. a piece of, a local unstable or local stable manifold and If L is part of a local unstable manifold and T n L has no self-intersections with L then the extremal index is one. The proofs we give in the case of the hyperbolic toral automorphism for this scenario break down but the techniques of the recent preprint [42] probably extend to this case. If L contains a periodic point ζ of period q then the extremal index would be roughly is the expansion in the unstable direction at ζ with a correctional factor due to the conditional measure on the unstable manifold which contains L. If L does not contain a periodic point but its continuation in the unstable manifold does contain a periodic point of period q then as in case (5) of theorem 2.1, if T q L ∩ L = ∅ then θ = 1, otherwise we expect θ to lie roughly in the range 1 − 1 |DT u (ζ)| q θ 1 (with all values of θ being realizable depending on the length and placement of L). If L is part of a local stable manifold and T n L has no self-intersections with L then the extremal index θ = 1. If L contains a periodic point ζ of period q then the extremal index would be roughly θ ∼ 1 − |DT s (ζ)| q where DT s (ζ) is the expansion in the stable direction at ζ. If L does not contain a periodic point but its continuation in the unstable manifold does contain a periodic point of period q then as in case (6) of theorem 2.1, if T q L ∩ L = ∅ then θ = 1, otherwise we expect θ to lie roughly in the range 1 − |DT s (ζ)| q θ 1 (with all values of θ being realizable depending on the length and placement of L).

Coupled systems of uniformly expanding maps
Now we consider a simple class of coupled mixing expanding maps of the unit interval, similar to those examined in [13]. In fact we were motivated by the comprehensive work of [13] (which uses sophisticated transfer operator techniques) to develop in this paper an alternate probabilistic approach in a coupled maps setting. The recent preprint [28] presents similar results to ours in the case of returns to the diagonal {x 1 = x 2 = . . . = x n }. Let T be a C 2 uniformly expanding map of S 1 and suppose that T has an invariant measure μ with density h bounded above and below from zero. In [13] piecewise C 2 expanding maps were considered but we will limit our discussion to smooth maps. We use all-to-all coupling and first discuss the case of two coupled maps for clarity.
Let 0 < γ < 1 and define We assume that F has an an invariant measure μ on T 2 with densityh on T 2 bounded above and also bounded below away from zero almost surely. We will require also that there exists > 0 and 0 < α 1 such that The semi-norm |.| α and this notion of regularity was described in [13] and established in several of their examples. An invariant density for F cannot reasonably be assumed to be continuous or Lipschitz.
For example a slight perturbation of the doubling map of the unit circle T(x) = (2x) (mod 1) to the map T(x) = ((2 + )x) (mod 1) gives rise to a map with invariant density which is of bounded variation but not Lipschitz or even continuous. |.| α can be completed to a norm . osc,α by defining . osc,α = |.| α + . 1 . The value of 0 and α does not matter in our subsequent discussion. We note that the bounded variation norm and the quasi-Hölder norm . osc,α are particularly suited to handle dynamical systems with discontinuities or singularities.
We also assume a strong form of exponential decay of correlations in the sense that for all where · Lip denotes the Lipschitz norm and . ∞ denotes the L ∞ norm. The function Θ n (Φ, Ψ) is called the correlation function. Let φ(x, y) = −log |x − y|, a function maximized on the line segment (or circle) L = {(x, y) : y = x}. In this setting L is invariant under F and the orthogonal direction to L is uniformly repelling. Note that the projection of (x, y) onto L is the point ( x+y 2 , x+y 2 ) and the projection on L ⊥ is (x − x+y 2 , y − x+y 2 ). Close to L we have uniform expansion away from L in the L ⊥ direction under F. This is because . There is uniform repulsion away from the invariant line L. This observation simplifies many of the geometric arguments we use to establish extreme value laws.
In the more general case of m-coupled maps we define For these maps, we assume the following: (a) there exists a mixing invariant measure μ with densityh, h osc,α < ∞, on T m bounded above and below away from zero; (b) exponential mixing for Lipschitz functions versus L ∞ functions as in equation (2.3).

Remark 2.7.
Using the spectral analysis of the transfer operator of this system as in [13] and standard perturbation theory it can be shown that (A) and (B) hold if γ is sufficiently small as the uncoupled system is uniformly expanding. An example explicitly given in [13, example 2.2] is to take T(x) = 3x (mod 1) with sufficiently small coupling γ > 0. Note that Lipschitz functions are quasi-Hölder in the sense of [13].
We consider a function maximized on where we have used twice-differentiability and the fact that m i=1 Δp i = 0. Hence again there is uniform expansion in a sufficiently small neighborhood of L in the direction of the n − 1 dimensional subspace orthogonal to L.
For y > 0 define u n (y) by nμ(φ > u n (y)) = y, and U n = {φ > u n (y)}. It can be seen that if where s is the natural co-ordinatization of the one dimensional subspace L.
We may also consider blocks of synchronization, as in [13, section 7.2] where we take the observable maximized on a set L consisting of synchrony on subsets of distinct lattice sites, for example of form L = {(x 1 , x 2 , . . . , x m ) : The main purpose of this section is to illustrate our geometric approach, so we will give one result of this type. Recall that for y > 0 the scaling constants u n (y) are defined by the condition nμ(φ > u n (y)) = y. Theorem 2.9. Let F : T m → T m be a coupled system of expanding maps satisfying (A) and (B). Let 0 < k m and choose k distinct lattice sites where y is the natural co-ordinatization of the m − k + 1 dimensional subspace L.

Extreme value scheme of proof
Our proofs are based on ideas from extreme value theory. We will use two conditions, adapted to the dynamical setting, introduced in the important work [22] that are based on D(u n ) and D 2 (u n ) but also allow a computation of the extremal index.
Let X n = φ • T n and define For s, l ∈ N and a set B ⊂ M, let Next we describe the two conditions introduced in [22]. In the following recall that u n ≡ u n (τ ) is the sequence defined in equation (1.1).

Condition
: we say that holds for the sequence X 0 , where γ(q, n, t) is decreasing in t and there exists a sequence (t n ) n∈N such that t n = o(n) and nγ(q, n, t n ) → 0 when n → ∞. We consider the sequence (t n ) n∈N given by condition and let (k n ) n∈N be another sequence of integers such that as n → ∞, k n → ∞ and k n t n = o(n).

Condition
: we say that holds for the sequence X 0 , X 1 , . . . if there exists a sequence (k n ) n∈N as above and such that We note that, taking U n := {X 0 > u n } for A (q) n , which corresponds to non-periodic behavior, in condition corresponds to condition D (u n ) from [35]. We will abuse notation and consider U n := {X 0 > u n } as the case of A (q) n with q = 0.

Now let
whenever this limit exists. In the settings we consider the limit is easily seen to exist.
Remark 3.1. In a dynamical setting verifying these two conditions picks up the main underlying periodicity or more generally recurrence properties of the system, for example returns to a periodic point of prime period q, and determines the extremal index. However, as we demonstrate, other recurrent phenomena may give rise to an extremal index not equal to unity. We show below that the self-intersection of a line segment L, T(L) ∩ L = 0 (none of whose points are periodic) may lead to a nontrivial extremal index for functions maximized on L. For a more detailed discussion of extremal indices see [22].
From [23, corollary 2.4], it follows that to establish theorem 2.1 it suffices to prove conditions and for q = 0 in the non-recurrent case θ = 1 and for q > 0 corresponding to the 'period' of the cases where there is some recurrence phenomena (θ < 1). In both cases lim n→∞ μ(M n u n (y)) = e −θy .
The scheme of the proof of condition is itself somewhat standard [10,24] and is a consequence of suitable decay of correlation estimates. We outline it for completeness, indicating the modifications that need to be made for the different geometries of A (q) n . The main work will be in establishing condition .

Proof of theorem 2.1
In the first instance we check condition . We recall some useful statistical properties of hyperbolic toral automorphisms. In the case where Φ and Ψ are Lipschitz continuous functions, it is known for hyperbolic toral automorphisms that there exists C > 0, τ 0 ∈ (0, 1) such that Furthermore if Ψ is constant on local stable leaves corresponding to a Markov partition, then the Lipshitz norm of Ψ on the right-hand side of equation (3.1) can be replaced by the L ∞ norm [43, section 4]. This fact will be useful when checking , see proposition 3.2 in section 3.2 below.
Consider now a set D, whose boundary ∂D is a union of a finite number of smooth curves, so that μ(∂D) = 0. Let W s 1 (x) denote the local stable manifold through x. We define In section 3.2 we show roughly that μ(H k (D)) decreases exponentially in k.

Checking condition
This argument is a minor adjustment of similar estimates in [10,24]. We state the following proposition Proposition 3.2. For every , t, n ∈ N, there exists λ 0 ∈ (0, 1), and C > 0 such that Condition immediately follows from this. We can take t n = ( log n) 5 so that nγ(q, n, t) → 0.
The proof of proposition 3.2 is as follows. To check condition we use decay of correlations. The main problem in estimating the correlation function Θ n (Φ, Ψ) [recall equation (2.3)] is that the relevant indicator functions Φ = 1 A (q) n and Ψ = 1 W 0, (A (q) n) of the sets A (q) n and W 0, A (q) n are not Lipschitz continuous. Standard smoothing methods can be used to approximate Φ, but Ψ cannot be uniformly approximated by a Lipschitz function: the level set Ψ = 1 has a geometry that becomes increasingly complex (i.e. with multiple connectivity) as increases. Fortunately, we can employ a further trick to approximate Ψ. This is done using a function that is constant on local stable manifolds. This allows us to use a decay of correlations estimate using the L ∞ norm. As part of this approximation we first estimate μ(H k (D)) with D = A (q) n . The geometry of the set A (q) n will be important in calculating this estimate.

Lemma 3.3. Consider the set D = A (q)
n . Then there exists C > 0 such that, for all k,

3)
where λ −1 < 1 is the (uniform) contraction rate along the stable manifolds for the hyperbolic toral automorphism.
Proof. We follow [10, proposition 4.1], and consider also the geometrical properties of D. Since the local stable manifolds contract uniformly there exists C 1 > 0 such that dist(T n (x), T n (y)) C 1 |λ| −n for all y ∈ W s 1 (x). This implies that |T k (W s 1 (x))| C 1 λ −k . Therefore, for every x ∈ H k (D), the leaf T k (W s 1 (x)) lies in an tubular region of width 2/|λ| k around ∂D. To measure of the size of this tube we note that m(D )  Then there exists 0 < τ 1 < 1 such that for all j 0 Proof. Following lemma [10, lemma 4.2], we take a version Ψ of Ψ that is constant on local stable manifolds, for example by taking a distinguished point x * on each local stable manifold W s 1 (x) and defining Ψ(y) = Ψ(x * ) for all y ∈ W s 1 (x). We let Ψ j = Ψ • T j , and again denote Ψ j as the relevant version of Ψ j (constant on local stable manifolds). A simple application of the triangle inequality gives the following bound: [recall that Θ j is defined in equation (2.3)]. To estimate μ{Ψ j/2 = Ψ j/2 }, we consider points x 1 , x 2 on the same stable manifold, and such that x 1 ∈ W i, A (q) n , but x 2 / ∈ W i, A (q) n , (for i j/2). This set is contained in ∪ i+ −1 k=i H k (A q n ). Hence The conclusion of lemma 3.4 follows.
To continue with the proof of proposition 3.2, and hence verify condition , we approximate the characteristic function of the set A (q) n by a suitable Lipschitz function. The key estimate is to bound the Lipschitz norm of the approximation.
Let A n = A (q) n and D n := x ∈ A (q) n : d H x, A c n n −2 , whereĀ c n denotes the closure of the complement of the set A n . Define Φ n : X → R by (3.6) Note that Φ n is Lipschitz continuous with Lipschitz constant given by n 2 . Moreover Φ n − 1 A n L 1 (m) C/n 2 for some constant C. It follows that for some generic constant C. Thus where γ(n, j) = C n −2 + n 2 λ j/2 1 , and Thus if, for instance, we choose j = t n = ( log n) 5 , then nγ(n, t n ) → 0 as n → ∞. This completes the proof.

Checking condition
We make the following decomposition: where the sequence R n → ∞ (as n → ∞) will be chosen later. Recall that for q = 0, A (q) n = U n . By exponential decay of correlations and a suitable Lipschitz approximation the last sum tends to 0 as n → ∞, so it suffices to estimate the two sums where 1 j ( log n) 5 .

Case L transverse to stable and unstable directions.
Fix y > 0 and define u n (y) by the requirement nμ{x : φ(x) u n (y)} = y. Henceforth we will drop the dependence on y and write simply u n for convenience. We define U n := {x : φ(x) u n }. Geometrically U n resembles a parallel strip of width 2 n . We will verify the short return condition with q = 0. Consider the set T − j U n ∩ U n = {x : T j (x) ∈ U n , x ∈ U n }. T j U n is a union of parallelogram-like strips corresponding to each winding around the torus and such strip has width O( λ − j n ) and length O(1), the precise constants depending on the angle between T j L and L as T j L aligns to the unstable direction. There are approximately λ j such parallelogram strips. Each strip intersects U n in an area of measure O( λ − j n −2 ) by transversality (see figure 1).
There are 2 main cases, with some subcases. Case (a): first assume that the linep 1 + tv + , −∞ < t < ∞ contains no point with rational coordinates. This holds for a measure one set ofp 1 as the set of points in the plane with rational coordinates is countable. In this case T n L, n 1, has no intersections with L. To see this suppose p ∈ L and there exists an n such that T n p = q ∈ L. If we take a line segmentL in direction v + of length 2l(L) centered at p we see by expansion thatL ⊂ T nL (since d(p, q) l(L)) and hence T n restricted toL has a fixed pointp inL. However, this implies the liftp 1 + tv + , −∞ < t < ∞ contains a point with rational coordinates, which is a contradiction.
Since p 1 is not periodic by assumption andp 1 is not in the direction of v + (otherwise the point (0, 0) would be contained inp 1 + tv + , −∞ < t < ∞) the iterates T j U n are disjoint for large n, for small j i.e. there exists R n → ∞ such that μ(T − j U n ∩ U n ) = 0 for j < R n . Corollary 2.2 of the recent preprint [42] implies in this case that the extremal index is one. We include an alternate proof for completeness.
For large n the set T j U n comprises λ j parallel rectangles (aligned with the unstable direction) of width O( λ − j n ). Identifying T 2 with the unit square the set We now estimate μ(T − j U n ∩ U n ). The set T j U n has approximately [λ j ] windings around the torus and we now estimate the fraction of these that intersect U n .
Note that γ is a quadratic irrational. This implies that γ has low discrepancy in the sense that there exists a constant C > 0 such that see [38]. Hence for j > R n n (log n) 5 This implies that a standard EVL holds with θ = 1 (see figure 2). Case (b): assume thatp 1 + tv + , −∞ < t < ∞ contains a point with rational coordinates, note that it will contain at most one as the slope of v + is irrational. Such a point projects to a point p per periodic under T with period q say. Case (b1): assume now that L itself contains p per , a periodic point of period q.  There will be only one periodic point in L as the slope of v + is irrational. Without loss of generality we take q = 1 by considering T q . It is easy to see that θ = lim n→∞ The same discrepancy argument as in the case of no periodic orbits shows that there exists an Hence θ = 1 − 1 λ q (see figure 3). Case (b2): L does not contain a periodic point.
We first consider the simplest case where the origin is the fixed point andp 1 parallel to v + so thatp 1 + tv + , −∞ < t < ∞ contains the fixed point (0, 0) but L does not contain (0, 0). The linep 1 + tv + , 0 t < ∞ has a natural ordering by distance from the origin (0, 0). If λp 1 >p 2 then it is easy to see all iterates of T n L on the torus are disjoint and the arguments given in case (a) apply giving θ = 1.
Suppose now λp 1 <p 2 . We take q = 1 and calculate as the stable manifolds are sent strictly into the region of intersection U n ∩ TU n (see figure 4).
By varyingp 1 andp 2 we may obtain all values in this range. Hence (1 − 1 λ ) θ 1. In the general case of a periodic point p per of period q contained in π(p 1 + tv + ), −∞ < t < ∞ we consider T q and the analysis proceeds in the same way by considering the expansion on the line segment [p 1 −p per ,p 2 −p per ]. We infer that for general q 1, with all values of θ in this range being realizable The verification of condition is similar to case (b1).

Case L is aligned with the stable direction.
Suppose now that L aligns with the stable direction v − . See figure 5. The analysis is similar to the case where L is aligned with the unstable direction, and again we consider the liftL =p 1 + tv − , t ∈ [0, l(L)], withp 1 ∈ R 2 , andp 2 denoting the other endpoint ofL, i.e.p 2 =p 1 + l(L)v − . We will make use of the timereversibility of the system in case (a) below.
We have the following cases. Case (a): first assume that the linep 1 + tv − , −∞ < t < ∞ contains no point with rational coordinates. Let S = T −1 . Then L is aligned with the unstable direction for S. As in the case where L aligned with the unstable direction for T, it follows again that S n (L) has no intersections with L, for all n 1. Hence T n (L) has no intersections with L for all n 1.
Thus all the iterates T j U n are disjoint for small j, i.e. there exists R n → ∞ such that μ(T − j U n ∩ U n ) = 0 for j < R n . Note that the definition of U n is the same for T and S and that by measure-preservation. The argument of case (a) when L is aligned with the unstable direction shows that Thus θ = 1. Case (b): assume thatp 1 + tv − , −∞ < t < ∞ contains a pointp per with rational coordinates. There will be only one such point as the slope of v − is irrational. The pointp per projects to a point p per periodic under T with period q say. We cannot use time-reversibility in this case as the set A q n depends upon the consideration of T or T −1 as the transformation. Case (b1): assume now that L contains the periodic point p per of period q.
Without loss of generality we (again) take q = 1 by considering T q . We have θ = lim n→∞ Geometrically A (q) n consists of two strips within U n . Both of these have length 1/n, (i.e. the same as U n ), but their width relative to U n is 1 2 1 − 1/λ q (see figure 6). The same argument as in the case of no periodic orbits shows that there exists an R n → ∞ such that T − j A (q) n ∩ A (q) n = ∅ for j < R n . We have uniform expansion of A q n in the unstable direction and the discrepancy argument of case (b1) of the previous section (alignment with the unstable direction) shows that We therefore have θ = 1 − 1 λ q . Case (b2): L does not contain a periodic orbit. Again, we illustrate by considering the simplest case ofp 1 parallel to v − so thatp 1 + tv − , −∞ < t < ∞ contains the fixed point (0, 0) but L does not contain (0, 0). For the lifted linê p 1 + tv − , 0 t < ∞, we use the natural ordering by distance from the origin (0, 0). Ifp 1 > λ −1p 2 , then all iterates of T n L on the torus are disjoint and the arguments given in case (a) apply giving θ = 1.
Suppose nowp 1 < λ −1p 2 . We take q = 1 and calculate See figure 7. The general case wherep 1 is not parallel v − proceeds the same way by considering the expansion of T orthogonal to the segment [p 1 −p per ,p 2 −p per ]. We infer that for general q 1, with all values of θ in this range being realizable. The verification of condition is similar to case (b1).

Proof of theorem 2.5
We will show that conditions and hold with q = 0 so that the extremal index θ = 1. We shall drop the subscript q in this section. The proof of follows the same strategy as in the hyperbolic toral automorphism case, the differences necessary in the planar dispersing billiard setting are addressed in [27, theorem 2.1] and [4, proposition 5 and lemma 6]). To simplify the exposition we will consider the case L = {x : r = r 0 } (see figure 8).

Checking condition
Before checking , we note that as the case of hyperbolic automorphisms of the twotorus we need only to consider the sum up to time ( log n) 1+δ , for δ > 0 because of exponential decay of correlations as in lemma 3.4. The analogous result to lemmas 3.3 and 3.4 in the case of Sinai dispersing billiards follows with no essential modification the proofs of proposition 5 [4] and lemma 6 [4] respectively (with (X 0 > u n ) equal to A q n ). Thus the remaining sum n k n /n j=(log n) 1+δ (Note, here, we work with A (0) n ≡ U n ). The set {r = r 0 } corresponds to a line (call it L) which is transverse to the discontinuity set S + for T and the discontinuity S − for T −1 . Let U n be the rectangle centered at L with length π and of width roughly τ πn corresponding to the set {φ > u n } so that μ(U n ) = τ n .
3.5.1. Short returns. Let S n = ∪ n−1 j=0 T − j S + . The number of smooth connected components of S n is bounded above by κ n for some κ > 0. Let C = 1 4 log κ and then the number of smooth connected components in S [C log n] is bounded above by n 1/4 . Let p i = (r 0 , ϑ i ) ∈ L be the intersection points S [C log n] ∩ L, ordered from lowest ϑ value to highest and let and is of length n −1/2 in the ϑ direction and width 1/n in the r-direction. If R i ∈ G then T [C log n] R i is a connected 'rectangle' which has expanded in the unstable direction, contracted in the stable direction and may wind around the phase space at most once. T [C log n] R i intersects U n transversely (since L is transverse to the unstable cone) in a connected component of measure

Intermediate returns.
The proof of this section is similar to that in the toral automorphism case but with additional complications due to the presence of discontinuities for T, causing the unstable manifolds to fragment into small pieces. A scenario which needs to be ruled out is that a large number of small pieces of fragmented unstable manifolds may find themselves again in U n . To overcome this we use the following property satisfied by the planar dispersing billiard map: One-step expansion. For α ∈ (0, 1], where the supremum is taken over regular unstable curves W ⊂ X, |W| denotes the length of W, and V n , n 1, the smooth components of T(W ), α ∈ (0, 1]. The class of regular curves includes our local unstable manifolds [6]. The expansion by DT is unbounded and this may lead to different expansion rates at different points on the local unstable manifold W u (x). To overcome this effect and obtain uniform estimates on the densities of conditional SRB measure it is common to define homogeneous local unstable and local stable manifolds. This is the approach adopted in [2,3,7,43]. Fix a large k 0 and define for k > k 0 : and In our setting we call a local unstable (stable) manifold W u (x), (W s (x)) homogeneous if for all n 0T n W u (x) (T −n W s (x)) does not intersect any of the line segments in ∪ k>k 0 (I k ∪ I −k ) ∪ I k 0 . Homogeneous local unstable manifolds W u (x) have almost constant conditional Sinai-Bowen-Ruelle (SRB) densities dμ W u (x) dμ W u (y) in the sense that there exists C > 0 such that 1 C for all z 1 , z 2 ∈ W u (x) (see [6, corollary 5.30]). Here dμ W u (x) dμ W u (y) = lim n→∞ where J W u T −n is the restriction of the map T n to the one-dimensional curve W u [6, theorem 5.2]. The conditional density, apriori only defined up to a constant, is then determined uniquely by the requirement that W u dμ W u (y)dy = 1. These conditional densities are the densities of the conditional measures induced on the measurable partition into local unstable manifolds by the invariant measure μ. Conditional measures are important in the study of billiard systems as they allow us to go from from knowledge of behavior of quantities on unstable manifolds to the whole phase space by Fubini type arguments. For more details of the construction of the conditional measures on local unstable manifolds and their densities we refer the reader to section 5.2 of [6]. From this point on all the local unstable (stable) manifolds that we consider will be homogeneous. We may as well suppose all such curves are contained in a set R i ∈ G as μ(B) < n −5/4 .
We now take care of the times [C log n] < j < ( log n) 1+δ . If W u (x) ∩ U n ⊂ R i ∈ G then T [C log n] has expanded W u (x) by a factor Λ C log n = n C log Λ = n β for some β > 0 and the iterates of the components of W u (x) ∩ U n have not hit an extremal set in the first [C log n] iterates. Let γ n (x) = W u (x) ∩ U n . By [7, theorem 5.7] Now we consider μ(U n ∩ T − j (U n )) for C log n j ( log n) 1+δ . Note that T j (γ n (x)) consists of a connected curve for j C log n. Recall by expansion under the map we have |T j γ n (x)| n β |γ n (x)| > n −1+β/2 . If we iterate this component further such that T i+ j γ n (x), i > 0 intersects an extremal line then we may decompose T i+ j γ n (x) into smooth connected components V n and their preimages Y n ⊂ T j γ n (x) so that T i maps Y n onto V n diffeomorphically and with uniformly bounded distortion. Applying one-step expansion to T j γ n (x) gives Fix T j γ n (x) and for every point p ∈ T j γ n (x) let Y n (p) denote the unique Y n ⊂ T j γ n (x) such that p ∈ Y n and let V n (p) denote T i Y n (p). Let ν(p) = |Y n (p)| |T j γ(x)| be the probability mass function of a probability measure ν on T j γ n (x) where C is the length of a rectifiable one dimensional be a function on this probability space. By one-step expan- We choose ε sufficiently small so that ρ 1 := 1 − (1 − ε)β/2α > 0 (since β < 1) and define ρ = min{ρ 1 , β/2αε}. With our choice of ε, if |V n | n −1+ β/2 then, By bounded distortion of the map T i+ j , after throwing away the V n such that |V n | n −1+εβ/2 we have This provides a bound on the length γ n (x) ∩ T −r (U n ) for c log n r ( log n) 1+δ . We may now use the fact that μ decomposes as a product measure of conditional SRB measures on γ n (x) ∩ U n to estimate Putting these results together implies Condition follows.
Remark 3.5. Using essentially the same analysis it is standard to show that the return time statistics to L = {(r, ϑ) : r = r 0 } is standard simple Poisson. To see this we need verify condition D * q (u n ) of [4, section 2], but the proof of this is a minor modification of . In contrast suppose (r 0 , ϑ 0 ) is a periodic point of period q, then we would obtain a compound Poisson process as given in [4, theorem 2].

Proof of theorems 2.8 and 2.9
We give the proof in detail only for the case of two coupled maps, as the proofs in the other cases are the same with obvious modifications. The uniform expansion away from the invariant subspace plays the same role in each setting. Note that the subspace L of theorem 2.9 is invariant, and we will show that there is uniform expansion in the directions orthogonal to L. Recall Furthermore A n is a union of two rectangles and A n ∩ F −2 A n = ∅ as a result of uniform expansion away from the invariant line L.
Condition follows easily by an approximation argument using exponential decay of correlations of Lipschitz versus L ∞ functions taking t n = ( log n) 5 say. Now we prove condition (for q = 1), namely Note that by uniform repulsion from the invariant line L there exists C 4 such that for j = 1, . . . , C 4 log n, μ(A n ∩ F − j A n ) = 0. This follows since F −1 A n ∩ A n = ∅ (by definition) and uniform repulsion from the invariant line ensures also F − j A n ∩ A n = ∅ for a certain number of Figure 9. Illustration of the set A n (given in dark gray) and its expansion under the map (given in light gray) which leaves a neighborhood of L in the first iterate and does not return for C 4 log n iterates.
iterates j = 1, . . . , C 4 log n until for all (x, y) in A n , |F j (x, y)| = O(1) (i.e. until the expansion in the L ⊥ direction is O(n)) (see figure 9). As DF is bounded and uniformly expanding, in all directions A n has been expanded by the map F [C 4 log n] by at least n α for some 0 < α < 1. To see this, note that for any expanding map the expansion of A n by the map F [C 4 log n] is given by at least C 5 |DT| C 4 log n min ∼ n α . Choose C 3 C 4 large enough that μ(A n ∩ F − j (A n )) 1 n 3/2 , this is possible by exponential decay of correlations and a Lipschitz approximation to 1 A n .
Thus for C 4 log n j C 3 log n, μ(A n ∩ F − j (A n )) 1 n 1+α . For 1 j C 4 log n, μ(F − j (A n ) ∩ A n ) = 0 for C 4 log n j C 3 log n, μ(A n ∩ F − j A n ) 1 n 1+α and for j C 4 log n, μ(A n ∩ F − j (A n )) 1 n 3/2 . This implies for q = 1 (corresponding to the fact that L is fixed). Finally we compute the extremal index, changing coordinates to v = x−y √ 2 , u = x+y √ 2 we have However, . We may assume thath is regularized along the diagonal in the sense that for Lebesgue almost every u,h(u, u) is the average of the limits ofh(u, v) andh(u, −v) as v → 0. Thus, as expansion along v at v = 0 is given by (1 − γ)DT(u), andh is essentially bounded: Remark 3.6. Our techniques allow us to obtain similar results to that of [13] in a simpler setting through a pure probabilistic approach and extend these results to blocks of synchronization discussion in [13, section 7.2].

Numerical results for the extremal index
In this section we provide numerical estimates for the extremal index to support the theoretical results for the coupled uniformly expanding map and the hyperbolic automorphism of the twotorus given in theorems 2.8 and 2.9 and theorem 2.1, respectively. We begin by verifying that the numerical estimates we obtain from the coupled systems agree with that of [13]. Then, we extend these results to include estimates for the extremal index over blocks of synchronization where each block introduces a new invariant direction and changes the value of the extremal index. We end with a numerical investigation for Arnold's cat map where the alignment of the line L and the existence of periodic orbits along L or its continuation determine the value of the extremal index.

Coupled systems of uniformly expanding maps
Numerical barriers in computing trajectories in piecewise uniformly expanding maps are given by the fact that (a) the periodic orbits are dense making long trajectories not easily computable; (b) round off errors may produce unreliable results.
To overcome (a) we employ a numerical technique adapted from [36] to prevent trapping of the orbit near the fixed point by adding a small ε = O(10 −2 ) perturbation to the trajectory. Arguments for this technique are typically given in the form of a shadowing lemma which states the existence of a true orbit that is -close to the computed orbit. We first note that [14] proves the existence of an EVL for randomly perturbed piecewise expanding maps provided this perturbation ε > 10 −4 . Further, [13] provides evidence that the extremal index is qualitatively robust under small ε = 10 −2 additive noise. To overcome (b), in light of our long trajectories (t = 10 6 ), we refer to [15] where the round off error resulting from double precision computation was shown to be equivalent to the addition of random noise of order 10 −7 . Estimating the EI for the coupled map system over the whole extremal set. We estimate the extremal index in a similar way to that of [13] for φ(x) = − log( p ⊥ ) using the formula provided by Süveges [41]. The code for this estimate can be found in [36]. From theorem 2.8 we expect We compute the extremal index for fixed m = 2 and varying values of γ, and varying values of both γ and m. Our results coincide with that of [13]; higher values of m and lower values of γ produce an extremal index near 1. Low values of γ give higher weights to the non-coupled components of the map resulting in a system which behaves more independently. Lower values of m result in a more dependent system since the coupled term is more affected by changes while larger values of m result in a coupled term which is averaged over a larger number of maps and less affected by individual changes. For results see figure 10.
for L = {(x 1 , x 2 , . . . , x m ) : x 1 = x 2 = · · · = x m−1 , x m }. Defining L in this way reduces the spatial dimension in which expansion away from L can occur. This results in an extremal index equivalent to that of an m − 1 coupled system. We give results in the case when m = 5 (see figure 11).
We also consider blocks of successive indices in the general setting of block synchronization so that L can be defined as any combination of block sequences. From theorem 2.9 we expect the value of the extremal index to be determined by the spatial dimension of expansion for the system. In the following numerical examples we consider m = 5 and note that the extremal index for that of L = {(x 1 , . . . , x 5 ) : x 1 = x 2 = x 3 = x 4 , x 5 } ( figure 11(b)) is equivalent to that of L = {(x 1 , . . . , x 5 ) : x 1 = x 2 = x 3 , x 4 = x 5 } ( figure 12(b)). This is expected since they share the same number of non-invariant directions of expansion.

Hyperbolic toral automorphisms
We compute trajectories for increasing time intervals of Arnold's cat map given by T x 1 x 2 = 2 1 1 1 x 1 x 2 mod 1.
The uniformly hyperbolic structure of this map allows us to calculate long trajectories without the risk of points being trapped in a few time steps. The stability of this map ensures that the  qualitative behavior is unaffected by small perturbations. We use this to argue the accuracy of the calculated orbit up to t = 10 4 under double precision.
From theorem 2.1 we expect the value of the extremal index θ to depend on both the alignment of v in the observable φ(x) = −log d(x, L), with d the usual Euclidean metric, and the existence of a periodic orbit along L. Figure 13(a) shows the extremal index estimation given by [41] for 10 different initial values where v aligns with the unstable direction and contains a two-periodic point. Hence, θ = 1 − 1 λ 2 . Figure 13(b) shows the extremal index estimation for 10 different initial values where v is not aligned with the stable or unstable direction. In this setting we expect θ = 1. The variation from the expected value for each realization is at most O(10 −2 ).

Discussion: towards more general observables and non-uniformly hyperbolic systems
In this article we have focused on hyperbolic systems with invariant measures absolutely continuous with respect to Lebesgue measure and considered observables whose level sets S shrink to a non-trivial extremal set S, such as a line segment. We recall that S = {x ∈ X : d H (x, S) }, and d H (x, S) is the Hausdorff distance from x to S. Thus if S is a smooth curve, then for this metric d H we see that S is a thin tube of width around S. The observable φ : X → R we have assumed to be given by φ(x) = f (d H (x, S)), for some smooth function f : [0, ∞) → R, maximized at 0, e.g. f(u) = −log u.
As explained in section 2, our methods apply to the case where S is a line segment. We conjecture that our methods extend to the case where S is a smooth curve, assuming some transversality conditions of S relative to the global stable/unstable manifolds of the system. We have also considered seemingly non-generic geometrical cases, e.g. where the line segment S aligns precisely with the global stable/unstable manifolds. For hyperbolic automorphisms of the two-torus, we established the limit laws that arise in these scenarios. More generally, it is natural to consider observables whose extremal set S is no longer (strictly) transverse to the global stable/unstable manifolds, i.e. there exist points of tangency between S and the global manifolds.
For (non-uniformly) hyperbolic systems ( f, Λ, μ) where Λ is an attractor the SRB measure μ may not be equivalent to Lebesgue. These systems include Axiom A systems, or Hénonlike attractors whose statistical properties (such as mixing rates) are established in [43]. As outlined in section 1.1, there is an established literature on extreme value theory in the nonuniformly hyperbolic setting for observables whose extremal set S is a point. Recently some progress has been made on more complicated geometries for S [28] but in a very axiomatic way. In the case where S is a line (or in higher dimensions a planar set), then we expect S to (generically) intersect a fractal attractor Λ in a Cantor-like set. For such a set, there are various difficulties that arise when trying to find the limit extreme value distribution distribution, in the sense of establishing (1.2), or in particular the limit law given by (1.4). If we suspect that a limit law of the form given in equation (1.2) is going to exist, then finding the scaling sequence u n is a first problem. For a specified observable φ (i.e. through specifying f), the properties of the sequence u n depend on the asymptotic properties of μ(S ) as → 0. To estimate this measure, we cannot use local dimension estimates, and finer arguments are required based on the geometric properties of μ. Furthermore, existence of a GEV limit of the form (1.4) is not guaranteed, as this requires μ(S ) to satisfy conditions of regular variation in (as → 0), see [36, chapter 3]. Axiomatic approaches, e.g. [5,28,32] suggest that once we've found these scaling laws then an extreme value law holds in the sense of equation (1.2). However, verification of these axioms still requires fine analysis. This includes verification of axiomatic conditions involving transversality of S with Λ, and conditions involving how μ behaves on certain shrinking sets (such as thin annuli) on a case-by-case basis.