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Extreme value distributions for one-parameter actions on homogeneous spaces

In this paper we study extreme value distributions for one-parameter actions on homogeneous spaces of Lie groups. We study both shortest vectors in unimodular lattices, maximal distance excursions and closest distance returns of a one-parameter action. For certain sparse subsequences of the one-parameter action and by taking the maximum over a moving interval of indices we prove non-trivial estimates for the limiting distribution in all cases.


Introduction
Say we have a dynamical system consisting of a probability space (X, µ), a flow ϕ t : X → X, t ∈ Ê and a metric d. It is of interest to study the maximal distance that a typical orbit of the system gets away from a fixed point in the space in a certain period of time. That is, for R ∈ Ê and some fixed point x 0 ∈ X. In particular, we are keen to understand the behaviour of this maximal distance as R → ∞. The growth rate of M R is well understood in many interesting cases. A well-known result of this type is Sullivan's logarithm law for the maximal excursions of the geodesic flow on a cocompact, finite volume quotient of the real hyperbolic space, [14]. Kleinbock and Margulis [11] generalized the result of Sullivan by proving that the logarithm law holds for one-parameter actions on a certain class of cocompact, finite volume homogeneous spaces and for a more general class of functions than d(·, x 0 ). Later, Athreya, Ghosh and Prasad [1], [2] proved ultrametric analogues of Kleinbock and Margulis' logarithm law.
In this paper we focus on the setting of Kleinbock and Margulis' logarithm law, which we formally define in Subsection 1.1. Since the growth rate of M R is already well understood in this case it is natural to ask more precise questions about the behaviour of M R . We do this by studying the distribution of M R , that is the function F M R : Ê → Ê given by Again we are keen to understand what happens when R → ∞, that is, to determine the existence and form of the limit lim R→∞ F M R . A result of this kind was proven by Pollicott This research was supported by ERC grant 239606. [13] who was able to give an exact asymptotic formula for the distribution of the maximal distance excursions of the geodesic flow on the modular surface. On the one hand Pollicott's statement is more precise than that of Sullivan, but on the other hand, it holds true only in a special case of Sullivan's setting.
The result of Pollicott is known as an extreme value distribution (EVD). One way to approach the task of determining EVD's in dynamical systems is to establish quantitative mixing properties of the system and use these along with some classical results from the field of extreme value theory. In the following subsection we elaborate on this idea.
Extreme value theory and quantitative mixing. For a real-valued stochastic process we may study two types of extremes, namely the minimum or the maximum among a certain collection of random variables in the process. The limiting distribution of either extreme is known as an EVD. The conventional approach to determining an EVD is through the techniques of extreme value theory. This is a branch of statistics which deals with the distributional properties of extreme events in stochastic processes. When we consider identically distributed stochastic processes, the theory roughly splits into two parts, the case of independent and the case of dependent stochastic processes. The independent case is simple and EVD's of independent, identically distributed processes are well understood. The dependent case is significantly more complicated and complex. If we make the stronger assumption that our stochastic process is stationary, we obtain EVD's when the dependence is sufficiently weak. Here "sufficiently weak" means that the stochastic process satisfies two conditions commonly known as Condition D and D ′ . For further details on these conditions and extreme value theory, see [12]. For a short resume on extreme value theory, see [9]. Consider a dynamical system consisting of a probability space (X, µ), a flow ϕ t : X → X, t ∈ Ê and an observable D : X → Ê. Assume that µ is ϕ t -invariant. We are interested in the stochastic process defined by the random variables ξ t = D(ϕ t (x)), t ∈ Ê which we refer to as the stochastic process arising from the dynamical system. Notice that ϕ t -invariance of µ implies that ξ t is stationary. The prospect of proving Condition D and D ′ for ξ t depends on the type and rate of mixing known for the system as well as the nature of the observable D. For many interesting dynamical systems, EVD's have been established. The first example came through the work of Collet [4] who proved an EVD for the closest distance returns of typical orbits of a certain transformation of a closed interval. Collets result holds for systems satisfying certain hyperbolicity assumptions. It is known that systems satisfying these assumptions are exponentially mixing and Collet used this rate of mixing to prove that the aforementioned conditions D and D ′ were satisfied. Collet's work provided a blueprint for how to apply extreme value theory to dynamical systems using quantitative mixing properties. Since then, EVD's have been determined for many other interesting dynamical systems, see for example [3], [5], [6], [7], [8].
As mentioned previously, we are interested in determining EVD's in the setting of Kleinbock and Margulis' logarithm law. It is known that the rate of mixing in this setting is exponential. Despite this relatively fast rate, it proved beyond our capabilities to verify the aforementioned condition D in this setting. The problem we encounter when trying to verify Condition D is that the exponential mixing only holds true for observables that satisfy restrictive smoothness assumptions. This causes problems in our calculations and creates error terms which we are unable to control. Note that we are not suggesting that condition D cannot be proven in this setting, we are merely saying that we were unable to do so.
Without Condition D we do not have the results of extreme value theory available to us. Instead we take a more direct approach to estimating the EVD's of the system. Our idea is to apply the mixing property of the system directly to the distribution function of the extreme event. This results in an error term which we can control when we look at a special case. In the following we introduce our setup, including the special case mentioned.
1.1. Setup and main results. Let G denote a connected semi-simple Lie group with finite center and let Γ < G be an irreducible lattice such that G/Γ is not compact. Set X = G/Γ and let a t denote a one-parameter subgroup of G. Let µ denote the normalized Haar measure on X and let d denote the dimension of G. Let d be the Riemannian metric on X chosen by fixing a right invariant Riemannian metric on G which is bi-invariant with respect to a maximal compact subgroup of G. Let D : X → Ê denote a measurable function.
In order to control the the aforementioned error terms we need to restrict ourselves to the following special case. The first restriction is that we only look at a sparse sequence of a t . The second restriction is that the maximum which we study is taken over an interval of indices whose endpoints are increasing. More specifically, let m j ∈ Ê be a sequence and let α n < β n be sequences of natural numbers both going to ∞ with n. Instead of studying the maximum M R , we are looking at the maximum max mα n ≤i≤m βn D(a i x).
From here and onwards we fix the notation I n = {m αn , m αn+1 , . . . , m βn } for given sequences of natural numbers α n < β n and m j ∈ Ê. We then write the above maximum as We also fix the notation N n := β n − α n + 1 throughout the paper. The necessary conditions on m j , α n and β n vary depending on the specific setting and will be made explicit in each of the forthcoming theorems.
We are also interested in the k'th largest element for some k ∈ AE. That is, instead of studying the maximum, i.e. the largest among D(a mα n x), . . . , D(a m βn x) , we study the k'th largest element in this collection. We will denote the k'th largest element by max (k) and write M (k) We are now ready to present the main results of the paper. We prove results for maximal distance excursions as well as closest distance returns in the full generality of the setup described above. Furthermore, in the special case of X being the space of unimodular lattices in Ê d we are able to obtain a more accurate result.
1.2. Maximal distance excursions. We study the maximal distance excursions by looking at the observable for a fixed point x 0 ∈ X. We prove the following.
where C ∈ (0, 1] is an explicit constant. Also, let α n < β n be sequences in AE for which α n → ∞ and N n → ∞. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable. Then there exist positive constants w 1 , w 2 and v such that for every where u n (r) = r + 1 v log N n . For the k'th largest element we obtain the following result.
where C ∈ (0, 1] is an explicit constant. Set ρ = sup j∈AE (m j−1 /m j ) −1 . Also let α n < β n be sequences in AE for which α n → ∞ and N n → ∞ in such a way that where σ > 0 is an explicit constant. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable. Then there exist positive constants w 1 , w 2 and v such that for every where u n (r) = r + 1 v log N n .

1.3.
Closest distance returns. We study closest distance returns by looking at the observable for a fixed point x 0 ∈ X. We prove the following.
where C ∈ (0, 1] is an explicit constant. Set ρ = sup j∈AE (m j−1 /m j ) −1 . Also let α n < β n be sequences in AE for which α n → ∞ and N n → ∞ in such a way that where σ > 0 is an explicit constant. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable. Let u n (r) = r + 1 d log N n . Then there exists a positive constant w such that for every x 0 ∈ X lim n→∞ µ (M In ≤ u n (r)) = e −we −dr .
A simple example for which the conditions of Theorem 1.3 are satisfied is β n = 2n, α n = n and m j = q j for some sufficiently large q ∈ Ê. For some w > 0 we then have for For the k'th largest element we obtain the following result.
where C ∈ (0, 1] is an explicit constant. Set ρ = sup j∈AE (m j−1 /m j ) −1 . Also let α n < β n be sequences in AE for which α n → ∞ and N n → ∞ in such a way that where σ > 0 is an explicit constant. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable. Set u n (r) = r + 1 d log N n . Then there exists a positive constant w such that for every and set D = ∆. We choose to study the observable ∆ because it is of great importance to the connections between flows on L d and Diophantine approximation. Note that up to a change of variables ∆ returns the length of the shortest non-zero vector in the lattice, hence the title of this subsection. We obtain the following result.
Theorem 1.5. Let m j ∈ Ê be a fixed sequence such that where C ∈ (0, 1] is an explicit constant. Also, let α n < β n be sequences in AE for which α n → ∞ and N n → ∞. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable.
For the k'th largest element we obtain the following result. Theorem 1.6. Let m j ∈ Ê be a fixed sequence satisfying where C ∈ (0, 1] is an explicit constant. Set ρ = sup j∈AE (m j−1 /m j ) −1 . Also let α n < β n be sequences in AE for which α n → ∞ and N n → ∞ in such a way that where σ > 0 is an explicit constant. Assume that a t is partially hyperbolic and that Ad(a t ) is diagonalizable. Set u n (r) = r + 1 d log N n . Then for w = V d 2ζ(d) we have

Structure of the paper
We begin with Section 3 which contains all the necessary concepts, definitions and previous results. We also prove some basic estimates which will be used many times throughout the paper. In Section 4 we prove two general results from which Theorem 1.1, 1.3 and 1.5 will follow almost directly. In Section 5 we again prove two general results, which in this case imply Theorem 1.2, 1.4 and 1.6. Much of this section is repetition from Section 4 hence many repetitive details are left out and only the new ideas and differences are elaborated on.

Preliminaries
3.1. Asymptotic volume estimates. Let (X, µ) be a probability space and let D : X → Ê denote a measurable function. We define the tail distribution function of D as In order to prove the main results it is necessary to have some estimate on the tail distribution function of the observable in question. We define two types of observables according to the accuracy with which the asymptotics of their tail distribution function is known.
Definition 3.1. For positive constants w 1 , w 2 and v, we say that D is (w 1 , w 2 , v)-DL ("Distance-Like") if it is uniformly continuous and satisfies For positive constants w and v, we say that D is (w, v)-SDL ("Strong-Distance-Like") if it is uniformly continuous and satisfies The notion of distance-like functions was introduced by Kleinbock and Margulis in [11], and distance-like properties for two observables of particular interest are also proven therein. The first of these is the Riemannian distance to a fixed point which is a natural choice of observable when studying cuspital excursions. As seen in the following theorem, the setup of Kleinbock and Margulis is sligthly more general than that introduced in section 1.1.
Theorem 3.2 (Kleinbock and Margulis, [11]). Let G be a connected semisimple Lie group and Γ a non-uniform irreducible lattice in G. Let µ be the normalized Haar measure on G/Γ and d a Riemannian metric on G/Γ chosen by fixing a right invariant Riemannian metric on G bi-invariant with respect to a maximal compact subgroup of G. Let x 0 be an arbitrary fixed point in G/Γ. Then there exist positive constants w 1 , w 2 and v such that the observable d(·, The second observable considered in [11] is ∆ : L d → Ê, introduced in Section 1.4. Recall the well-known fact that L d is isomorphic to the homogeneous space SL(d, Ê)/SL(d, ).
We therefore write and in accordance with the focus of this paper, we think of L d as a homogeneous space.
The following is known about the tail distribution function of ∆.
Let (X, µ) denote a probability space which is also a metric space with a metric d. Another observable which we will study in this paper is − log d(·, x 0 ) with x 0 ∈ X an arbitrary fixed point. This is the observable we use to study closest distance returns. It is clear that small values of d(x, x 0 ) correspond to large values of − log d(x, x 0 ). Hence, by looking at the successive maxima of − log d(·, x 0 ) along some trajectory, we are actually looking at the closest distance returns of said trajectory. The choice of − log is not canonical and indeed we could have chosen any continuous functions f with the property that f (x) → sup x∈X f (x) for x → 0. However, − log turns out to be convenient from a technical point of view which is the reason why we make this choice.
The observable − log d(·, x 0 ) is not uniformly continuous and hence it fails to be neither SDL nor DL. However, we can still determine the asymptotics of its tail distribution function. Indeed, the following lemma is easy to prove. By a smooth measure we will mean a C ∞ function times the Lebesgue measure.
Lemma 3.4. Assume that X is a Riemannian manifold of dimension d with a smooth measure µ having positive density. Let x 0 ∈ X be an arbitrary fixed point and set Proof. Since X is a Riemannian manifold we can find a coordinate chart σ : X → Ê d such that σ(B r (x 0 )) = B r (0). Let ρ denote the density of µ with respect to the Lebesque measure m on Ê d . Then we can write Since ρ is smooth, we know that we can write ρ(x) = ρ(0) + O(r). Using this we get We conclude that In particular, with w = ρ(0) as z → ∞.
Remark 3.5. The assumption that the measure µ is smooth is stronger than necessary. However, in the settings where we apply this lemma, we are equipped with a smooth measure, hence we made this assumption in the lemma to simplify the proof.

3.2.
Exponential mixing for one-parameter flows. Decay of correlations often plays a central role in developing stochastic properties for a dynamical system. In this section we describe the rate of decay known for the one-parameter actions which we study.
We remind ourselves of the main setup introduced in Section 1.1. G denotes a connected semi-simple Lie group with finite center and Γ < G is an irreducible lattice such that G/Γ is not compact. a t denotes a one-parameter subgroup of G and µ is the normalized Haar measure on X := G/Γ induced by the Haar measure m on G. The dimension of G is denoted d and g is the Lie algebra of G.
The type of mixing which is known in this setting is exponential decay of correlations against the Sobolev norm of the observables. To precisely formulate this version of exponential mixing we need to define the Sobolev norm on the Sobolev space of X for which in turn we need to define what we mean by a derivative on X. We therefore define as follows Definition 3.6 (Derivative on X). The derivative of a function f : X → Ê in the direction of an element ζ ∈ g will be denoted D ζ f and is defined by S k (f ) and is given by Let {ζ 1 , . . . , ζ d } be a fixed basis of g. Forl = (n 1 , . . . , n l ) ∈ {1, . . . , d} ×l := L l we define the differential operator Dl = D ζn 1 . . . D ζn l . We may then define the set W (2,k) (X) = f ∈ L 2 (X) : Dlf ∈ L 2 (X) for alll ∈ L l and every 0 ≤ l ≤ k .
We then call the space W (2,k) (X), equipped with the norm S k , the Sobolev space of X. Definition 3.8 (Exponential mixing). We say that a t has exponential decay of correlations against W (2,k) Sobolev observables if for all f, g ∈ W (2,k) (X) there exist constants δ > 0 and c > 0 such that (3.1) In the setup described above we have exponential mixing. This is a consequence of different works by Harish-Chandra, Howe, Cowling and Katok-Spatzier. In the context of our work it is most convenient to use the following formulation of the exponential mixing property for a t . First we need to define what we mean by a t being partially hyperbolic. Definition 3.9 (Partially hyperbolic). We say that the one-parameter subgroup a t ⊂ G is partially hyperbolic if the adjoint action Ad(a t ) has at least one eigenvalue different from 1 in absolute value.

3.2.1.
Smooth approximations of characteristic functions. Theorem 3.10 provides us with a fast rate of decay, but the upper bound being given by Sobolev observables causes a variety of problems when trying to estimate EVD's for one-parameter actions. The precise nature of these problems will be clear later, however, in general terms, the problem is that the observables to which we want to apply exponential mixing are not Sobolev functions but characteristic functions. Hence, we can not use the exponential mixing property of a t directly. First we need to approximate the characteristic functions by smooth functions. The smooth functions that we use is the convolution of a particular smooth function and a characteristic function. In the following we discuss some of the technicalities necessary for constructing suitable smooth approximations.
Recall that for functions ϕ : G → Ê and ψ : X → Ê, we define the convolution ϕ * ψ : X → Ê as Also, for any ζ ∈ g we have Proof. The first claim is an easy consequence of Fubini's Theorem, the definition of convolution and the fact that the action of G on X is measure preserving. Namely, we see that To prove the second part let ζ ∈ g and look at the integral We do a change of variables by setting h = exp(−tζ)g. Then Using this we can calculate the derivative of the convolution It clearly follows that Taking the L 2 -norm and using the Young inequality for convolutions we get It is then clear from the definition of the Sobolev norm that S k (ϕ * ½ A ) ≤ S k (ϕ)µ(A).
Let ε > 0 and A ⊂ X be measurable. We will approximate characteristic functions by where ρ denotes the density of the measure m with respect to the Lebesque measure on This function is easily seen to have the desirable properties that G ϕ ε dm = 1 and supp(ϕ ε ) ⊂ B (e, ε). Notice also that by doing a change of variables we get that for ε > 0 sufficiently small, We will often need the following estimate.
Lemma 3.12. For sufficiently small ε > 0 we have Proof. For arbitrary ζ r ∈ g we have In the following letx = ( Using the chain rule we compute the derivative, and notice that ψ is a smooth function. Let again ρ denote the density of µ with respect to the Lebesque measure on Ê d . From the above we get that It is easy to see that for higher order derivatives we get and from the definition of the Sobolev norm,

General estimates for EVD's of one-parameter actions
In this section we establish the general theory from which Theorem 1.1, 1.3 and 1.5 will follow. First we introduce the necessary notation and definitions. 4.1. Notation and setup. Let G denote a Lie group, Γ < G a lattice and set X := G/Γ. Let µ denote the Haar measure on X which is induced by the Haar measure m on G. We assume that Γ is chosen in such a way that X has finite measure with respect to µ. Assume also that µ has been renormalized to a probability measure. Let a t denote a one-parameter subgroup of G and let d denote the dimension of G. Let g denote the Lie algebra of G. As usual Ad(a t ) denotes the adjoint action of a t . If Ad(a t ) is diagonalizable, then there exists an eigenbasis W = {ζ 1 , · · · , ζ d } of g such that for every ζ r ∈ W there exists γ r ∈ Ê such that Ad(a t )ζ r = e γrt ζ r . (4.1) Let m j denote a sequence in Ê. Also, let α n and β n denote sequences of natural numbers for which α n < β n , α n → ∞ and β n − α n → ∞. Recall that N n := β n − α n + 1, . . , m βn } and for an observable D : We make the following assumptions throughout the section.
(1) Assume that Ad(a t ) is diagonalizable and fix an eigenbasis W := {ζ 1 , · · · , ζ d } for Ad(a t ). Set also throughout where γ i satisfies (4.1) (2) Assume throughout that the Sobolev norm S k is defined with respect to the fixed eigenbasis W . (3) Assume that the one-parameter action a t has exponential decay of correlations in the sense of Definition 3.1.

Maximal excursions.
In this subsection we prove a general result from which 1.1 and Theorem 1.5 will follow. In those theorems we were concerned with the observables d(·, x 0 ) and ∆(·). In this section we look at a general observable D : X → Ê which we characterize according to the asymptotic behavior of its tail distribution function. We state the first main theorem of this subsection.
where k, δ are as in (3.1) and γ as in Remark 4.2. For notational simplicity we will write u n instead of u n (r) whenever the dependence on r is not important.
Preparations for the proof of Theorem 4.1. In the following we prepare for the proof of Theorem 4.1. While the preparations are on the heavy side in terms of technicalities, they serve to lighten the proof of the theorem itself. The overall structure of the proof is described by the following three steps.
Step 1) Rewriting µ (M In ≤ u n ) as an integral of a product of characteristic functions and approximating the characteristic functions by smooth functions.
Step 2) Applying exponential decay of correlations to the integral of a product of smooth functions in order to obtain an expression for the error term between the integral of the product and the product of integrals.
Step 3) Deriving an estimate for the error term which shows that the error term vanishes in the limit and that the product of integrals of smooth approximations and the integral of the product of characteristic functions agree in the limit.
The technical and computationally tedious parts of these three steps are taken care of in this section.
For k ∈ AE and C > 0 we say that ψ ∈ C ∞ (X) is (C, k)-regular if For some function D : X → Ê, set V (r) = {x : D(x) ≤ r}. We are interested in approximating the characteristic functions ½ V (un) by (C, k)-regular functions. This is done in the following lemma. Lemma 4.3. Assume that D : X → Ê is uniformly continuous. Then for any k ∈ AE and any δ > 0 there exist ε > 0, C ε = O ε −( d 2 +k) > 0 and two (C ε , k)-regular functions g n,ε and h n,ε such that Proof. For ε > 0, define the sets Clearly V ′ (u n , ε) ⊂ V (u n ) ⊂ V ′′ (u n , ε). Let δ > 0 be given. It follows from uniform continuity of D that we can find an ε = ε(δ) > 0 such that d(x, y) < ε implies |D(x) − D(y)| < δ. This implies that V (u n − δ) ⊂ V ′ (u n , ε). To see this, assume first that x ∈ V (u n − δ), i.e. D(x) ≤ u n − δ. Let y ∈ ∂V (u n ) which means that D(y) = u n . Together this gives δ ≤ D(y) − D(x) which implies d(x, y) ≥ ε by uniform continuity. This shows that x ∈ V ′ (u n , ε). It follows by the same argument that V ′′ (u n , ε) ⊂ V (u n + δ). So we have that for any δ > 0 there exists an ε > 0 such that For every δ > 0 we define the functions g n,ε and h n,ε as with ϕ ε as defined in (3.3). For notational simplicity we will omit writing the dependency of ε on δ. By taking measures in (4.3) and using Lemma 3.11 we get The definition of g n,ε and h n,ε also implies g n,ε ≤ ½ V (un) ≤ h n,ε ≤ 1.

(4.4)
To see this, we first rewrite g n,ε as an integral, If x / ∈ gV ′ (u n , ε) then the first inequality is trivial. Assume therefore that x ∈ gV ′ (u n , ε). This means that we can write x = gy where y ∈ V ′ (u n .ε) and hence d(y, ∂V (u n )) ≥ ε. Since g ∈ B(e, ε) we see that gy ∈ V (u n ) which means that gV ′ (u n , ε) ⊂ V (u n ). This proves the first inequality in (4.4) and the second is proved the same way while the third is obvious.
The (C ε , k)-regularity of g n,ε follows directly from Lemma 3.11 with C ε = S k (ϕ ε ) since By the same argument h n,ε is (C ε , k)-regular. By Lemma 3.12 we have that for sufficiently Exponential mixing, as we define it in Definition 3.1, is also known as exponential 2mixing since it describes the exponential rate of decay of correlations between two observables. As will be clear later in the proof of Theorem 4.1, we will be given an integral of a product of multiple observables and we will be interested in estimating the decay of correlations of the entire product. In the following we show which estimate is obtained by repeatedly applying 2-mixing to said product.
Proof. For any C > 0 and k ∈ AE let ψ denote a (C, k)-regular function such that X ψ dµ ≤ 1. Set ψ t (x) = ψ(a mt x) and for any integers i 1 < i 2 , set The idea is to write and apply exponential decay of correlations to this product. Doing this we get We again apply decay of correlations, this time to F (αn,βn−1) (x) = F (αn,βn−2) (x)ψ(a m βn x) and we obtain the estimate Inserting this in (4.5) gives We continue by estimating X F (αn,βn−s) dµ for all 1 ≤ s ≤ β n −α n +1 using the exponential 2-mixing. The calculation terminates for s = β n − α n − 1, i.e. when we have the estimate Inserting all the estimates, one after the other, into (4.5) we get that Since we have assumed X ψ dµ ≤ 1, we can simplify the expression above to get Finally, for ε > 0 sufficiently small g n,ε and h n,ε are both (C ε , k)-regular functions with integral bounded by 1 and Lemma 3.11 and Lemma 3.12 imply that S k (g n,ε ) and S k (h n,ε ) are both bounded by O ε −( d 2 +k) . Hence, setting ψ = g n,ε or ψ = h n,ε completes the proof.
We need estimates on the error terms in Lemma 4.4 that are explicit in n and ε. For this we need to understand the Sobolev norms of the functions G (αn,c) and H (αn,c) for α n < c ≤ β n . This is the content of the next lemma.
Lemma 4.5. Assume that m j is increasing. Then for ε > 0 sufficiently small for any α n < c ≤ β n .
Proof. Throughout the proof, let ψ = ϕ ε * ½ A where A ⊂ X is measurable and ϕ ε is defined as in (3.3). Notice that ψ ≤ 1 since ϕ ε dm = 1. Set again ψ t (x) = ψ(a mt x) and for any integer α n < c ≤ β n , set In order to obtain an upper bound on S k (F ) we derive an upper bound on D ζn 1 · · · D ζn l F 2 which is independent of n 1 , . . . , n l . However, we begin by proving the lemma in the special case of k = 1 which we do by deriving an upper bound on D ζr F 2 which is independent of r. The k = 1 case is simpler in terms of notation but still demonstrates the main ideas of the general estimate. Let ζ r ∈ W be arbitrary. Using the product rule we get We can rewrite ψ s (exp(tζ r )x) as follows. = ψ(exp(te γr ms ζ r )a ms x).
Recall also the well-know fact that for any a ∈ Ê we have D aζr = aD ζr . Using this we get = D e γr ms ζr ψ(a ms x) = e γrms D ζr ψ(a ms x). (4.7) Inserting this in (4.6) gives, Taking the L 2 -norm we get where the last inequality holds since ψ ≤ 1. Recall that γ = max 1≤i≤d γ i and recall also from the proofs of Lemma 3.11 and 3.12 that D ζ ψ 2 ≤ D ζ ϕ ε 2 = O ε −( d 2 +1) . Then It follows from this that Having understood how to estimate the first Sobolev norm of F , we can proceed to estimate the k'th Sobolev norm of F . In this case we are looking at uniformly estimating the L 2norm of all derivatives of the type D ζn 1 · · · D ζn l F . Writing down an exact expression for the result of applying the product rule to F , l times, is notationally complicated and provides little enlightenment. Instead, consider that since F is a product of (c − α n + 1) functions, D ζn 1 · · · D ζn l F becomes a sum containing a total of (c − α n + 1) l terms. Fix a number 1 ≤ p ≤ (c − α n + 1) l . Then the p'th term of this sum is of the form D p equals l, or in other words, q(α n ) + · · · + q(c) = l, we get

It follows from equations (3.2), (3.4) and the Young inequality for convolutions that
Notice that for c−α n +1 ≥ l, at most l of the operators D = O e lγmc ε −l( d 2 +1) .
Since this bound holds for all 1 ≤ p ≤ (c − α n + 1) l we see that and consequently Since g n,ε and h n,ε both satisfy the definition of ψ, the bound on the Sobolev norm holds for both F = G (αn,c) and F = H (αn,c) .
Combining Lemma 4.4 and Lemma 4.5 we get the following estimates for the increasing sequence m j and all ε > 0 sufficiently small.  Notice that the assumption that lim j→∞ m j−1 m j < min 1, δ kγ means that m j is increasing for j ≥ j 0 for some sufficiently large j 0 . It then follows from (4.9) and (4.10) that for sufficiently large n and sufficiently small ε > 0, The task is now to determine the limit, if it exists, of the upper and lower bound. First we take care of the error term.
The assumption that lim s→∞ m s−1 ms < min 1, δ kγ means that there exist some s 0 ∈ AE and ρ > 1 such that for all s ≥ s 0 we have From this we see that Consequently, for all s ≥ s 0 , Then, since α n → ∞ for n → ∞, we see that for The ratio test shows that the series converges and hence the right hand side goes to 0 as n → ∞. This proves that for n → ∞.
Returning to the main terms, we claim that To see this we first look at the limits for n → ∞. Recall from Lemma 4.3 that for any δ > 0 we can find ε = ε(δ) > 0 such that and hence Recall that u n (r) = r + 1 v log N n . Using the (w, v)-SDL property of D we get, To compute the limit of the upper and lower bound we first rewrite as follows, We then estimate the right hand side using the second order Taylor expansion of log(1 + x), that is, log(1 + x) = x + O(x 2 ). Inserting the resulting estimate and taking the limit for n → ∞ gives Therefore the claim is proved by letting δ → 0. It then follows from (4.14) and (4.13), that when we take the limit for n → ∞ and then the limit for δ → 0 in (4.12), we get lim n→∞ µ (M In ≤ u n ) = e −we −vr which proves part A) of the theorem.
The proof of part B) is essentially identical to the proof of part A). The only real difference occurs following equation (4.15) where we apply the DL property instead of the SDL property. For some positive constants w 1 , w 2 and v we then get which gives the desired inequalities when raising to the power N n and taking the lim and lim for n → ∞. In fact, the proof of part B) by itself would be easier as we can leave ε fixed since we do not have hopes of an exact limit. This would simplify the estimates of the Sobolev norms in Lemma 4.5 as we would not have to keep track of the dependency on the parameter ε.

4.3.
Closest distance returns. In this subsection we prove a general result from which Theorem 1.3 will follow. Recall that, for a fixed point x 0 ∈ X, does not satisfy the criteria for being a (w, d)-SDL function since it is not uniformly continuous. However, recall from Lemma 3.4 that for some w > 0 as z → ∞. The following result is a closest return analogue of Theorem 4.1, but the lack of uniform continuity means that we need to make stronger assumptions. The statement is as follows. Since the conditions of Theorem 4.6 are rather technical, we give a simple example for which the conditions are met.
Example 4.7. Let q ∈ Ê and set m j = q j . It is then clear that for sufficiently large q ∈ Ê we have the inequality With this choice of m j we see from condition (4.16) that the assumptions of Theorem 4.6 are satisfied when we make the simple choice β n = 2n and α n = n. It then holds true that for some w > 0, Notice that we can pick α n and β n such that N n grows much faster. However, we made the given choice to obtain a simple and intuitive statement. Lemma 4.8. For every k ∈ AE, any ω > 0 and any n ∈ AE we can find two (C n,ω , k)-regular functions g n,ω and h n,ω such that Proof. For any ω > 0 we have Set ε = ε n,ω = e −un (1 − e −ω ) and let ϕ ε be defined as in (3.3). We then define the functions g n,ω and h n,ω as g n,ω = ϕ ε * ½ B(x 0 ,e −un+ω ) c and h n,ω = ϕ ε * ½ B(x 0 ,e −un−ω ) c .
Hence gy ∈ B(x 0 , e −un ) c and gB(x 0 , e −un+ω ) c ⊂ B(x 0 , e −un ) c . The second inequality of (4.18) is proved similarly and the third inequality is trivial. The (C n,ω , k)-regularity of g n,ω follows directly from Lemma 3.11 with C n,ω = S k (ϕ ε ) since By the same argument h n,ω is (C n,ω , k)-regular.  For any integers i 1 < i 2 , set

Lemma 4.8 and equation (4.19) then imply that
For g n,ω and h n,ω the analogues of (4.9) and (4.10) are, that for sufficiently large n, This implies Again, we want to determine the limit, if it exists, of the upper and lower bound as n → ∞ and ω → 0 and we begin by looking at the error term. First we make the trivial estimate We now look at the sum in the error term. Since sup s∈AE m s−1 ms = 1 ρ we see that for all s ∈ AE m s ≥ ρm s−1 ≥ · · · ≥ ρ s m 0 , and consequently, Set σ ′ = −m 0 kγ ρ − δ > 0. Now, since ρ > 1, we know that for s ∈ AE sufficiently large we have that ρ s+i ≥ ρ s + i and hence Consequently, for n ∈ AE sufficiently large we can write (4.21) Finally we look at the rest of the error term. Recalling from Lemma 4.8 that ε n,ω = e −un (1 − e −ω ), we get Here we considered r and ω constants since we are currently focusing on the limit as n → ∞. Hence for n ∈ AE sufficiently large we have, for n → ∞. The convergence follows from the assumption in (4.16).
For the main terms we claim that Recall that u n (r) = r + 1 d log n. Raising to the power N n we get X g n,ω dµ To compute the limit of the two right hand sides above, we first rewrite as follows We then estimate the right hand side using the second order Taylor expansion of log(1 + x), that is, log(1 + x) = x + O(x 2 ). Inserting the resulting estimate and taking the limit for n → ∞ gives, The claim is then proved by letting ω → 0. It then follows from (4.23) and (4.24) that when we take the limit for n → ∞ and then for ω → 0 in (4.20), we get This proves the theorem.

General theory for exceedances by one-parameter actions
We now proceed to the general theory from which Theorem 1.2, 1.4 and 1.6 will follow. Many aspects of the proofs are similar to the proofs presented in Section 4, but some alterations and new ideas are necessary.
First we need to rewrite the distribution function of the k'th largest maximum of a general stationary process ξ i . The computation is basic and essentially only contains straight forward set and measure theoretic arguments. However, the resulting expression from this derivation will be used often throughout this section, hence we have dedicated a separate section to it. 5.1. The distribution of the k'th largest maximum for a stationary process. Let (X, P) denote a probability space. Let ξ i be a stationary sequence of random variables on X. For any subset I ⊂ AE we will denote the k'th largest maximum by by which we mean the k'th largest ξ i for i ∈ I where k ≤ |I|. We are interested in the limiting distribution of M We can write this set as a union of disjoint sets as well, but first we need to introduce some additional notation. In the following let m j be a subsequence in AE, let a < b be integers As an example we see that for i = 2 this union simplifies to As the sets in the union are clearly disjoint we get Re-inserting this in (5.1) gives This way of writing the distribution of the k'th largest maximum will be used several time throughout the section. We notice that a way of deriving an estimate on the limiting distribution of the k'th largest maximum is to derive an estimate on the probability which is independent of the choice of J ∈ S (i) I .

Notation and setup.
We return to the setup introduced in Section 4.1 which we briefly recall. m j denotes a fixed sequence in Ê. α n and β n denote sequences for which α n < β n , α n → ∞ and β n − α n → ∞. Set N n := β n − α n + 1 and I n = {m αn , m αn+1 , . . . , m βn }. We also recall the general assumptions of the section, namely (1) Assume that Ad(a t ) is diagonalizable and fix an eigenbasis W := {ζ 1 , · · · , ζ d } for Ad(a t ). That is for every ζ r ∈ W there exists γ r ∈ Ê such that Ad(a t )ζ r = e γrt ζ r .
Set also throughout where γ i satisfies (5.3) (2) Assume that the Sobolev norm S k is defined with respect to the fixed eigenbasis W . (3) The one-parameter action a t will be assumed to have exponential decay of correlations as defined in Definition 3.1.

Maximal excursions.
In this section we prove a general result from which Theorem 1.2 and 1.6 will follow. Let D : X → Ê denote a measurable function and set In this section we will look at the l'th largest maximum to avoid notational confusion with the degree of the Sobolev norm for which we use the parameter k.
We are ready to state the first main theorem of the section.
The idea is to derive upper and lower bounds on the probability that are independent of the choice of J ⊂ I n . We do this similarly to how we estimated µ ξ mα n ≤ u n , . . . , ξ m βn ≤ u n in the proof of Theorem 4.1. The procedure is to a great extent identical, however, some alterations are necessary. Set V (u n ) = {ξ 0 ≤ u n } and U(u n ) = V (u n ) c = {ξ 0 > u n }. In Lemma 4.3 we approximated the function ½ V (un) by smooth functions. In this case we need to approximate ½ U (un) as well. Fortunately, this follows almost immediately from the proof of Lemma 4.3.
Lemma 5.2. Assume that D : X → Ê is uniformly continuous. Then for any k ∈ AE and any δ > 0 there exist ε > 0 and C ε = O ε −( d 2 +k) > 0 such that for every r ∈ Ê one can find two (C ε , k)-regular functions g ′ n,ε and h ′ n,ε on X such that g ′ n,ε ≤ ½ U (un(r)) ≤ h ′ n,ε ≤ 1 and Proof. The proof is identical to the proof of Lemma 4.3. With ε = ε(δ) > 0 being as in Lemma 4.3, the approximating functions are defined as if j ∈ J h n,ε if j ∈ I n \J. and also, for integers i 1 < i 2 The following is an adaptation of Lemma 4.4. Proof. Let J ⊂ I n be an arbitrary subset of cardinality i − 1. In the following, let ϕ and ψ denote two (C, k)-regular functions on X. Define the function Define also the function As in Lemma 4.4, the idea is to write and use the exponential decay of correlations on this product. Indeed, the proof of the lemma is identical to that of Lemma 4.4 with the only exception being that we do not apply the estimate X η (j) J dµ ≤ 1. We leave out the details to avoid unnecessary repetition. We need estimates on the error terms in Lemma 5.3 that are explicit in n and ε. For this we need to understand the Sobolev norm of the functions G J (αn,s−1) and H J (αn,s−1) . The following is an adaptation of Lemma 4.5.
Lemma 5.5. Let J ⊂ I n be any subset of cardinality i−1 and let i 1 = |J ∩ {m αn , . . . , m c }|. Then for any α n < c ≤ β n we have that for ε > 0 sufficiently small, Proof. This proof bears many similarities to the proof of Lemma 4.5, hence in the following we leave out several details which have already been discussed. First we prove the statement for S k (G J (αn,c) ).
From this expression and the fact that µ(U(u n )) is decreasing with n it is clear that the error term is maximal when i 1 is maximal, which happens for choice J = {m αn , . . . , m αn+i−1 }. Picking i 1 = i − 1, i 2 = 0 and i 3 = 0 then gives us an error term which is independent of J and is given by Since we consider ε > 0 fixed, the result follows.
This completes the necessary adaptations and we are ready to prove Theorem 5.1.   The task is now to determine the limit of the upper and lower bound. We first take care of the error term. Recall that S → we −v(δ+r) i−1 (i − 1)! e −we −vr for n → ∞.
Here we estimated the second integral similarly to the computation following equation (4.14). By equivalent calculations we get we v(δ−r) i−1 (i − 1)! e −we −vr for n → ∞.
Looking again at (5.5) we see that we need to sum over i and take the limit for n → ∞ in (5.15). This gives  The proof proceeds by making the same derivations as in equation (5.11) and through to equation (5.15). The only difference is in the estimate of the error term which in this case is The rest of the proof is identical to the to last part of the proof of Theorem 5.1.

5.5.
Finalizing proofs of main results. We already discussed in Section 4.4 that in the setups of Theorem 1.2, 1.4 and 1.6, a t has exponential decay of correlations. This follows from Theorem 3.10. Also, Theorem 3.3 gives that ∆ is (w, d)-SDL for w = V d 2ζ(d) and Theorem 3.2 gives that d(·, x 0 ) is (w 1 , w 2 , v)-DL for some positive constants w 1 , w 2 and v. Hence Theorem 1.2 and 1.6 follow from Theorem 5.1 part B) and A) respectively while Theorem 1.4 follows from Theorem 5.7.