• Ешқандай Нәтиже Табылған Жоқ

Escape of mass and entropy for diagonal flows in real rank one situations

N/A
N/A
Protected

Academic year: 2023

Share "Escape of mass and entropy for diagonal flows in real rank one situations"

Copied!
40
0
0

Толық мәтін

(1)

IN REAL RANK ONE SITUATIONS

M. EINSIEDLER, S. KADYROV, AND A. POHL

Abstract. LetGbe a connected semisimple Lie group of real rank 1 with finite center, let Γ be a non-uniform lattice in Gand a any diagonalizable element inG. We investigate the relation between the metric entropy of a acting on the homogeneous space Γ\G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration ofa) which miss a fixed open set is not full.

Contents

1. Introduction 1

2. Fundamental domains in the cusps 3

3. The height function 4

4. Coordinate system for G 6

5. Variation of height 7

6. Common cusp excursions of nearby points 11

7. Estimate of metric entropy and proof of Theorem A 16 8. Hausdorff dimension of orbits missing a fixed open subset 27

9. Modification of the partition from [EL10] 38

References 40

1. Introduction

LetGbe a connected semisimple (real) Lie group of R-rank 1 with finite center and Γ a lattice inG. Suppose that

X:= Γ\G

denotes the associated homogeneous space. LetAbe a one-parameter subgroup consisting of R-diagonalizable elements. Pick an element ˜a ∈ Ar{id} and

2010 Mathematics Subject Classification. Primary: 37A35, Secondary: 28D20, 22D40.

Key words and phrases. escape of mass, entropy, diagonal flows, Hausdorff dimension.

M.E. acknowledges the support by the SNF (Grant 200021-127145). S.K. acknowledges the support by the EPSRC. A.P. acknowledges the support by the SNF (Grant 200021-127145) and the Volkswagen Foundation.

1

(2)

consider the right action

T:

X → X x 7→ x˜a

of ˜aonX. Further let (µn)n∈Nbe a sequence ofT-invariant probability measures on Xwhich converges in the weak* topology to the measure ν.

If ν is itself a probability measure (which is always the case if Γ is cocompact), then upper semi-continuity of metric entropy is well-known, that is

lim sup

n→∞ hµn(T)≤hν(T).

In this article we investigate the case that Γ is non-cocompact and ν is not a probability measure. We show that if upper semi-continuity does not hold, the amount by which it fails is controlled by the escaping mass. More precisely, the main result can be stated as follows.

Theorem A. Let hm(T) denote the maximal metric entropy ofT and suppose that ν(X)>0. Then

ν(X)h ν

ν(X)(T) +12hm(T)·(1−ν(X))≥lim sup

n→∞ hµn(T).

In [KP] it is shown that the factor 12 is sharp. A consequence of this theorem is the following result about escape of mass, which is of interest on its own.

Corollary. Suppose that lim suphµn(T)≥c. Then ν(X)≥ 2c

hm(T) −1.

Thus, if the entropy on the sequence (µn) is high, meaning at least 12hm(T) +ε, then not all of the mass can escape and the remaining mass can be bounded quantitively.

ForX= SL2(Z)\SL2(R) andT being the time-one map this control on escape of mass is already shown in [ELMV12]. For recent results of this kind in different settings and their applications we refer to [EK12,Kad12, KKLM].

In case of equality in the corollary above, Theorem A yields the following con- sequence for the remaining normalized measure.

Corollary. If lim suphµn(T)≥c and ν(X) = 2c

hm(T) −1>0, then h ν

ν(X)(T) =hm(T) and ν(νX) is the Haar measure on X.

As an application of these results and the methods for their proofs we show in Section 8 the following observation, thereby answering a question of Barak Weiss. Its positive solution is already used in [HW13].

Theorem B. Let O be an open nonempty subset of X, and let E be the set of points in X whose forward trajectories (forward A-orbits) do not intersect O. Then the Hausdorff dimension of E is strictly smaller than the (Hausdorff ) dimension of X.

(3)

We outline the strategy of proof for Theorem A. The key tool for its proof is the existence of a finite partition η of X such that for each T-invariant probability measure µ on Xthe entropy of µ, the entropy of the partition η and the mass

“high” in the cusps ofXare seen to be related as in Theorem A. More precisely, if X>s denotes the part of X above height s(the notion of height is defined in Section3 below), then

hµ(T)≤hµ(T, η) +cs+12hm(T)µ(X>s)

with a global constant cs such that cs → 0 as s → ∞. We remark that η is independent of µ. To achieve this we use a partition of Xinto a fixed compact part, the partX>s above heights, and the strip between the compact part and X>s. The compact part is refined into very small sets, depending on the width of the strip, such that this part and the strip do not contribute to entropy.

The entropy of µis estimated from above using the Brin-Katok Lemma, which reduces this task to counting Bowen balls needed to cover some set of fixed positive measure. In Lemma 7.4 below we provide a non-trivial bound for this number. In order to be able to establish this result, we translate the situation to Siegel sets inG(which is possible thanks to a result of Garland and Raghunathan [GR70] on fundamental domains), and conduct a detailed study how nearby trajectories behave high up in the cusp.

These investigations do not use the classification ofR-rank 1 simple Lie groups.

Rather we take advantage of the uniform and easy to manipulate construction of rank 1 symmetric spaces of noncompact type provided by [CDKR91] and [CDKR98] and the coordinate system of the associated Lie groups adapted to their geometry.

Acknowledgment. We thank the anonymous referees for many valuable com- ments that helped to improve the presentation of the paper.

2. Fundamental domains in the cusps

Let A be a one-parameter R-diagonalizable subgroup in G containing the di- agonalizable element ˜a defining the transformation T on X via x 7→ x˜a. Let C = CA(G) denote the centralizer of A in G and let c be its Lie algebra. Let g denote the Lie algebra of G. Since G is of R-rank 1, there exists a group homomorphism α:A→(R>0,·) such that with

gj :=n X ∈g

∀a∈A: AdaX=α(a)j2Xo

, j∈ {±1,±2}, we have the direct sum decomposition

(1) g=g−2⊕g−1⊕c⊕g1⊕g2.

We choose the homomorphism α such that α(˜a)>1. The Lie algebra g is the direct product of a simple Lie algebra and a compact one. Unless this simple Lie algebra is isomorphic to so(1, n), the homomorphism α is then unique and (1) is the restricted root space decomposition of g. If the simple factor of g is isomorphic to so(1, n) for some n∈N, n≥2, then there are two choices for α.

Depending on the choice, either g2 or g1 is trivial. In this case (1) simplifies to g=g−1⊕c⊕g1 resp. g=g−2⊕c⊕g2,

(4)

each of which is the restricted root space decomposition of g. The first one corresponds to the Cayley-Klein models of real hyperbolic spaces, the second one to the Poincar´e models. Definen:=g2⊕g1 and letN be the connected, simply connected Lie subgroup of G with Lie algebra n. By the theorem concerning Iwasawa decompositions of G, there exists a maximal compact subgroupK of G such that

N ×A×K →G, (n, a, k) 7→nak is a diffeomorphism. Let

M :=K∩C.

For any s >0 we set

As:={a∈A|α(a)> s}.

Moreover, for anys >0 and any compact subsetη ofN we define the Siegel set Ω(s, η) :=ηAsK.

Garland and Raghunathan provide the following result on fundamental domains for the non-cocompact lattice Γ in G.

Proposition 2.1 (Theorem 0.6 and 0.7 in [GR70]). There exists s0 > 0, a compact subset η0 of N and a finite subset Ξ of G such that

(i) G= ΓΞΩ(s0, η0),

(ii) for all ξ∈Ξ, the group Γ∩ξN ξ−1 is a cocompact lattice inξN ξ−1, (iii) for all compact subsets η of N the set

{γ ∈Γ|γΞΩ(s0, η)∩Ω(s0, η)6=∅}

is finite,

(iv) for each compact subset η of N containing η0, there exists s1 > s0 such that for all ξ1, ξ2 ∈ Ξ and all γ ∈ Γ with γξ1Ω(s0, η)∩ξ2Ω(s1, η) 6= ∅ we have ξ12 and γ ∈ξ1N M ξ1−1.

For the remainder of this article we fix s1 > s0 >0, a compact subsetη0 of N and a finite subset Ξ of Gwhich satisfy (i)-(iv) of Proposition 2.1with η:=η0. The elements of Ξ are a minimal set of representatives for the cusps of

X:= Γ\G,

and for each ξ ∈Ξ, the Siegel set ξΩ(s1, η) modulo Γ∩ξN M ξ−1 is a neighbor- hood of the corresponding cusp ofX. In the following we will often identify this cusp with its neighborhood (Γ∩ξN M ξ−1)\ξΩ(s1, η)⊆X, and also refer to the latter one as the cusp represented byξ.

3. The height function

For each ξ ∈Ξ, we introduce a height function which measures how far a point x∈Xis “in the cusp represented by ξ”. More precisely, theξ-height ofx is the maximal valueα(a) for anx-representativeξnakinG=ξN AK. The maximum over allξ-heights gives the total height ofx∈X. For a coordinate-free definition of the height functions, we introduce a representation derived from the adjoint representation. This representation was also used in [Dan84].

(5)

For each ξ ∈Ξ we set

Lξ :=ξN M ξ−1

and denote its Lie algebra by lξ. Set ℓ:= dimlξ (which in fact is independent of ξ) and letV be theℓ-th exterior power ofg,

V :=^

g.

Let ̺be the right1 G-action onV given by theℓ-th exterior power of Ad◦ (·)−1:G→End(g), g7→Adg−1,

hence

̺:=^

Ad◦ (·)−1

:G→End(V).

We fix a non-zero element vξ in the one-dimensional space Wξ:=^

lξ and let

θξ:ξN M Aξ−1→R>0

be the unique group homomorphism into the multiplicative group (R>0,·) such that for allg∈ξN M Aξ−1 we have

vξ̺(g) =θξ(g)vξ.

One easily shows thatθξ(g) = 1 for g in the connected component of Lξ, and θξ(ξaξ−1) =α(a)(12dimg1+dimg2)

fora∈A. Let

q:= 12dimg1+ dimg2.

We choose a̺(K)-invariant inner producth·,·ionV (e.g. induced by the Killing form) and denote its associated norm by k · k.

For ξ∈Ξ, the ξ-height ofx∈Xis defined as (2) htξ(x) := sup

(kvξ̺(g)k kvξ̺(ξ)k

1q

g∈G, x= Γg )

.

If g ∈ G is represented as g = ξnak with n ∈ N, a ∈ A and k ∈ K, then by definition

kvξ̺(g)k kvξ̺(ξ)k

1q

=α(a).

Hence this value only depends on the A-components of g when represented in ξN AK(=G), of which we may think as an Iwasawa decomposition ofGrelative to ξ.

Theheight of x∈Xis

ht(x) := max

htξ(x)

ξ ∈Ξ . For s >0 andξ ∈Ξ we set

X(ξ, s) :={x∈X|htξ(x)> s}

1When applying̺(g) forgGtovV we will writev̺(g) instead of̺(g)vto stress that it is a right action.

(6)

and

(3) X>s:={x∈X|ht(x)> s}= [

ξ∈Ξ

X(ξ, s).

In the following we will see that the points inX(ξ, s) correspond to the elements in the Siegel set ξΩ(s, η). To that end let Bδ denote the open k · k-ball in V with radiusδ >0, centered at 0. We define

δξ(s) :=s−qkvξ̺(ξ)k.

Proposition 3.1 (Corollary 2.3 in [Dan84]). Let ξ ∈ Ξ, s > 0, and g ∈ G.

Then Γg∈Γ\ΓξΩ(s, η) if and only if vξ̺(γg)∈Bδξ(s) for some γ ∈Γ. Further, if s≥s1 and γ1, γ2 ∈Γ satisfy vξ̺(γjg) ∈ Bδξ(s) for j = 1,2, then vξ̺(γ1g) ∈ {±vξ̺(γ2g)}.

Thus

X(ξ, s) = Γ\ΓξΩ(s, η)

for all ξ ∈Ξ and s >0. If s≥s1, the supremum in (2) is attained. Moreover, by Proposition 2.1(iv),

X(ξ, s)∩X(ξ, s) =∅

if ξ 6= ξ ∈ Ξ. Hence the sets X(ξ, s) are then disjoint neighborhoods of the cusps of X, and the union in (3) is disjoint.

4. Coordinate system for G

Recall that the Lie algebra g is the direct sum of a simple Lie algebra of rank 1 and a compact one. Since the height function is right-̺(K)-invariant and all further considerations are right-̺(K)-invariant, we can restrict to g being simple. [CDKR91] and [CDKR98] provide a classification-free construction of all Riemannian symmetric spaces of noncompact type and rank one. Their results rely on the choice of a certain coordinate system for real simple Lie groupsGof real rank 1, which allows us to treat all these groups without refering to their classification. In the following we recall this coordinate system, the one for the associated symmetric spaces and some essential formulas.

The semidirect productN A is parametrized by

R>0×g2×g1 →N A, (s, Z, X)7→exp(Z+X)·as,

where we may assume s := α(as). The (left) action of as = (s,0,0) ∈ A on n= (1, Z, X)∈N is then given by

asn= (s, sZ, s1/2X).

We define an inner product on n=g2⊕g1 as follows. Letk be the Lie algebra of K. Let θ be a Cartan involution of g such that k is its 1-eigenspace. For X, Y ∈nwe define

hX, Yi:=− 1

dimg1+ 4 dimg2

B(X, θY)

whereB is the Killing form ofg. It is well-known thath·,·i is an inner product on n. As in [CDKR91,CDKR98], we identifyG/K∼=N A∼=R>0×g2×g1 with

D:=

(t, Z, X)D ∈R×g2×g1

t > 14|X|2

(7)

via

R>0×g2×g1 →D, (t, Z, X) 7→(t+14|X|2, Z, X)D.

We will include the subscriptDwhen denoting elements (·,·,·)D of the symmet- ric spaceDto avoid confusion with elements of the groupN A. The (left) action of an element s= (ts, Zs, Xs)∈N A on a pointp= (tp, Zp, Xp)D ∈Dbecomes s.p= tstp+14|Xs|2+12t1/2s hXs, Xpi, Zs+tsZp+ 12t1/2s [Xs, Xp], Xs+t1/2s Xp

D. These coordinates of G/K enable us to use [CDKR91, CDKR98], and they simplify some of the expressions below, in particular the one for the geodesic inversion. To state the geodesic inversion, we define the linear map

J: g2→End(g1), Z 7→JZ, via

hJZX, Yi=hZ,[X, Y]i for allX, Y ∈g1.

Then the geodesic inversionσ ofDato:= (1,0,0)D is given by (see [CDKR98]) σ(t, Z, X)D = 1

t2+|Z|2 t,−Z,(−t+JZ)X

D.

We identifyσwith an element inKwhich acts as geodesic inversion onD=G/K at o. Then Ghas the Bruhat decomposition ([CDKR98, Theorem 6.4])

G=N AM ∪N AM σN.

Multiplying this withξ ∈Ξ from the left and σ from the right, we get G=ξN AM σ∪ξN AM U

withU :=σN σ. This decomposition provides a coordinate system onGadapted to the cusp represented by ξ. The set ξN AM σ we call the small ξ-Bruhat cell and ξN AM U thebig ξ-Bruhat cell.

The group M is parametrized by the pairs (ϕ, ψ) consisting of the orthogonal endomorphisms ϕ on g2 resp. ψ on g1 such that ψ(JZX) = Jϕ(Z)ψ(X) for all (Z, X)∈g2×g1. The action of (ϕ, ψ)∈M on p= (t, Z, X)D ∈D is given by

(ϕ, ψ).p= (t, ϕ(Z), ψ(X))D.

By [CDKR98, Proposition 7.1],|JZX|=|Z||X|for all Z ∈g2,X∈g1. 5. Variation of height

Suppose that the pointx∈Xis of big height and its trajectory stays far out for some time. In this section, we provide non-trivial bounds on the unstable com- ponents of a group elementg∈Grepresentingx. In Proposition6.3below, this bound implies constraints on the perturbation allowed forxwithout destroying the qualitative behavior of its trajectory during this time.

Lemma 5.1. Let at, ar ∈A, m ∈ M and n ∈N with n= (1, Z, X) such that σmnat∈N arK. Then

r = t

t+14|X|22

+|Z|2.

(8)

Proof. By Iwasawa decomposition we know that σmnat = nark for suitable n ∈ N, k∈K and r ∈R>0. Suppose that m = (ϕ, ψ). Applying both σmnat and narkto the base pointo= (1,0,0)D inD, we find

σmnat·o=nark·o=nar·o.

In the coordinates ofD one easily calculates that σmnat·o=

= 1

t+14|X|22

+|Z|2 t+14|X|2,−ϕ(Z), −t−14|X|2+Jϕ(Z) ψ(X)

D. Suppose that n= (1, Z, X). Then

nar·o= r+14|X|2, Z, X

D. Thus

X = 1

t+14|X|22

+|Z|2 −t−14|X|2+Jϕ(Z) ψ(X),

|X|2 = 1 t+14|X|22

+|Z|22

t+14|X|22

|X|2+

Jϕ(Z)ψ(X) 2

−2 t+ 14|X|2 ψ(X), Jϕ(Z)ψ(X)

= |X|2

t+14|X|22

+|Z|2, and

r = t+14|X|2 t+14|X|22

+|Z|214|X|2 = t t+14|X|22

+|Z|2.

Lemma 5.2. Let ξ ∈ Ξ and g ∈ G. If g = ξnasmσ with n∈ N and m ∈ M,

then

kvξ̺(gat)k kvξ̺(ξ)k

1q

= s t. If g=ξnasmσ(1, Z, X)σ with n∈N and m∈M, then

kvξ̺(gat)k kvξ̺(ξ)k

1q

=s·

1 t 1

t +14|X|22

+|Z|2.

Recall the identification of the cusp represented by ξ ∈ Ξ with the cusp neigh- borhood (Γ∩ξN M ξ−1)\ξΩ(s1, η) from Section2. Let us note that the first case corresponds to a trajectory pointing straight out of the cusp represented by ξ.

In the second case, the element u = σ(1, Z, X)σ determines the perturbation to the trajectory pointing straight into the cusp. If (Z, X) = (0,0), the second case correspond to a trajectory pointing straight into the cusp, and the formula simplifies to

kvξ̺(gat)k kvξ̺(ξ)k

1q

=st.

(9)

Proof of Lemma 5.2. At first we suppose thatg=ξnasmσ. Then gat=ξnas/tmσ=ξnas/tξ−1ξmσ ∈ ξN Aξ−1

ξK . Hence

kvξ̺(gat)k=θξ(ξas/tξ−1)kvξ̺(ξ)k=s t

−q

kvξ̺(ξ)k.

Suppose now that g = ξnmasu with u = σnσ and n = (1, Z, X). Then (for somem∈M)

kvξ̺(gat)k=s−qkvξ̺(ξσmnσat)k=s−qkvξ̺(ξσmna1/tσ)k

=s−qkvξ̺(ξσmna1/t)k.

Lemma 5.1yields

σmna1/t=n′′ark for somen′′∈N,k∈K and

r =

1 t 1

t +14|X|22

+|Z|2. Thus,

kvξ̺(gat)k=

1 t 1

t +14|X|22

+|Z|2

!−q

s−qkvξ̺(ξ)k.

The following proposition describes the amount of time a trajectory spends in a neighborhood of the cusp represented by ξ.

Proposition 5.3. Let ξ ∈Ξand g ∈G. Write δ :=kvξ̺(g)k. If g∈ξN AM σ, then vξ̺(gat) ∈ Bδ if and only if t < 1. If g = ξnasmu ∈ ξN AM U with u=σ(1, Z, X)σ, then vξ̺(gat)∈Bδ if and only if

t∈ 1

1

16|X|4+|Z|2,1

!

∪ 1, 1

1

16|X|4+|Z|2

! .

If u= id, then 161|X|4+|Z|2−1

is to be understood as ∞.

Proof. The first part of the statement follows immediately from Lemma5.2. Sup- pose now that g=ξnmasu withu=σnσ and n = (1, Z, X). By Lemma 5.2, (4) kvξ̺(gar)k=

1 r 1

r +14|X|22

+|Z|2

!−q

s−qkvξ̺(ξ)k.

Applying (4) for r= 1 and r =t, we see that kvξ̺(gat)k<kvξ̺(g)k if and only if

1 1 +14|X|22

+|Z|2 <

1 t 1

t +14|X|22

+|Z|2, which is equivalent to

1−1

t −1 t + 1

16|X|4+|Z|2

<0.

(10)

This is the case if and only if

|Z|2+ 1

16|X|4 < 1

t <1 or |Z|2+ 1

16|X|4> 1 t >1.

Suppose that kvξ̺(gat)k = kvξ̺(γgat)k for some g ∈ G, γ ∈ Γ and all t in a non-trivial interval (ie., an interval which contains at least two points). Then Lemma5.2yields thatgandγghave the sameA-component inξN AKand they are in the same ξ-Bruhat cell. If moreover, g and γg are in the big ξ-Bruhat cell, then also the norms of theirU-components are equal. The following lemma shows that far out in the cusp much more is true.

Lemma 5.4. Let ξ∈Ξ and suppose that g∈G and γ ∈Γ are such that kvξ̺(g)k=kvξ̺(γg)k< δξ(s1).

Then γ ∈ξN M ξ−1. In particular, if g=ξnamσ resp. g=ξnamu with n∈N, a ∈ A, m ∈ M and u ∈ U, then γg = ξnamσ resp. γg = ξnamu for some n ∈N, m∈M.

Proof. By [Dan84, Lemma 2.2] (see also Proposition 3.1), for each s > 0 we have

LξξAsK=n g∈G

vξ̺(g)∈Bδξ(s)o .

Henceg, γg∈LξξAs1K. By [Dan84, Remark 1.3] (withηas in Proposition2.1), LξξAs1K = (Γ∩Lξ)ξηAs1K.

Hence there existγ1, γ2∈Γ∩Lξ,h1, h2 ∈ηAs1K such that g=γ1ξh1, γg=γ2ξh2.

Therefore

g∈γ1ξΩ(s1, η)∩γ−1γ2ξΩ(s1, η).

Proposition2.1(iv) yields γ1−1γ−1γ2∈ξN M ξ−1. Thus,γ ∈ξN M ξ−1. For the proof of the following proposition we recall that the supremum in the definition of ξ-height (2) is realized if htξ(x)≥s1.

Proposition 5.5. Let s > s1 and x∈X. Suppose that there exists an interval I in R such that ht(xat) > s for all t ∈ I. Then there exists a unique cusp representative ξ ∈Ξ and a (non-unique) element g∈Gwith x= Γg such that

ht(xat) = htξ(xat) =

kvξ̺(gat)k kvξ̺(ξ)k

1q

for all t ∈ I. Moreover, if 1 ∈ I and if there exists t ∈ I with t > 1 and ht(xat) > ht(x), then g = ξnarmu for some r > 0, n ∈ N, m ∈ M and u ∈ U. The elements ar and u do not depend on the choice of g. Finally, if u=σ(1, Z, X)σ, then

|X|<2t−1/4 and |Z|< t−1/2.

(11)

Proof. If y ∈Xand ξ ∈Ξ such that htξ(y) > s1, then there exists h∈ Gsuch that y= Γh and

htξ(y) =

kvξ̺(h)k kvξ̺(ξ)k

1q

. Since the function

R>0 → R

r 7→ kvξ̺(gar)k

is continuous, there exists an open neighborhood J of 1 in R>0 such that htξ(yar)> s1 for all r∈J. Forξ ∈Ξ let

Jξ:=

t∈I

htξ(xat)> s .

These sets are pairwise disjoint, open in I and cover I. Since I is connected, there exists a uniqueξ ∈Ξ with I =Jξ. Thus

ht(xat) = htξ(xat)

for all t∈I. For each t∈I pick an element gt∈Gsuch that x= Γgt and htξ(xat) =

kvξ̺(gtat)k kvξ̺(ξ)k

1q

.

Let Jtbe the set of p∈I such that htξ(xap) =

kvξ̺(gtap)k kvξ̺(ξ)k

1q

.

ThenI is covered by the sets Jt, and these are open inI by Proposition 3.1. If Jt and Jr overlap for some t, r∈I,t6=r, then Lemma 5.4 and 5.2 imply that Jt=Jr. In turn,Jt =I for each t∈I.

The remaining statements follow immediately from Proposition5.3and Lemma5.4.

6. Common cusp excursions of nearby points

For s >0 we define

X≤s:=X\X>s. Further we let

r0 :=α(˜a) and recall that r0 >1 by our choice of α.

Each connected component of Xof height above s1 can essentially be identified with a Siegel set (cf. Proposition 2.1). For the proof of the main theorem, trajectories of points x ∈ X are only considered time-discretized by the map T. In the following lemma we construct a height level s2 above which we can identify pieces of these discretized trajectories with trajectory segments in a Siegel set. More specifically, as soon as we know that two consecutive points of the discretized trajectory stay above height s ≥ s2, then the (continuous) trajectory segment of the corresponding geodesic also stays above height sand, in particular, does not visit the compact set X≤s

1. Then we construct a second height levels3 > s2such that any discretized trajectory enteringX>s

3 can locally be identified with a continuous trajectory segment in the Siegel set. In Section8

(12)

below this will be crucial to effectively determine the behavior of nearby starting trajectories. Of special importance for Section 7 below is the item (v) of the following lemma, which states that if we start to descend somewhere high in a cusp, then we actually descend up to below heights3.

Lemma 6.1. There exist s3 > s2 > s1 such that we have the following proper- ties:

(i) If x∈X>s

2, then ht(xat)>2s1 for all t∈[r−10 , r0].

(ii) If s≥s2 andx, T x∈X>s, then ht(xat)> s for allt∈[1, r0].

(iii) Let s > s3. If x ∈ X≤s

3 and Tjx ∈ X>s for some j ∈ N, then there exists n∈ {0, . . . , j −1} such that ht(Tnx) ≤s3 and ht(xat) > s2 for all t∈[r0n, rj0].

(iv) Let s > s3. If x∈X>s and Tjx∈X≤s

3 for some j ∈N, then there exists n∈ {1, . . . , j} such that ht(Tnx)≤s3 and ht(xat)> s2 for all t∈[1, rn0].

(v) Let s > s3. If x ∈ X>s and T x∈ X≤s, then there exists n ∈N such that Tnx∈X≤s

3 and Tkx∈X≤s for allk= 1, . . . , n.

Proof. We will choose s2 > s1 below. Let x ∈ X>s

2. We wish to prove that xat∈X>2s

1 for all t∈[r0−1, r0]. Since s2 > s1, there exist by Proposition3.1 a unique ξ∈Ξ and an element g∈Gsuch that x= Γg and

ht(x) = htξ(x) =

kvξ̺(g)k kvξ̺(ξ)k

1q

.

Further, for all t∈[r−10 , r0], we have ht(xat)≥htξ(xat)≥

kvξ̺(gat)k kvξ̺(ξ)k

1q

.

However, now it is clear that if s2 is sufficiently big2 or equivalently kvξ̺(g)k is sufficiently small, this will force kvξ̺(gat)k fort∈[r−10 , r0] sufficiently small to get the claim in (i).

For the proof of the remaining properties we will use the (quite natural) mono- tonicity properties of the functions appearing in Lemma 5.2. So assume that

ht(x) = htξ(x) =

kvξ̺(g)k kvξ̺(ξ)k

1q

> s1

forx= Γg (with the cusp representativeξ and±vξ̺(g) uniquely determined by Proposition 3.1). If g =ξnasmσ is as in the first part of Lemma5.2, then the trajectory comes straight out of the cusp. Hence ht(xat) = st is monotonically decreasing until it reaches the value s1 (at which point Proposition 3.1will not apply any longer). In the more general case, if g =ξnasmσ(1, Z, X)σ is as in the second part of Lemma 5.2, then the height ofxat is given by the formula

ht(xat) =s·

1 t 1

t + 14|X|22

+|Z|2 =· s

1

t + 14|X|2+

|Z|2+|X16|4 t

,

at least for all t for which the right hand side is≥s1. If X = 0 andZ = 0 the right hand side equalsstand the orbit points straight into the cusp. However, in

2A more careful analysis using Lemma5.2reveals thats2>2r20s1suffices.

(13)

general the right hand side has a unique maximum, is monotonically increasing left to the maximum and monotonically decreasing to the right of the maximum.

Property (i) and these monotonicity properties imply (ii).

We choose s3 in the same way as s2 but with s2 replacing s1 in (i). Assume now s > s3,x∈X≤s

3 and Tjx ∈X>s for some j ∈N. We choose the maximal integer n < j with ht(Tnx) ≤ s3. By our choice of s3 we have ht(xat) > 2s2 for t∈[r0n−1, r0n+1]. Using the above monotonicity properties now implies (iii).

Property (iv) follows in the same way using the first n≤j with ht(Tnx)≤s3. Property (v) follows directly from the monotonicity properties.

Given a point x ∈ X whose orbit stays near the cusp represented by ξ for the next S steps, Proposition 6.3 below provides non-trivial constraints on small perturbations ofxwhich do not destroy the qualitative behavior of the orbit for these next S steps. The following lemma is needed for its proof.

Lemma 6.2. LetDU be a bounded subset ofU. Letξ∈Ξandg=ξnarmu∈G with n∈N, ar∈A, m∈M and u=σ(1, Z, X)σ ∈DU. Suppose that

kvξ̺(gat)k kvξ̺(ξ)k

1q

> λ

kvξ̺(g)k kvξ̺(ξ)k

1q

for some t > 1 and λ >0. Then there exist c1, c2 > 0, only depending on DU and λ, such that

|X|< c1t−1/4 and |Z|< c2t−1/2.

Proof. Forλ≥1, the statement is already proven in Proposition5.5. So suppose 1> λ >0. Invoking Lemma5.2we find

(5) t

"

1 t +1

4|X|2 2

+|Z|2

#

< 1 λ

"

1 +1 4|X|2

2

+|Z|2

# .

Thus,

t 1

t +1 4|X|2

2

< 1 λ

1 +1

4|X|2 2

+ (λ−1−t)|Z|2. For t > λ−1, it follows that

t 1

t +1 4|X|2

2

< λ−1

1 +1 4|X|2

2

.

Therefore,

|X|2 <4(tλ−1)12 −1 t12 −λ12 t12. Hence, for t > λ−1+ 1,we have

|X|< c1t14

for some constant c1 >0. Since |X|is bounded, by possibly choosing a larger c1, this estimate holds for all t >1. To deduce the bound for |Z|we note that

(14)

(5) yields

(t−λ−1)|Z|2 < λ−1

1 +1 4|X|2

2

−t 1

t + 1 4|X|2

2

< λ−1(1 + 1

4c21)2 =c3. Suppose that t > λ−1+ 1. Then

|Z|2 < c3

t−λ−1 = c3

1−(tλ)−1t−1. The factor in front oft−1 is bounded. Thus,

|Z|< c2t12

for some constant c2 >0. As before, since |Z| is bounded, this estimate holds for all t >1 after possibly choosing a larger c2. This completes the proof.

Let d be a the left-G-invariant metric on G induced from a left-invariant Rie- mannian metric that is induced by an inner product on g. For r > 0 let BrG denote the open d-ball in G centered at the identity of G with radius r. For κ >0 letDUκ denote the subset of U consisting of the elementsu=σ(1, Z, X)σ with|Z|< κand |X|< κ, and let DκN AM :=BκG∩N AM. Further let

(6) Dκ :=DκUDκN AM.

ThenDκ is open. We chooseκ >0 such that for all h∈Dκ we have (7) k̺(h)k,k̺(h−1)k ≤

s1

s2 −q

.

We considerκ to be fixed throughout and will shrink it if necessary (e.g. in the paragraph before Lemma 7.3).

Proposition 6.3. There existc3, c4>0such that the following holds: Letx∈X, S ∈N, h∈Dκ be such that ht(Tjx)> s2 and ht(Tj(xh))> s2 for j= 0, . . . , S, ht(TSx)>ht(x) andht(TS(xh))>ht(xh). Suppose that h=σ(1, Z, X)σnarm.

Then

|X| ≤c3r−S/40 and |Z| ≤c4r0−S/2.

Proof. By Lemma 6.1 we have ht(xat) > s2 and ht(xhat) > s2 for all t ∈ [1, r0S]. Since s2 > s1, Proposition 5.5 shows that there exist a unique cusp representative ξ∈Ξ and an element g∈Gsuch that x= Γg and

ht(xat) =

kvξ̺(gat)k kvξ̺(ξ)k

1q

for allt∈[1, r0S]. Moreover, there exist a unique cusp representativeξ1 ∈Ξ and an elementg1 ∈G such thatxh= Γg1h and

ht(xhat) =

kvξ1̺(g1hat)k kvξ1̺(ξ1)k

1q

for all t∈[1, rS0]. In the following we show thatξ =ξ1 and that we can choose g1 =g. We have

kvξ̺(ghat)k=kvξ̺(gatat−1hat)k ≤ kvξ̺(gat)k · k̺(at−1hat)k.

(15)

Now, at−1hat ∈ Dκ for t near 1, say in the non-trivial interval I. By (7), for t∈I this yields

k̺(at−1hat)k ≤ s1

s2 −q

. Thus, for t∈I,

kvξ̺(ghat)k ≤ kvξ̺(gat)k s1

s2 −q

<

s1 s2

−q

s−q2 kvξ̺(ξ)k

=s−q1 kvξ̺(ξ)k.

Hence, fort∈I,

kvξ̺(ghat)k kvξ̺(ξ)k

1q

> s1.

The uniqueness of ξ1 yields ξ1 = ξ. Moreover, we can choose g1 =g for t∈ I.

As in the proof of Proposition 5.5, we see that we can choose g1 = g for all t∈[1, r0S].

Proposition 5.5 shows that g ∈ ξN AM U, say g = ξn4ar1m1u1 with u1 = σ(1, Z1, X1)σ, and that

(8) |X1|<2r−S/40 and |Z1|< r0−S/2. Suppose that h=u2n3ar2m2 and seth2 :=n3ar2m2. Then

kvξ̺(gh)k=kvξ̺(gu2h2)k ≤ kvξ̺(gu2)kk̺(h2)k ≤ kvξ̺(gu2)k s1

s2

−q

and

kvξ̺(gu2aS)k=kvξ̺(ghaSa−Sh−12 aS)k

≤ kvξ̺(ghaS)kk̺(a−Sh−12 aS)k ≤ kvξ̺(ghaS)k s1

s2 −q

.

This yields

kvξ̺(gu2aS)k kvξ̺(ξ)k

1q

≥ s1 s2

kvξ̺(ghaS)k kvξ̺(ξ)k

1q

= s1

s2 ht(xhaS)

> s1 s2

ht(xh) = s1 s2

kvξ̺(gh)k kvξ̺(ξ)k

1q

≥ s1

s2 2

kvξ̺(gu2)k kvξ̺(ξ)k

1q

.

Let u2 =σ(1, Z2, X2)σ. Then

u1u2 =σ(1, Z1+Z2+12[X1, X2], X1+X2)σ.

From (8) andu2∈DκU it follows that

|X1+X2| ≤ |X1|+|X2|<2 +κ.

Moreover, using triangle inequality and [Poh10, Lemma 2.12, Proposition 3.3]

we find

|Z1+Z2+12[X1, X2]| ≤ |Z1|+|Z2|+12|X1||X2|<1 + 2κ.

(16)

Thus, u1u2 is contained in the bounded set DU2+2κ. Note that this set only depends onκ. Then Lemma6.2gives

|X1+X2|< c1r0−S/4 and |Z1+Z2+12[X1, X2]|< c2r−S/20 , where the constants c1, c2 only depend ons1, s2 and κ. It follows that

|X2|< c1r0−S/4+|X1|<(c1+ 2)r−S/40 and

|Z2| ≤ |Z1+Z2+ 12[X1, X2]|+|Z1|+12|X1||X2|

≤c2r0−S/2+r0−S/2+ (c1+ 2)r0−S/2.

This completes the proof.

7. Estimate of metric entropy and proof of Theorem A This section, in which we prove Theorem A, can be understood independently from the previous ones if one is willing to accept the following facts previously shown: The height levels3is chosen such that the connected parts ofX>s

3 (thus, cuspidal ends of uniform “length”) can be identified with (Γ∩P)\C, whereC is the cylindrical setC =ξAs3N K at the cusp represented byξ of the considered end andP is the corresponding minimal parabolic subgroup inG. In particular, this means that connected parts of geodesic trajectories inX>s

3 can be identified with any representing geodesic trajectories inC. As a consequence we know (see Lemma6.1) that (discretized) geodesic trajectories inX>s

3 which start to move out of the cusp actually descend to below height level s3, and geodesics in X which move from one of these cuspidal ends to another one necessarily have to pass through the compact part X≤s

3. Moreover, if the trajectories of two nearby points x, xh in X (h ∈ G) stay together near a cusp (meaning in the same connected component of X>s

3) for “time”t, then the unstable component ofhis restricted (up to a multiplicative constant) byt−1/2 in the direction of the long root and byt−1/4 in the direction of the short root (see Proposition 6.3).

Let M1(X)T denote the set of T-invariant probability measures on X. Let µ∈ M1(X)T and suppose that Pis a partition of X (consisting of measurable sets).

We denote the static entropy ofP with respect toµby

(9) Hµ(P) =−X

PP

µ(P) logµ(P).

For n∈N0 let Pn0 :=

_n

j=0

T−jP=

Pj0 ∩T−1Pj1∩. . .∩T−nPjn

Pji ∈P .

Then

hµ(T,P) = inf

n∈N

1

nHµ Pn−1

0

is the dynamical entropy of (T,P) with respect toµ. Finally, hµ(T) = sup{hµ(T,P)|Ppartition of X,Hµ(P)<∞}

= sup{hµ(T,P)|Pfinite partition of X}

(17)

is the (metric) entropy of T with respect toµ.

In our set-up there exists a unique maximal entropy measure forT. We provide a reference for this statement and recall how to calculate its value in the following proposition. Set p1 := dimg1,p2 := dimg2 and recall that ea=ar0.

Proposition 7.1. The maximal entropy ofT is achieved by the Haar measurem on Xand is given by

hm(T) = max

hµ(T)|µ∈M1(X)T =p1

2 +p2

logr0.

Moreover, the Haar measure is the only T-invariant probability measure that achieves this maximal entropy.

Proof. The statement follows from a combination of the proposition in Sec- tion 9.3 in [MT94] and Lemma 9.5 and Proposition 9.6 in [MT94]. If G is algebraic, a more accessible reference is [EL10, Theorem 7.6]. Note that

−log det Ada|g−1⊕g−2

=p1 2 +p2

logr0.

For r >0 we call

(10) BL:=BL(r) :=

L−1\

j=0

˜

ajBrG˜a−j

a (forward) Bowen L-ball inGwith (radius) parameter r. Further, any subset of Xof the form

(11) xBL=xBL(r)

withx∈Xis called a Bowen L-ball inXwithcenter xand (radius) parameter r.

Through the work of Brin–Katok [BK83] it is well known that entropy is strongly related to the decay rate of the measure of BowenL-balls. For the Haar measure this can be established quite directly and in the following strong form (which will be used in many covering arguments below).

Lemma 7.2. Let r >0 be sufficiently small (depending only onG) and L∈N. Then

rdimGe−hm(T)L≪m(BL(r))≪rdimGe−hm(T)L, where the implied constants only depend on Gand ˜a.

Proof. Recall from (6) the definition of Dr. We find r1, r2 > 0 (uniform for small r) such that

Dr1r⊆BrG⊆Dr2r. Then

D(L)(r1r) :=

L−1\

j=0

˜

ajDr1r˜a−j ⊆BL(r) =

L−1\

j=0

˜

ajBrG˜a−j

L−1\

j=0

˜

ajDr2r˜a−j. One easily checks that

D(L)(r) = ˜aL−1DrU˜a−(L−1)DrN AM

(18)

and

˜

aL−1DUr˜a−(L−1) =n

σ(1, Z, X)σ

|Z|< rr0−(L−1)/2, |X|< rr0−(L−1)/4o .

Let du, dn, da and dm be Haar measures on U, N, A and M, respectively, and letdg denote the Haar measure onG. With appropriate normalizations we have ([Hel00, Chapter I, Proposition 5.21], and [Hel00, Chapter I, Corollary 5.2] for

the change of order of integration) Z

G

f(g)dg = Z

U×N×A×M

f(unam)dudndadm

for allf ∈Cc(G). Further, we recall from [Hel00, Chapter I, Theorem 1.14] that if the support of f ∈Cc(G) is contained in the canonical coordinate neighbor- hood of G, then

(12)

Z

G

f(g)dg = Z

g

f(expW) det

1−eadW adW

dW,

wheredW is the Euclidean measure on gwhich coincides with (dg)id.

We now use the coordinates (Z, X)∈g2×g1for the Lie algebrauofU. Sinceuis two-step nilpotent, the Jacobian determinant in (12) (applied forG=U) equals 1 for allW ∈u. With an appropriate global constantcU, the Haar measure du is then

mU(f) :=

Z

U

f(u)du=cU

Z

u

f(σ(1, Z, X)σ)dZdX.

Thus,

mU

˜

aL−1DrU˜a−(L−1)

=cUrdimUe−hm(T)(L−1). Hence

m D(L)(r1r)

=cUr1dimUrdimUe−hm(T)(L−1)mN AM DN AMr1r ,

where mN AM := dn⊗da⊗dm. We may assume that DN AMr = BrN AM. For sufficiently small r >0, the parameter space in n×a×m for the setDrN AM is the spherical normal neighborhoodVr={W ∈n×a×m| kWk< r}(see [Hel01, Chapter I, Proposition 9.4]). On this neighborhood, the Jacobian determinant (12) (applied to G = N AM) is bounded from above and from below by some

positive constants. Hence, (12) yields

rdimN AM ≪mN AM DN AMr

≪rdimN AM.

This completes the proof.

We pickλ >0 such thatr0λis an injectivity radius ofX≤s

3and use it throughout as radius parameter for Bowen balls. Recall the setDκ and the choice ofκfrom (6)-(7). We may choose λso small such thatBλG⊆Dκ.

In Lemma 7.4 below we will estimate how many Bowen L-balls are needed to cover P ∈η0L−1,P ⊆X≤s

3, for certain partitionsη of X.

Lemma 7.3. Let s > s3. Then there exists kmax ∈ N such that whenever x∈X>s satisfies T x, . . . , Tkx∈X≤s∩X>s

3, then k≤kmax.

Сурет

Figure 1. An excursion V j into X &gt;s containing an excursion into X &gt;s ′ and contained in the excursion ˜V j into X &gt;s

Ақпарат көздері

СӘЙКЕС КЕЛЕТІН ҚҰЖАТТАР

With this thermal LB model, we simulated the following two types of thermal flows: one is thermal flows in a cubic cavity with hot and cold vertical walls and located spherical