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

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

consider the right action

T:

X → X x 7→ x˜a

of ˜aonX. Further let (µ_n)_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→∞ h_µn(T)≤h_ν(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 h_m(T) denote the maximal metric entropy ofT and suppose that ν(X)>0. Then

ν(X)h ^ν

ν(X)(T) +¹₂h_m(T)·(1−ν(X))≥lim sup

n→∞ h_µn(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 suph_µn(T)≥c. Then ν(X)≥ 2c

h_m(T) −1.

Thus, if the entropy on the sequence (µ_n) is high, meaning at least ¹₂h_m(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 suph_µn(T)≥c and ν(X) = 2c

h_m(T) −1>0, then h ^ν

ν(X)(T) =h_m(T) and _ν(^ν_X) 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.

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

h_µ(T)≤h_µ(T, η) +c_s+¹₂h_m(T)µ(X_>s)

with a global constant c_s such that c_s → 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 = C_A(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

g_j :=n X ∈g

∀a∈A: Ad_aX=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,

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) :=ηA_sK.

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 ξ1 =ξ2 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].

For each ξ ∈Ξ we set

L_ξ :=ξN M ξ⁻¹

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

V :=^ℓ

g.

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

hence

̺:=^ℓ

Ad◦ (·)⁻¹

:G→End(V).

We fix a non-zero element v_ξ in the one-dimensional space W_ξ:=^ℓ

l_ξ and let

θ_ξ: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

v_ξ̺(g) =θ_ξ(g)v_ξ.

One easily shows thatθ_ξ(g) = 1 for g in the connected component of L_ξ, and θ_ξ(ξaξ⁻¹) =α(a)⁻(¹₂^dimg1+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) ht_ξ(x) := sup

(kvξ̺(g)k kvξ̺(ξ)k

−¹_q

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 kv_ξ̺(ξ)k

−¹_q

=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

ht_ξ(x)

ξ ∈Ξ . For s >0 andξ ∈Ξ we set

X(ξ, s) :={x∈X|ht_ξ(x)> s}

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

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 B_δ denote the open k · k-ball in V with radiusδ >0, centered at 0. We define

δ_ξ(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 v_ξ̺(γg)∈B_δξ(s) for some γ ∈Γ. Further, if s≥s1 and γ1, γ2 ∈Γ satisfy v_ξ̺(γjg) ∈ B_δξ(s) for j = 1,2, then v_ξ̺(γ1g) ∈ {±v_ξ̺(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)·a_s,

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

a_sn= (s, sZ, s^1/2X).

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

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= (t_s, Z_s, X_s)∈N A on a pointp= (t_p, Z_p, X_p)_D ∈Dbecomes s.p= t_st_p+¹₄|X_s|²+¹₂t^1/2_s hX_s, X_pi, Z_s+t_sZ_p+ ¹₂t^1/2_s [X_s, X_p], X_s+t^1/2_s X_p

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→J_Z, 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(J_ZX) = J_ϕ(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 σmna_t∈N a_rK. Then

r = t

t+¹₄|X|²2

+|Z|².

Proof. By Iwasawa decomposition we know that σmnat = n^′ark for suitable n^′ ∈ N, k∈K and r ∈R_>0. Suppose that m = (ϕ, ψ). Applying both σmna_t and n^′a_rkto the base pointo= (1,0,0)_D inD, we find

σmna_t·o=n^′a_rk·o=n^′a_r·o.

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

= 1

t+¹₄|X|²2

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

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

n^′a_r·o= r+¹₄|X^′|², Z^′, X^′

D. Thus

X^′ = 1

t+¹₄|X|²2

+|Z|² −t−¹₄|X|²+J_ϕ(Z) ψ(X),

|X^′|² = 1 t+¹₄|X|²2

+|Z|²2

t+¹₄|X|²2

|X|²+

J_ϕ(Z)ψ(X) ²

−2 t+ ¹₄|X|² ψ(X), J_ϕ(Z)ψ(X)

= |X|²

t+¹₄|X|²2

+|Z|², and

r = t+¹₄|X|² t+¹₄|X|²2

+|Z|² −¹₄|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(ga_t)k kv_ξ̺(ξ)k

−¹_q

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

kvξ̺(ga_t)k kvξ̺(ξ)k

−¹_q

=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(ga_t)k kvξ̺(ξ)k

−¹_q

=st.

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

ξK . Hence

kvξ̺(ga_t)k=θ_ξ(ξa_s/tξ⁻¹)kvξ̺(ξ)k=s t

−q

kvξ̺(ξ)k.

Suppose now that g = ξnma_su with u = σn^′σ and n^′ = (1, Z, X). Then (for somem^′∈M)

kv_ξ̺(gat)k=s^−qkv_ξ̺(ξσm^′n^′σat)k=s^−qkv_ξ̺(ξσm^′n^′a_1/tσ)k

=s^−qkvξ̺(ξσm^′n^′a_1/t)k.

Lemma 5.1yields

σm^′n^′a_1/t=n^′′a_rk for somen^′′∈N,k∈K and

r =

1 t 1

t +¹₄|X|²2

+|Z|². Thus,

kv_ξ̺(gat)k=

1 t 1

t +¹₄|X|²2

+|Z|²

!−q

s^−qkv_ξ̺(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 :=kv_ξ̺(g)k. If g∈ξN AM σ, then v_ξ̺(ga_t) ∈ B_δ if and only if t < 1. If g = ξna_smu ∈ ξN AM U with u=σ(1, Z, X)σ, then v_ξ̺(gat)∈B_δ 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=ξnma_su withu=σn^′σ and n^′ = (1, Z, X). By Lemma 5.2, (4) kvξ̺(ga_r)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(ga_t)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.

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 kv_ξ̺(ga_t)k = kv_ξ̺(γga_t)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< δ_ξ(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 = ξn^′am^′σ resp. γg = ξn^′am^′u for some n^′ ∈N, m^′∈M.

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

L_ξξA_sK=n g∈G

v_ξ̺(g)∈B_δξ(s)o .

Henceg, γg∈L_ξξA_s1K. By [Dan84, Remark 1.3] (withηas in Proposition2.1), L_ξξAs1K = (Γ∩L_ξ)ξηAs1K.

Hence there existγ1, γ2∈Γ∩L_ξ,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 ht_ξ(x)≥s₁.

Proposition 5.5. Let s > s1 and x∈X. Suppose that there exists an interval I in R such that ht(xa_t) > 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(xa_t) = ht_ξ(xa_t) =

kv_ξ̺(ga_t)k kvξ̺(ξ)k

−¹_q

for all t ∈ I. Moreover, if 1 ∈ I and if there exists t ∈ I with t > 1 and ht(xa_t) > ht(x), then g = ξna_rmu for some r > 0, n ∈ N, m ∈ M and u ∈ U. The elements a_r 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.

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

ht_ξ(y) =

kv_ξ̺(h)k kvξ̺(ξ)k

−¹_q

. Since the function

R_>0 → R

r 7→ kvξ̺(ga_r)k

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

J_ξ:=

t∈I

ht_ξ(xa_t)> s .

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

ht(xa_t) = ht_ξ(xa_t)

for all t∈I. For each t∈I pick an element g_t∈Gsuch that x= Γg_t and ht_ξ(xa_t) =

kvξ̺(g_ta_t)k kvξ̺(ξ)k

−¹_q

.

Let J_tbe the set of p∈I such that ht_ξ(xa_p) =

kvξ̺(g_ta_p)k kvξ̺(ξ)k

−¹_q

.

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

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(xa_t)> s for allt∈[1, r₀].

(iii) Let s > s₃. If x ∈ X_≤s

3 and T^jx ∈ X_>s for some j ∈ N, then there exists n∈ {0, . . . , j −1} such that ht(Tⁿx) ≤s₃ and ht(xa_t) > s₂ for all t∈[r₀ⁿ, r^j₀].

(iv) Let s > s₃. If x∈X_>s and T^jx∈X_≤s

3 for some j ∈N, then there exists n∈ {1, . . . , j} such that ht(Tⁿx)≤s₃ and ht(xa_t)> 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 T^kx∈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) = ht_ξ(x) =

kvξ̺(g)k kv_ξ̺(ξ)k

−¹_q

.

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

kvξ̺(ga_t)k kvξ̺(ξ)k

−¹_q

.

However, now it is clear that if s₂ is sufficiently big² or equivalently kvξ̺(g)k is sufficiently small, this will force kv_ξ̺(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) = ht_ξ(x) =

kvξ̺(g)k kv_ξ̺(ξ)k

−¹_q

> s₁

forx= Γg (with the cusp representativeξ and±v_ξ̺(g) uniquely determined by Proposition 3.1). If g =ξna_smσ is as in the first part of Lemma5.2, then the trajectory comes straight out of the cusp. Hence ht(xa_t) = ^s_t 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 =ξna_smσ(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.

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 T^jx ∈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(xa_t) > 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. LetD^U be a bounded subset ofU. Letξ∈Ξandg=ξnarmu∈G with n∈N, a_r∈A, m∈M and u=σ(1, Z, X)σ ∈D^U. Suppose that

kv_ξ̺(gat)k kvξ̺(ξ)k

−¹_q

> λ

kv_ξ̺(g)k kvξ̺(ξ)k

−¹_q

for some t > 1 and λ >0. Then there exist c₁, c₂ > 0, only depending on D^U 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 −λ⁻¹² t⁻¹². Hence, for t > λ⁻¹+ 1,we have

|X|< c1t⁻¹⁴

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

(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|< c2t⁻¹²

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 B_r^G denote the open d-ball in G centered at the identity of G with radius r. For κ >0 letD^U_κ denote the subset of U consisting of the elementsu=σ(1, Z, X)σ with|Z|< κand |X|< κ, and let D_κ^{N AM} :=B_κ^G∩N AM. Further let

(6) D_κ :=D_κ^UD_κ^{N AM}.

ThenD_κ is open. We chooseκ >0 such that for all h∈D_κ 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∈D_κ be such that ht(T^jx)> s₂ and ht(T^j(xh))> s₂ for j= 0, . . . , S, ht(T^Sx)>ht(x) andht(T^S(xh))>ht(xh). Suppose that h=σ(1, Z, X)σna_rm.

Then

|X| ≤c3r^−S/4₀ and |Z| ≤c4r₀^−S/2.

Proof. By Lemma 6.1 we have ht(xa_t) > s₂ and ht(xha_t) > s₂ for all t ∈ [1, r₀^S]. 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(xa_t) =

kvξ̺(ga_t)k kv_ξ̺(ξ)k

−¹_q

for allt∈[1, r₀^S]. Moreover, there exist a unique cusp representativeξ1 ∈Ξ and an elementg₁ ∈G such thatxh= Γg₁h and

ht(xhat) =

kvξ1̺(g₁ha_t)k kv_ξ1̺(ξ₁)k

−¹_q

for all t∈[1, r^S₀]. In the following we show thatξ =ξ1 and that we can choose g₁ =g. We have

kvξ̺(gha_t)k=kvξ̺(ga_ta_t⁻¹ha_t)k ≤ kvξ̺(ga_t)k · k̺(a_t⁻¹ha_t)k.

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

k̺(a_t⁻¹ha_t)k ≤ s₁

s₂ −q

. Thus, for t∈I,

kv_ξ̺(gha_t)k ≤ kv_ξ̺(ga_t)k s₁

s₂ −q

<

s₁ s₂

−q

s^−q₂ kv_ξ̺(ξ)k

=s^−q₁ kv_ξ̺(ξ)k.

Hence, fort∈I,

kvξ̺(gha_t)k kvξ̺(ξ)k

−¹_q

> 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, r₀^S].

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

kv_ξ̺(gu2a^S)k=kv_ξ̺(gha^Sa^−Sh⁻¹₂ a^S)k

≤ kv_ξ̺(gha^S)kk̺(a^−Sh⁻¹₂ a^S)k ≤ kv_ξ̺(gha^S)k s₁

s₂ −q

.

This yields

kv_ξ̺(gu₂a^S)k kvξ̺(ξ)k

−¹_q

≥ s₁ s₂

kv_ξ̺(gha^S)k kvξ̺(ξ)k

−¹_q

= s₁

s₂ ht(xha^S)

> s₁ s2

ht(xh) = s₁ s2

kvξ̺(gh)k kv_ξ̺(ξ)k

−¹_q

≥ s1

s₂ 2

kvξ̺(gu₂)k kvξ̺(ξ)k

−¹_q

.

Let u2 =σ(1, Z2, X2)σ. Then

u₁u₂ =σ(1, Z₁+Z₂+¹₂[X₁, X₂], X₁+X₂)σ.

From (8) andu₂∈D_κ^U 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.

Thus, u1u2 is contained in the bounded set D^U_2+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 =ξA_s3N 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

P∈P

µ(P) logµ(P).

For n∈N₀ let Pⁿ₀ :=

_n

j=0

T^−jP=

P_j0 ∩T⁻¹P_j1∩. . .∩T⁻ⁿP_jn

P_ji ∈P .

Then

h_µ(T,P) = inf

n∈N

1

nH_µ Pⁿ⁻¹

0

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

= sup{hµ(T,P)|Pfinite partition of X}

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=a_r0.

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 Ad_a|g−1⊕g−2

=p₁ 2 +p₂

logr₀.

For r >0 we call

(10) B_L:=B_L(r) :=

L−1\

j=0

˜

a^jB_r^G˜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

r^dimGe^−hm(T)L≪m(B_L(r))≪r^dimGe^−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

D_r1r⊆B_r^G⊆D_r2r. Then

D^(L)(r₁r) :=

L−1\

j=0

˜

a^jD_r1r˜a^−j ⊆B_L(r) =

L−1\

j=0

˜

a^jB_r^G˜a^−j ⊆

L−1\

j=0

˜

a^jD_r2r˜a^−j. One easily checks that

D^(L)(r) = ˜a^L−1D_r^U˜a^−(L−1)D_r^{N AM}

and

˜

a^L−1D^U_r˜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 ∈C_c(G). Further, we recall from [Hel00, Chapter I, Theorem 1.14] that if the support of f ∈C_c(G) is contained in the canonical coordinate neighbor- hood of G, then

(12)

Z

G

f(g)dg = Z

g

f(expW) det

1−e^−adW 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 constantc_U, the Haar measure du is then

mU(f) :=

Z

U

f(u)du=cU

Z

u

f(σ(1, Z, X)σ)dZdX.

Thus,

m_U

˜

a^L−1D_r^U˜a^−(L−1)

=c_Ur^dimUe^{−hm(T)(L−1)}. Hence

m D^(L)(r₁r)

=c_Ur₁^dimUr^dimUe^{−hm(T)(L−1)}m_{N AM} D^{N AM}_r1r ,

where mN AM := dn⊗da⊗dm. We may assume that D^{N AM}_r = B_r^{N AM}. For sufficiently small r >0, the parameter space in n×a×m for the setD_r^{N 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

r^{dimN AM} ≪m_{N AM} D^{N AM}_r

≪r^{dimN 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 setD_κ and the choice ofκfrom (6)-(7). We may choose λso small such thatB_λ^G⊆D_κ.

In Lemma 7.4 below we will estimate how many Bowen L-balls are needed to cover P ∈η₀^L−1,P ⊆X_≤s

3, for certain partitionsη of X.

Lemma 7.3. Let s > s₃. Then there exists k^max ∈ N such that whenever x∈X_>s satisfies T x, . . . , T^kx∈X_≤s∩X_>s

3, then k≤kmax.