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∈}^{N}be 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_{µ}_{n}(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) +^{1}_{2}h_{m}(T)·(1−ν(X))≥lim sup

n→∞ h_{µ}_{n}(T).

In [KP] it is shown that the factor ^{1}_{2} 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_{µ}_{n}(T)≥c. Then
ν(X)≥ 2c

h_{m}(T) −1.

Thus, if the entropy on the sequence (µ_{n}) is high, meaning at least ^{1}_{2}h_{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_{µ}_{n}(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}+^{1}_{2}h_{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_{a}X=α(a)^{j}^{2}Xo

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

(1) g=g_{−2}⊕g_{−1}⊕c⊕g_{1}⊕g_{2}.

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⊕g_{1} resp. g=g_{−2}⊕c⊕g_{2},

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_{s}K.

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 η_{0} of N and a finite subset Ξ of G such that

(i) G= ΓΞΩ(s_{0}, η_{0}),

(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(s_{0}, η)∩Ω(s_{0}, η)6=∅}

is finite,

(iv) for each compact subset η of N containing η_{0}, there exists s_{1} > s_{0} such
that for all ξ_{1}, ξ_{2} ∈ Ξ and all γ ∈ Γ with γξ_{1}Ω(s_{0}, η)∩ξ_{2}Ω(s_{1}, η) 6= ∅ we
have ξ1 =ξ2 and γ ∈ξ1N M ξ_{1}^{−1}.

For the remainder of this article we fix s_{1} > s_{0} >0, a compact subsetη_{0} of N
and a finite subset Ξ of Gwhich satisfy (i)-(iv) of Proposition 2.1with η:=η_{0}.
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(s_{1}, 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(s_{1}, 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 right^{1} 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_{ξ}̺(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)^{−}(^{1}_{2}^{dim}^{g}1+dimg2)

fora∈A. Let

q:= ^{1}_{2}dimg_{1}+ dimg_{2}.

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_{ξ}̺(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^{−q}kvξ̺(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_{ξ}̺(γg)∈B_{δ}_{ξ}_{(s)} for some γ ∈Γ. Further,
if s≥s1 and γ1, γ2 ∈Γ satisfy v_{ξ}̺(γjg) ∈ B_{δ}_{ξ}_{(s)} for j = 1,2, then v_{ξ}̺(γ1g) ∈
{±v_{ξ}̺(γ2g)}.

Thus

X(ξ, s) = Γ\ΓξΩ(s, η)

for all ξ ∈Ξ and s >0. If s≥s_{1}, 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}×g_{2}×g_{1} →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_{s}n= (s, sZ, s^{1/2}X).

We define an inner product on n=g_{2}⊕g_{1} 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}×g_{2}×g_{1} with

D:=

(t, Z, X)_{D} ∈R×g_{2}×g_{1}

t > ^{1}_{4}|X|^{2}

via

R_{>0}×g_{2}×g_{1} →D, (t, Z, X) 7→(t+^{1}_{4}|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_{s}t_{p}+^{1}_{4}|X_{s}|^{2}+^{1}_{2}t^{1/2}_{s} hX_{s}, X_{p}i, Z_{s}+t_{s}Z_{p}+ ^{1}_{2}t^{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: g_{2}→End(g_{1}), Z 7→J_{Z},
via

hJZX, Yi=hZ,[X, Y]i for allX, Y ∈g_{1}.

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

t^{2}+|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 g_{2} resp. ψ on g_{1} such that ψ(J_{Z}X) = J_{ϕ(Z)}ψ(X) for all
(Z, X)∈g_{2}×g_{1}. 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 ∈g_{2},X∈g_{1}.
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_{r}K. Then

r = t

t+^{1}_{4}|X|^{2}2

+|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_{r}kto the base pointo= (1,0,0)_{D} inD, we find

σmna_{t}·o=n^{′}a_{r}k·o=n^{′}a_{r}·o.

In the coordinates ofD one easily calculates that
σmna_{t}·o=

= 1

t+^{1}_{4}|X|^{2}2

+|Z|^{2} t+^{1}_{4}|X|^{2},−ϕ(Z), −t−^{1}_{4}|X|^{2}+J_{ϕ(Z)}
ψ(X)

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

n^{′}a_{r}·o= r+^{1}_{4}|X^{′}|^{2}, Z^{′}, X^{′}

D. Thus

X^{′} = 1

t+^{1}_{4}|X|^{2}2

+|Z|^{2} −t−^{1}_{4}|X|^{2}+J_{ϕ(Z)}
ψ(X),

|X^{′}|^{2} = 1
t+^{1}_{4}|X|^{2}2

+|Z|^{2}2

t+^{1}_{4}|X|^{2}2

|X|^{2}+

J_{ϕ(Z)}ψ(X)
^{2}

−2 t+ ^{1}_{4}|X|^{2} ψ(X), J_{ϕ(Z)}ψ(X)

= |X|^{2}

t+^{1}_{4}|X|^{2}2

+|Z|^{2},
and

r = t+^{1}_{4}|X|^{2}
t+^{1}_{4}|X|^{2}2

+|Z|^{2} −^{1}_{4}|X^{′}|^{2} = t
t+^{1}_{4}|X|^{2}2

+|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_{ξ}̺(ξ)k

−^{1}_{q}

= s
t.
If g=ξna_{s}mσ(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 +^{1}_{4}|X|^{2}2

+|Z|^{2}.

Recall the identification of the cusp represented by ξ ∈ Ξ with the cusp neigh-
borhood (Γ∩ξN M ξ^{−1})\ξΩ(s_{1}, 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/t}mσ=ξna_{s/t}ξ^{−1}ξmσ ∈ ξN Aξ^{−1}

ξK . Hence

kvξ̺(ga_{t})k=θ_{ξ}(ξa_{s/t}ξ^{−1})kvξ̺(ξ)k=s
t

−q

kvξ̺(ξ)k.

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

kv_{ξ}̺(gat)k=s^{−q}kv_{ξ}̺(ξσm^{′}n^{′}σat)k=s^{−q}kv_{ξ}̺(ξσm^{′}n^{′}a_{1/t}σ)k

=s^{−q}kvξ̺(ξσm^{′}n^{′}a_{1/t})k.

Lemma 5.1yields

σm^{′}n^{′}a_{1/t}=n^{′′}a_{r}k
for somen^{′′}∈N,k∈K and

r =

1 t 1

t +^{1}_{4}|X|^{2}2

+|Z|^{2}.
Thus,

kv_{ξ}̺(gat)k=

1 t 1

t +^{1}_{4}|X|^{2}2

+|Z|^{2}

!−q

s^{−q}kv_{ξ}̺(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_{ξ}̺(g)k. If g∈ξN AM σ,
then v_{ξ}̺(ga_{t}) ∈ B_{δ} if and only if t < 1. If g = ξna_{s}mu ∈ ξN AM U with
u=σ(1, Z, X)σ, then v_{ξ}̺(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 _{16}^{1}|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_{s}u withu=σn^{′}σ and n^{′} = (1, Z, X). By Lemma 5.2,
(4) kvξ̺(ga_{r})k=

1 r 1

r +^{1}_{4}|X|^{2}2

+|Z|^{2}

!−q

s^{−q}kvξ̺(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 +^{1}_{4}|X|^{2}2

+|Z|^{2} <

1 t 1

t +^{1}_{4}|X|^{2}2

+|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_{ξ}̺(ga_{t})k = kv_{ξ}̺(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}(s_{1}).

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_{ξ}ξA_{s}K=n
g∈G

v_{ξ}̺(g)∈B_{δ}_{ξ}_{(s)}o
.

Henceg, γg∈L_{ξ}ξA_{s}_{1}K. By [Dan84, Remark 1.3] (withηas in Proposition2.1),
L_{ξ}ξAs1K = (Γ∩L_{ξ})ξηAs1K.

Hence there existγ1, γ2∈Γ∩L_{ξ},h1, h2 ∈ηAs1K such that
g=γ_{1}ξh_{1}, γg=γ_{2}ξh_{2}.

Therefore

g∈γ_{1}ξΩ(s_{1}, η)∩γ^{−1}γ_{2}ξΩ(s_{1}, η).

Proposition2.1(iv) yields γ_{1}^{−1}γ^{−1}γ2∈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)≥s_{1}.

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_{ξ}̺(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_{r}mu 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_{ξ}̺(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})> s_{1} 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_{t}a_{t})k
kvξ̺(ξ)k

−^{1}_{q}

.

Let J_{t}be the set of p∈I such that
ht_{ξ}(xa_{p}) =

kvξ̺(g_{t}a_{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

r_{0} :=α(˜a)
and recall that r_{0} >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 s_{2} 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 ≥ s_{2}, 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 levels_{3} > s_{2}such 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 heights_{3}.

Lemma 6.1. There exist s_{3} > s_{2} > s_{1} such that we have the following proper-
ties:

(i) If x∈X_{>s}

2, then ht(xat)>2s1 for all t∈[r^{−1}_{0} , r0].

(ii) If s≥s_{2} andx, T x∈X_{>s}, then ht(xa_{t})> s for allt∈[1, r_{0}].

(iii) Let s > s_{3}. If x ∈ X_{≤s}

3 and T^{j}x ∈ X_{>s} for some j ∈ N, then there
exists n∈ {0, . . . , j −1} such that ht(T^{n}x) ≤s_{3} and ht(xa_{t}) > s_{2} for all
t∈[r_{0}^{n}, r^{j}_{0}].

(iv) Let s > s_{3}. If x∈X_{>s} and T^{j}x∈X_{≤s}

3 for some j ∈N, then there exists
n∈ {1, . . . , j} such that ht(T^{n}x)≤s_{3} and ht(xa_{t})> s_{2} for all t∈[1, r^{n}_{0}].

(v) Let s > s_{3}. If x ∈ X_{>s} and T x∈ X_{≤s}, then there exists n ∈N such that
T^{n}x∈X_{≤s}

3 and T^{k}x∈X_{≤s} for allk= 1, . . . , n.

Proof. We will choose s_{2} > s_{1} below. Let x ∈ X_{>s}

2. We wish to prove that
xat∈X_{>2s}

1 for all t∈[r_{0}^{−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_{ξ}̺(ξ)k

−^{1}_{q}

.

Further, for all t∈[r^{−1}_{0} , r_{0}], we have
ht(xat)≥ht_{ξ}(xat)≥

kvξ̺(ga_{t})k
kvξ̺(ξ)k

−^{1}_{q}

.

However, now it is clear that if s_{2} is sufficiently big^{2} or equivalently kvξ̺(g)k is
sufficiently small, this will force kv_{ξ}̺(gat)k fort∈[r^{−1}_{0} , 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_{ξ}̺(ξ)k

−^{1}_{q}

> s_{1}

forx= Γg (with the cusp representativeξ and±v_{ξ}̺(g) uniquely determined by
Proposition 3.1). If g =ξna_{s}mσ 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 s_{1} (at which point Proposition 3.1will not
apply any longer). In the more general case, if g =ξna_{s}mσ(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 + ^{1}_{4}|X|^{2}2

+|Z|^{2} =· s

1

t + ^{1}_{4}|X|^{2}+

|Z|^{2}+^{|X}_{16}^{|}^{4}
t

,

at least for all t for which the right hand side is≥s_{1}. 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>2r^{2}_{0}s1suffices.

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 s_{3} in the same way as s_{2} but with s_{2} replacing s_{1} in (i). Assume
now s > s_{3},x∈X_{≤s}

3 and T^{j}x ∈X_{>s} for some j ∈N. We choose the maximal
integer n < j with ht(T^{n}x) ≤ s_{3}. By our choice of s_{3} we have ht(xa_{t}) > 2s_{2}
for t∈[r_{0}^{n−1}, r_{0}^{n+1}]. Using the above monotonicity properties now implies (iii).

Property (iv) follows in the same way using the first n≤j with ht(T^{n}x)≤s_{3}.
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_{ξ}̺(gat)k
kvξ̺(ξ)k

−^{1}_{q}

> λ

kv_{ξ}̺(g)k
kvξ̺(ξ)k

−^{1}_{q}

for some t > 1 and λ >0. Then there exist c_{1}, c_{2} > 0, only depending on D^{U}
and λ, such that

|X|< c_{1}t^{−1/4} and |Z|< c_{2}t^{−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})^{1}^{2} −1
t^{1}^{2} −λ^{−}^{1}^{2} t^{−}^{1}^{2}.
Hence, for t > λ^{−1}+ 1,we have

|X|< c1t^{−}^{1}^{4}

for some constant c_{1} >0. Since |X|is bounded, by possibly choosing a larger
c_{1}, 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

4c^{2}_{1})^{2} =c_{3}.
Suppose that t > λ^{−1}+ 1. Then

|Z|^{2} < c_{3}

t−λ^{−1} = c_{3}

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

|Z|< c2t^{−}^{1}^{2}

for some constant c_{2} >0. As before, since |Z| is bounded, this estimate holds
for all t >1 after possibly choosing a larger c_{2}. 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}^{U}D_{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

s_{2}
−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 existc_{3}, c_{4}>0such that the following holds: Letx∈X,
S ∈N, h∈D_{κ} be such that ht(T^{j}x)> s_{2} and ht(T^{j}(xh))> s_{2} for j= 0, . . . , S,
ht(T^{S}x)>ht(x) andht(T^{S}(xh))>ht(xh). Suppose that h=σ(1, Z, X)σna_{r}m.

Then

|X| ≤c3r^{−S/4}_{0} and |Z| ≤c4r_{0}^{−S/2}.

Proof. By Lemma 6.1 we have ht(xa_{t}) > s_{2} and ht(xha_{t}) > s_{2} for all t ∈
[1, r_{0}^{S}]. Since s_{2} > s_{1}, 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_{ξ}̺(ξ)k

−^{1}_{q}

for allt∈[1, r_{0}^{S}]. Moreover, there exist a unique cusp representativeξ1 ∈Ξ and
an elementg_{1} ∈G such thatxh= Γg_{1}h and

ht(xhat) =

kvξ1̺(g_{1}ha_{t})k
kv_{ξ}_{1}̺(ξ_{1})k

−^{1}_{q}

for all t∈[1, r^{S}_{0}]. In the following we show thatξ =ξ1 and that we can choose
g_{1} =g. We have

kvξ̺(gha_{t})k=kvξ̺(ga_{t}a_{t}^{−1}ha_{t})k ≤ kvξ̺(ga_{t})k · k̺(a_{t}^{−1}ha_{t})k.

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

k̺(a_{t}^{−1}ha_{t})k ≤
s_{1}

s_{2}
−q

. Thus, for t∈I,

kv_{ξ}̺(gha_{t})k ≤ kv_{ξ}̺(ga_{t})k
s_{1}

s_{2}
−q

<

s_{1}
s_{2}

−q

s^{−q}_{2} kv_{ξ}̺(ξ)k

=s^{−q}_{1} kv_{ξ}̺(ξ)k.

Hence, fort∈I,

kvξ̺(gha_{t})k
kvξ̺(ξ)k

−^{1}_{q}

> s1.

The uniqueness of ξ_{1} yields ξ_{1} = ξ. Moreover, we can choose g_{1} =g for t∈ I.

As in the proof of Proposition 5.5, we see that we can choose g_{1} = g for all
t∈[1, r_{0}^{S}].

Proposition 5.5 shows that g ∈ ξN AM U, say g = ξn4ar1m1u1 with u1 =
σ(1, Z_{1}, X_{1})σ, and that

(8) |X1|<2r^{−S/4}_{0} and |Z1|< r_{0}^{−S/2}.
Suppose that h=u2n3ar2m2 and seth2 :=n3ar2m2. Then

kvξ̺(gh)k=kvξ̺(gu_{2}h_{2})k ≤ kvξ̺(gu_{2})kk̺(h2)k ≤ kvξ̺(gu_{2})k
s_{1}

s2

−q

and

kv_{ξ}̺(gu2a^{S})k=kv_{ξ}̺(gha^{S}a^{−S}h^{−1}_{2} a^{S})k

≤ kv_{ξ}̺(gha^{S})kk̺(a^{−S}h^{−1}_{2} a^{S})k ≤ kv_{ξ}̺(gha^{S})k
s_{1}

s_{2}
−q

.

This yields

kv_{ξ}̺(gu_{2}a^{S})k
kvξ̺(ξ)k

−^{1}_{q}

≥ s_{1}
s_{2}

kv_{ξ}̺(gha^{S})k
kvξ̺(ξ)k

−^{1}_{q}

= s_{1}

s_{2} ht(xha^{S})

> s_{1}
s2

ht(xh) = s_{1}
s2

kvξ̺(gh)k
kv_{ξ}̺(ξ)k

−^{1}_{q}

≥ s1

s_{2}
2

kvξ̺(gu_{2})k
kvξ̺(ξ)k

−^{1}_{q}

.

Let u2 =σ(1, Z2, X2)σ. Then

u_{1}u_{2} =σ(1, Z_{1}+Z_{2}+^{1}_{2}[X_{1}, X_{2}], X_{1}+X_{2})σ.

From (8) andu_{2}∈D_{κ}^{U} it follows that

|X_{1}+X_{2}| ≤ |X_{1}|+|X_{2}|<2 +κ.

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

we find

|Z1+Z_{2}+^{1}_{2}[X_{1}, X_{2}]| ≤ |Z1|+|Z2|+^{1}_{2}|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

|X_{1}+X_{2}|< c_{1}r_{0}^{−S/4} and |Z_{1}+Z_{2}+^{1}_{2}[X_{1}, X_{2}]|< c_{2}r^{−S/2}_{0} ,
where the constants c_{1}, c_{2} only depend ons_{1}, s_{2} and κ. It follows that

|X2|< c_{1}r_{0}^{−S/4}+|X1|<(c_{1}+ 2)r^{−S/4}_{0}
and

|Z2| ≤ |Z1+Z_{2}+ ^{1}_{2}[X_{1}, X_{2}]|+|Z1|+^{1}_{2}|X1||X2|

≤c2r_{0}^{−S/2}+r_{0}^{−S/2}+ (c1+ 2)r_{0}^{−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_{s}_{3}N 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 M_{1}(X)^{T} denote the set of T-invariant probability measures on X. Let µ∈
M_{1}(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∈N_{0} let
P^{n}_{0} :=

_n

j=0

T^{−j}P=

P_{j}_{0} ∩T^{−1}P_{j}_{1}∩. . .∩T^{−n}P_{j}_{n}

P_{j}_{i} ∈P .

Then

h_{µ}(T,P) = inf

n∈N

1

nH_{µ} P^{n−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 p_{1} := dimg_{1},p_{2} := dimg_{2} and recall that ea=a_{r}_{0}.

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

hm(T) = max

hµ(T)|µ∈M_{1}(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

=p_{1}
2 +p_{2}

logr_{0}.

For r >0 we call

(10) B_{L}:=B_{L}(r) :=

L−1\

j=0

˜

a^{j}B_{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^{dim}^{G}e^{−h}^{m}^{(T}^{)L}≪m(B_{L}(r))≪r^{dim}^{G}e^{−h}^{m}^{(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_{r}_{1}_{r}⊆B_{r}^{G}⊆D_{r}_{2}_{r}.
Then

D^{(L)}(r_{1}r) :=

L−1\

j=0

˜

a^{j}D_{r}_{1}_{r}˜a^{−j} ⊆B_{L}(r) =

L−1\

j=0

˜

a^{j}B_{r}^{G}˜a^{−j} ⊆

L−1\

j=0

˜

a^{j}D_{r}_{2}_{r}˜a^{−j}.
One easily checks that

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

and

˜

a^{L−1}D^{U}_{r}˜a^{−(L−1)} =n

σ(1, Z, X)σ

|Z|< rr_{0}^{−(L−1)/2}, |X|< rr_{0}^{−(L−1)/4}o
.

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^{−}^{ad}^{W}
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−1}D_{r}^{U}˜a^{−(L−1)}

=c_{U}r^{dim}^{U}e^{−h}^{m}^{(T}^{)(L−1)}.
Hence

m D^{(L)}(r_{1}r)

=c_{U}r_{1}^{dim}^{U}r^{dim}^{U}e^{−h}^{m}^{(T}^{)(L−1)}m_{N AM} D^{N AM}_{r}_{1}_{r}
,

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^{dim}^{N AM} ≪m_{N AM} D^{N AM}_{r}

≪r^{dim}^{N 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 ∈η_{0}^{L−1},P ⊆X_{≤s}

3, for certain partitionsη of X.

Lemma 7.3. Let s > s_{3}. Then there exists k^{max} ∈ N such that whenever
x∈X_{>s} satisfies T x, . . . , T^{k}x∈X_{≤s}∩X_{>s}

3, then k≤kmax.