geodesic flows on negatively curved manifolds. ik1 - American ...

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 178, April 1973

GEODESIC FLOWSON NEGATIVELY CURVED MANIFOLDS.IK1) BY

PATRICK EBERLEIN ABSTRACT. Let M be a complete Riemannian manifold with sectional curvature KsO, SM the unit tangent bundle of M, T the geodesic flow on SM and il ç S/W the set of nonwandering points relative to 7',. T¡ is topologically mixing (respectively lopologically transitive) on SM if for any open

sets

0, U of SM there exisrs

0 (respectively there tor y e SM we define lfuu(!'), and we relate of Tt to the existence

A > 0 such that

|/| 2 A implies

T¡(0) n ¡/ *

exists z e R such that 7,(0) n V * 0). For each vecstable and unstable sets Ws(i), Wss(v), W"( i;) and topological mixing (respectively topological transitivity) of a vector i e .S/W such that lfss(i) (respectively

Ws(v)) is dense in SM. If M is a Visibility manifold (implied by K S r < 0) and if 0 = SM then 7, is topologically mixing on SM. Let Sn = (Visibility manifolds For each If M e S if and

A/ of dimension 72 such that T¡ is topologically mixing on .S/WI. n 2 2, Sn ¡s closed under normal (Galois) Riemannian coverings. we classify |i e SM: Wss( r) is dense in S/Wj, and M is compact

only

subset

if this

set

= .S/W. Wc also

Introduction.

Let

fold with sectional geodesies

H be any complete,

curvature

Unit vectors

v, w tangent

y

determine

that they

a family

the case

where

ß is a proper

of tangency and

if - v and

- w are weakly

(strongly)

Wuuiv)

be the collections

stable

asymptotes

of v.

stable

and strong

unstable

K < 0 may be represented uous group of isometries Received

by the editors

These

sets

of //.

by v.

H/D

If M = H/D

geodesies

y

Let

if, in

determined

by the

Wss(v),

strong

stable,

manifold

D is a properly

we define

asymp-

and strong

the stable,

where

unstably

Ws(v),

unstable

A complete

and

asymptotic

(strongly)

weak

will be called

as a quotient

x.

sphere

asymptotic. strong,

determined

x of asymptotic

if the

limit

mani-

with center

w ate strongly

w ate weakly

of weak, sets

class

asymptotic

lie on the same . v and

Riemannian

limit spheres

v and

y

W"iv),

connected

Each equivalence

to H ate weakly

class

of y

simply

of parallel

are asymptotic,

their points

equivalence totic

K < 0.

in H determines

addition,

consider

of S/W.

the stable

unun-

M with

discontin-

and unstable

August 2, 1971.

,LWS(MOS)subject classifications (1970). Primary 53A35, 53C20; Secondary 28A65, 34C35, 58F99. points,

Key words and phrases. stable and unstable

Visibility

(')

Geodesic flow, prolongational sets, topological transitivity,

limit sets, nonwandering topological mixing, Axiom

1,

manifold.

This research

was supported

in part by NSF Grants GP-11476 and GP-20096. Cupyrigh! © 197.*.American Mathematical Society

57 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

PATRICK EBERLEIN

58 sets

in the unit tangent

tion,

then

Wssin^v)

bundle

of M by projection;

if 7r: H —>M is the projec-

for

v tangent

= jr^Wssiv)

any unit vector

to H and similarly

for the other sets. The sets

Ws, Wss, Wu and

Wuu ate defined

manifold

K < 0 and coincide

to an arbitrary

unstable

manifolds

were first

studied

the geodesic

of variable unstable

as horocycles

flow of surfaces

negative manifolds,

stable

sets

sets

The connection transitivity

and topological

4.16.

In Theorem

4.15 we allow

of M. We considered simply

connected

and we showed

involved

in proving

to stable

and

than the geodesic of horocycles

to the

way to study

K < 0.

to the geodesic

VIs and

mixing

the

The relationship flow appears

Wss and the properties

occurs

manifolds

covering

transitive

to kinds

4.6 and 4.11. in Theorems

4.14,

of topo4.15 and

of the unit tangent

M without

H satisfies

that if fi = SM, the unit tangent

flow is topologically ficulties

M with

Q to be any subset

in [6] those

Riemannian

and

Wss)

and certain

to be a natural

manifold

the sets

logical

curvature

The generalization

are not manifolds)

between

stable

(primarily

were later generalized

in Propositions

v tangent

[12] and Grant [9] and applied

here appears

complete

(which possibly

in §4, especially

sets

for flows more general

K < 0.

defined

unit vector

with the usual

These

negative

Horocycles

with

flow on an arbitrary

of these

K < 0.

by Hedlund

which are defined manifold

and unstable

clearly

with

of constant

curvature.

flow on a compact

geodesic

M with

if M is compact

for each

conjugate

the uniform

bundle

points

mixing

whose

Visibility

axiom,

of M, then the geodesic

on SM. At the end of §4 we indicate the topological

bundle

of the geodesic

the difflow on SM

by the method used here. §5 contains such

that

results

on the existence

Q Ç "/"(i/).

that the geodesic

The main applications

flow is topologically

with some zero curvature

with infinite

volume

The stable

and metrical plete variant

with

for any

K < 0.

S M holds?

mixing Questions

Notation

the Riemannian

offer a promising

of the geodesic For example, v £ A. set

flow invariant

v £ Q. We note

M = H/D manifolds

way to investigate

a set

Is the geodesic A, either

measure

sort should

and preliminaries.

structure

are in §6. manifolds

flow on SM, where

we define

if the corresponding of this

for certain

6.4) and for many noncompact

sets

Ws invariant

to the natural

SM metrically

1.

and unstable

and results

mixing

of vectors

M

6.5).

if VIsiv) C A whenever

is true that,

relative

(Theorem

(Theorem

mixing problems

manifold

and classification

ergodic

M is any com-

A C SM to be Ws in-

flow in SM ergodic

if it

A or S M - A has measure

zero

on SM? Is the geodesic

property

for Wss invariant

flow in sets

A C

be answered.

For any Riemannian

by ( , ), the Riemannian

metric

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

manifold

M we denote

by d and the sectional

59

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II curvature

and

by K.

If p £ M let

be the unit sphere

in the tangent

let SM be the unit tangent bundle of M with the projection

v, w £ Sip) the angle

(v, w) = cos 6. such

that

If M is complete

fold of dimension

manifold.

y

it) = q where

The complete

of isometries

of U.

denote

We list discussed

some

1.1.

c > 0 such

ate asymptotic

if they

have

relation A point

Note that points.

of H.

class

For any point

Fp: Bp -

B

Fp(v)

H such

that

= yvi°°) F

(mostly

t > 0.

topology

defined later

sequel

we shall

that

are

I7j.

if there exists

a

y, a in H/D

on the set of geodesies

of the reverse

the distance

class

the asymptote

curve

between

for

of asymp-

/ —> y(- t).

any two of its

H, and let

H = H U zV(oo).

unit ball in H ; we define

Fo; each

p there

These

invariantly

define

proofs)

See also

a bijection

Let Fpiv) = expp (v/l - \\v\\) if \\v\\ < 1

if ||i/|| = 1.

We shall

without

Geodesies

relation

at infinity

be the closed

of H.

map of rt.

y in // let y(°°) denote

y realizes

with the cone

group

27: H —• M = H/D will

for H is an equivalence

of points

subset

at

in // that are asymptotic.

is a homeomorphism.

always

such a group;

y(- °°) the class

Tl and FpiSip)) = M~).

and let

that

manifolds

discontinous

ß in H ate asymptotic

For any geodesic

p £ H let

K < 0 are the quotient

is an equivalence

//(«>) be the set

a Hadamard

such

-£ (ztz, 72) subtended

in *x1 and §2 of 18].

a, ß

mani-

(O), Ybn(^)'-

27^: SH —>SM the differential

at infinity

of y and

denote

geodesic

D is a properly

ßt) < c fot all

y(°°) / y(- °°) since Let

The angle

and results

lifts

seg-

Riemannian

H will always

$ (y

and

a and

d(at,

The asymptote

totic geodesies

and

and

A geodesic

connected

be the unique

denote

with proofs

in H and in H^D.

equivalence

manifold

Geodesies that

y

t = dip, q).

definitions

detail

K < 0.

from p is

D will always

basic

in great

that

have unit speed

indicated.

simply

M with curvature

the projection

Definition number

manifolds

H is a Hadamard

p: SM — M. If

0 < 6 < n such

all geodesies

otherwise

in H, let

m, n of H distinct

zVl^,

interval.

72> 2 with curvature

(0) = p and

where

that

H is a complete,

If p / q ate points

p by points

assume

on a compact

manifold

number

space

and v £ SM, let y' : R — zM be the geodesic

real line unless

defined

A Hadamard

always

We shall

on the entire

ment is a geodesic

H/D

6 = %. iv, w) is the unique

v' ' V (0) = v.

ate defined

y

Sip)

is a unique

topologies

in §2 of [8], and

the horocycle

mean the cone topology

topology

coincide

topology

on

for each

H is a dense

p open

for H, but in the

if the topology

of zV is unspec-

ified. If p £ H and

^ and

Ypx^

= *•

x fc //(ro),

there

is a unique

If P e " is distinct

geodesic

from points

y

// £ H, then the angle

$

that

y

(O) =

a, b in H, then the angle

¿^(z?, b) subtended at p by a and ¿ is 4 (y¿a(0), y^(0)). set not containing

such

(S) subtended


If S Ç W is any at p by S is

60

PATRICK EBERLEIN

supW

ia, b): a £ S, b £ S\. Proposition

1.2.

V: A —♦ SH given

morphism

Let

The following

results

are basic

A = ¡(p, b) £ H x H: p / b\.

and natural.

The vector

by Vip, b) = y , (O) zs continuous,

and

function

V \ H x z7(°°) zs a homeo-

onto SH.

Proposition

H xHxH:

1.3.

The angle

function

$Aa,

b) is continuous

on {ip, a, b) £

p / a, p / b\.

Proposition

1.4.

The function if/: SH x [- œ, ] —>H given by if/iv, t) = y it)

is continuous.

It follows

sequence,

from Propositions

then

p

If cb is an isometry icb ° a)(oo)

where

homeomorphism comes

a group

corollary

LiD)

A properly

of homeomorphisms points

of the law of cosines

does

nonempty

not depend closed

subset

points

useful

are satisfied

exists

equivalent

single

LiD)

following

Proposition

under

in //( 0 there exists a number r = rip, c) if o: [a, b] — r, then

& ioa, oh) < c. P Property distant have

(3) is referred

geodesies the uniform

to as the Visibility

look small. Visibility

If r can be chosen

axiom in [6J, [7], and L8j since to depend

axiom.


only on c, then we

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II

61

Definition 1.6. M = H/D is a iuniform) Visibility manifold if H satisfies the (uniform)

Visibility

axiom.

Proof of Proposition in [8] in Proposition 4.3 of [6] states Hioo) such

To prove

The assertions

(3) =» (2) and (3) =» (D are proved

4.7 and the discussion

that if property

that

now follows

1.5.

(3) fails

4- ix, y) < rr/2

immediately.

a point

points

x / y be given.

be a neighborhood

for every

point

that

Let

exist

Lemma

points

The assertion

for any two points

%p(x, y) > by Proposition

p £ H.

4.4.

proof of this fact may be found in [15].

only show that

p in H such

Proposition

in H, then there

An independent

(2) =» (3) we need

exists

following

yO») = x, and let

z = y(- °°) and

by property

and let

y.

W

yit) — y() as

(2), if / is sufficiently

large,

then X-y (.z, y) < n/2 and %y¿(x, y) = n - &yt(z, y) > n/2.

Remark 1.7. (1)

Let H satisfy Axiom 1.

If r = r(p, () as in property

(3) and if a is a geodesic

in H such that

d(p, a) > r then Í (a) < e. (2)

i(aP

If a

2. list

is a sequence

of geodesies

in H such

that

dip, a

) —>°° then

)-0." n Limit spheres

some basic

these

and the horocycle topology.

properties

are classically

by Hedlund constant

negative

Proofs

referred

to study

Our approach

curvature.

of some

These

spheres

in H.

For example,

assertions

should

x £ H(°c),

center

There

radius

function

center

x.

property

notions

radius

he obtains

as level

of the space. and removes

The following

p and

function

M = H/D.

in §3 of [8].

spheres

By adapting

level

may be defined

x £ H(oo).

with centers p £ H and

p and the

proximity

of points

lie on the same

an idea

an essentially

sets

in a Hadamard

is useful.


sphere

of Buseunique

are the spheres

in any G-space

all difficulties.

result

manifolds

by a point

the relative

are not well behaved

The law of cosines

or nearly

with the point

by defining

x £ /V(oo) whose

sets

of finite

q should

from x. precise

systematically

of constant

is a point

determined

be some way to measure

for each point

Busemann's

the spheres

be uniquely

x £ Hi). Points

vague

"center"

vary continuously

x if they are "equidistant"

mann we make these

and may be found

in H whose

should

used

and

two

[5L

many of the properties

and it should

should

p, q in H to a point

properties

share

surface

his idea to arbitrary

are omitted

"spheres"

a sphere

the center

with center

We extend

we define

In dimension

and were first

flow on a compact

on that of Busemann

is to define

In this section

(horospheres).

to as horocycles

the geodesic

is modeled

Our goal

x.

of limit spheres

without

with

although

some nice

manifold

H is a nice

62

PATRICK EBERLEIN Proposition

desic

triangle

a, ß and

2.1.

If A, B, C are three

they determine

y satisfies

(1)

c b

that

0 > c — b cos a - a cos ß which proves

c

>a

+ b

- 2ab cos y.

- 2bc cos a. - 2zzc cos ß.

(1).

Thus

Simplifying

To prove

b

we find

(2) we note that

c

>

a2 + b2 - 2ab cos y = (a - b)2 + 2abil - cos y) > 2abil - cos y). It follows dip

r7i'

from (2) above

, q ) —• oo and

diq

jti

Definition 2.2.

, r ) is bounded

'n'

t —» dip,

and bounded

° a;

4-

iq

pn

' n'

y t) - t is monotone

below

by -dip,

that

, r ) —' 0. n

nonincreasing

by the

y O).

/: [a, b] —>R is corzvex on [a, b] if for any

f(\a + (1 - k)b) < kfia) + (1 - k)fib). j

then

in H such

Let B: SH x H -» R be defined by B(tz, p) =

inequality

A function

r 71 ate sequences *

above,

7i

limbec dip' yvl) - + 00

, x ) -

dip « , x n ) = diq'n' , y'Vn s 72) - s 72 + dip, x 72) - dip^77' , x 72) = B S„ iv 72', q^72 ) + dip, x 72) r ^ d(pn,

and

x)

B

—» B(v, q) = B(V(p,

• B uniformly

Definition

2.7.

x), q) = a(p,

on compact

If y = y

x, q) since

subsets

is a geodesic

fy: H —>R is given by fy(q) = B(v, q) = lim

v

—> v, p

—. p, q

by Lemma 2.4. in H then the Busemann

diq,yt) - /.


function

—, q

64

PATRICK EBERLEIN If v = Vip, x) then

fyiq)

= aip,

x, q).

Busemann

functions

have

geodesic

y, f

is a uniformly

the following

properties.

Proposition

vex function

such

(1)

that

For each

\fyip)

- fyiq)\

continuous

con-

< dip, q).

(2)

If y and

ß are asymptotic

geodesies

(3)

// y and

ß are asymptotic

geodesies,

every

then

f

then

- fg = fyißO)

fyißt)

in H.

- fyißs)

= s — t for

s £ R, t £ R. Proof.

and

2.8.

(1) is merely

q be points

We must

show that

x

is any sequence

lim

71 — „{dir, °°

aip,

(3)

of Proposition

x a point

for any point

in Hi°°)

= - t.

at x.

If /

then

limit

sphere

the

x, r) - aiq,

7Z~°°^

x, r) = aip, 2.6,

{dir, x 71) - diq,'

If y and

(2), let

and

p

ß = y

.

x, q).

If

aip,

x, r) =

x 71)\.

Hence

x, q) by Proposition

ß ate asymptotic

geodesies,

= s - t by (2).

If x £ W(°o) and if y is a geodesic

function

To prove

y = J.

x ) - dip, x )\ = a(p,

ß, f Aßt)

fyißt) - fyißs) = fßißt) - fßs)

2.3. that

to x, then by Proposition

aiq,* x, r) = lim

x, r) = lim _^^{diq,

For any geodesic

such

r in H, aip,

in H converging

x 71) - dip,r x 71)\ and

x, r) - aiq,

2.6.

a restatement

in H and

is a Busemann

representing

function

at x through

x, we call

/

a Busemann

at x and if p is any point

p is the set

Lip,

in H,

x) = {q £ H: fq = fp\.

The

limit ball at x determined by p is the set Nip, x) = {q £ H: fq < fp\.

By the

preceding

of the

proposition,

function

these

definitions

/ at x; the Busemann

stant.

It is easy

and that

to verify

that

The result

below

at x is unique

up to an additive

open set

at

follows

of the choice

Lip, x) is the topological

Nip, x) is the convex

is the open ball with center

osition

function

are independent

U

and radius easily

0 N iat)

boundary

where

con-

of Nip, x)

a = y

and

N iat)

t.

from the definition

of limit sphere

and Prop-

2.6.

Proposition

2.9.

Let

p, q be points

in H,

x a point

in H(°o).

(1) q £ Lip, x) if and only if p £ Liq, x). (2) a

q £ L(p, x) if and only

is any sequence

(3)

in H converging

// cb is an isometry

The following radius

function

sphere

/~

(0).

value

Let

then

from

\d(p, a ) - diq,

x; that

is, if

a )\ —• 0.

cbLip, x) = Licbp, cbx).

show that a Busemann at a point

(1) and (2) follow

/ be a Busemann

at x.

q are equidistant

to x,

of H then

observations

whose

(3) and (4) are Propositions sphere

if p and

p is the signed

from the discussion

function

/ is indeed

distance

to the limit

above

and Proposition

3.2 and 3.3 of [8]. function

at a point

x e //(no),

Then


and let

L be a limit

a

2.8.

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II (1)

The

(2)

Each

(3)

Let

which

est

y

limit

spheres

geodesic

at x are the level

y representing

x meets

rj, : H —> L be the function

meets

L.

Then

for any

sets

that

65

of /.

L precisely

sends

p £ H, r¡ Ap)

once.

p £ H to the unique

is the unique

point

point

at

of L near-

p. (4)

Limit

spheres

at x are parallel;

that is, if L

is another

limit sphere

at

x and if p £ L, p' £ L' ate arbitrary, then ÁL, L') = dip, L') = dip', L) = \fL -

fL'\ = \fp-fp'\. Next we show

that

a limit

sphere

Lip,

x) depends

continuously

on the points

p and x. Proposition

2.10.

For any open set

Let

Lip, x) be a limit

0 of H containing

sphere

in H, and let

q we can find open sets

q £ Lip,

A and

x).

B of H

and Hi°°) such that ip, x) £ A x B and if ip', x) £ A x B then Lip', x) O 0 ¿0. Proof.

0-(p, x, q) = 0 where

open set

0 of H containing

of radius

e and center

q.

and B of H and f/W \aip',

x , q)\ < (.

o = y xi.

dip',

a is the function

q, choose

c > 0 so that

By the continuity

2.5-

0 contains

Given

an

the open ball

of a. we may choose

open sets

A

such that (p, x) £ A x ß and if (p', x') £ A x B then

These

are the desired

x , at) = - t + aip',

then \tA < ( and aip',

of Definition

open sets.

Given

(p',

x , q) by Proposition

x)

2.8.

£ A x S,

let

If t. = a.(p',x',q)

x', otA = 0. Since at £ 0 fot \t\ < t, it follows that

otQ £ Lip', x') n 0. The

spect

Euclidean

and hyperbolic

to the geometric

limit sphere

properties

spaces

of their

Lip, x) is a hyperplane

represent

limit

the two extremes

spheres.

through

p orthogonal

and the corresponding

limit ball is the open half-space

rays

making

an angle

< 7r/2 with

limit

sphere

at

P(°°)

at

x is a Euclidean

x; the only geodesic

We now define

spheres

for

a first

lowing

in the limit

countable,

H = H U H(°°) where

ball at x £ H(°°), we call result

2.11.

T

P that ball

space

y

topology

H is any Hadamard

model),

h induces

for each

manifold.

If N is a limit

limit ball at x. The fol-

3.7 of [8].

There

the original

x £ f/(°°)

a

on limit

exists

a unique

(horocycle)

topology

h on H such

topology

of H, and

H is a dense

open subset

of H, (2)

x,

.

based

that (1)

a

to the unit sphere

is the geodesic

horocycle

space

of all geodesic

P (disk

is tangent

Nx = N U ixi an augmented

is Proposition

Proposition

in

with re-

to the direction

composed

y px . In hyperbolic

sphere

In Euclidean

the set of augmented

limit

basis for h at x.


balls

at x is a local

66

PATRICK EBERLEIN If D is a group of isometries

LhÍD);

r/(°°) of an orbit

pendent

Dip) with respect

of the point

of horocyclic

need

LhiD)

a horocyclic

under

D.

topology

OAd)

not be closed

in /7(°o) with respect

complicated

groups

D.

x £ Hi°°) and any cone neighborhood

4.8 and 4.9 of [8]).

horocycle

2.12.

Proof.

Let

in

is inde-

is the set

to the cone topology

If H satisfies

W of x there

The result

Let

topology.

limit ball

LAd)

= Hi°°) - LAD)

p

p £ H and let

pn £ N for n sufficiently

p £ H, dip

a=

large.

y

to x £ W(°°) in the

, Lip, x)) —* "».

For any number

. Since

c > 0 consider

pn —> x, in the horocycle

If / is the

limit

later.

in H converging

L = Lip, x).

x) where

an augmented

L¿(d) C LÍD) (see Prop-

will be useful

be a sequence

For any point

N = Niac,

below

and may

Axiom 1, then for

exists

ball Nx such that Nx C W. From this it follows that

Proposition

h.

points

points.

for even quite

ositions

limit set

is the set of accumulation

to the horocycle

p and invariant

ordinary

L¡ÍD)

be empty any

of H, we may define

if p is any point of H, then

Busemann

function

the

topology

fa at x, then

by

remark (4) following Proposition 2.9, dip , L) = \fp - fp\ > \fp | - |/p| > c - \fp\ tot

n sufficiently 3.

The sets

with curvature sets

large.

Wsiv),

stable

Wssiv),

For each

the sets

this

are smooth

In this

section

Let

c > 0 was arbitrary.

M = H/D

These

in modern

global

if M is compact

with

relationship.

of the geodesic

since

be any complete

v £ SM we define

Wuuiv).

manifolds

to suggest

for the study

Wu".

vector

Wuiv) and

for example

intended tools

follows

Ws, Wss, W" and

K < 0.

and unstable

situations,

The result

sets

generalize

analysis

the concept

p. 30].

The

and unstable

sets

flow on SM even though

manifold

and unstable

and coincide

K < 0 [l,

The stable

stable

of

in certain notation

is

are useful

it is not clear

that

manifolds. we define

the sets

and develop

some

basic

properties

of

them. Unit vectors

geodesies, piv)

and

v, w £ SH ate

v, w ate strongly piw)

(strongly)

(strongly)

asymptotic.

Passing asymptotic

asymptotic.

denote totes

Similarly in SM.

respectively of v.

and strong

These unstable

limit

the weak, sets sets

determined

asymptotic

to a quotient if they have

one defines For a vector

strong

SH) let

weak unstable

will be referred determined

manifold lifts

yw ate asymptotic

asymptotic

and if

by x = y (°°) = ywi"°i-

if - v and

asymptotes

v £ SM (or

strong,

it yv and

if they are weakly

sphere

unstably

v, w £ SM are weakly

asymptotes

asymptotic

asymptotic

lie on the same

v, w ate weakly

weakly

- w ate weakly

M = H/D,

unit vectors

v*, w* £ SH that are weakly and weak Vls(v),

and strong

to as the stable,

by v.


(strong)

Wssiv),

strong

unstable

Wuiv),

unstable stable,

Wuuiv)

asympunstable

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II If is an isometry of asymptotes

It follows

that

that the stable

in SH; that

is,

three

on the vectors

Proposition denote

from Proposition and

and unstable

the geodesic

2.9(3)

and the definition

= W"(c/j#i/)

for any

in SM ate projections

relations

v £ SH.

of those

v £ SH, and similar

All four asymptote

in SM and

3.1.

If A is any set

sets.

SM is a surjective

in SH onto open sets

= Ws(tt^v)

exists

_

in SM.

by keeping

local

(b)

it suffices

and

similarly

of the corresponding

Proof of Proposition

sets

3.1.

the sets

Wss,

to consider

in mind the fol-

homeomorphism

If T

again

flow in both S H and SM then 27+° T = T ° n^.

tions

w £ Wsiv))

Then there

(1), (2), (4) and (6) of the proposition

(a)

open sets

desic

£ U then

w.

If w £ Wssiv), then Wssiw) Ç WsAv).

Remark

lowing

if v

irespectively

_

(1), (2), (4), (6) and (7).

(2)

w £ Wss(v)

0/0).

(7)

ments

Let

and car-

denotes

the geo-

(c) For any v £ SH,

Wu and

Wuu in SM ate

projec-

in SH.

We assume

in (1), (2), (4) and (6) that

v £ SH.

(1) It suffices to prove that, for any t £ R, TWAv) Ç WAv) = WAT(v) and TWsAv) Ç WsATv). If w £ TWAv) then T_w £ WA\v) and the geodesies yT

and

WAv).

The same

y

Now let

tion at x.

are asymptotic. argument

w £ Wss(v);

By assumption

Therefore

shows let

fiy

that

y

x = y (e*) = ywi°°)

O) = fiy

and

y

are asymptotic

and w £

Ws(v) = WA.T v). and let / be a Busemann

O) and, by Proposition


2.8,

fiy

func-

t) =

68

PATRICK EBERLEIN

- t + fiy 0) = - t + fiy O) = fiy t). Since the points piT(w) and piT{v) lie on the same

limit sphere

determined

by x and since

T w £ Wsiv) = WsiT(v)

we see

that

Ttw £ WssiTv). (2)

Let

w £ Wsiv),

x = y («j) and / be a Busemann

fiyw0) - fiy 0). Then fiyj) lie on the same

definition

Let

sequence

determined

v be periodic such

Passing

7i

=t

that

v

let

^

This

equivalence

follows

relations

(5)

This

(6)

Let

v

cn

c

n

For each

n there

a „ such

=T

,«

ln~an

Then

T

f

cn

Let

exists

that

e W"(T

n

v

~an_

a number

of the definitions

p and

and let

q be points

By the continuity

(7 is an open set

choice

If v

of C and

such

n

by (1 ).

c

relations

are

in SH and SM.

2.10 we may choose

morphism.

t

,,tz) = W"(i;)

the asymptote

of the sets

Wu and

Wuu.

0 be an open set of SH containing

in H such

that

of V (Proposition

open sets

C and

in 5/i

£ U, then

D there

containing

D oí H and

v = Vip',

exists

g

v since

x)

V:

for some

w.

w = Viq, x), v = Vip, x) and

1.2), we may choose

open sets

/j(°o) such

e C x D and if (p', x') £ C x D, then L(p', x') O A / 0. then

be a

a n co — < t n < ia n + l)co

A and B of H and H(oo) such that iq, x) £ A x B and V(A x B) C 0. osition

by

—>T w £ Wssiv).

n

from the fact that

v £ SH, w £ Wss(v)

q £ Lip, x).

/ =

T w £ Wsiv) it follows

w £ Wsiv).

integers &

—> c.

71

immediately

is a restatement

If x = y (oo) let

co > 0 and let

—>w.

n

Choose

72

to a subsequence

e

(4)

by x and since

with period

- a co. Then F

n

Let

_

T tn v n £ \Vssiv) byj (2). \ i \ i

and let c

at x.

The points piT w) and piv)

T w £ Wssiv).

in Wsiv)

^

that

limit sphere

that

(3)

= -t + fiyw0) = fiyji).

function

By Propthat

(p, x)

If U = V(C x (B n D)),

H x //(oo) —> SH is a homeo-

ip',

x') £ C x (B O D).

e L(p , x) O A, and by the choice

By the

of A and

B, V(V, x') e Wssiv') n o. Now let

/ £ R such

ziz e Ws(v)

that

and let

T w £ Wssiv),

0 be an open

and let

set

such that if v' £ U then Wssiv' ) O T 0 / 0. T_tWssiv') no/ 0. _ set (of SM or SH) containing

an open

set

Since

(7 containing

zzz e |f"(t/)

there

Wss(zy ) n Q ^ 0.

trary, _W^%v)

exists

Since

Ç Wssjv).

w such v

v*.

that,

Since

£ Wssiv)

Let iz* £ Wss(w) Wssiw)

for every

0 was arbitrary,

Similarly

w.

Choose

v

If v £ U, then Wsiv') n 02 _

(7) Let v £ SM (or SH) and let w £ Wss(v). an open

of SH containing

U be an open set of SH containing

v

£ U, Wssiv')

n U, and therefore zv*e Wssiv),

one may use

and let 0 be

O O ^ 0, we may choose O O / 0. Wssiv)

and since

(6) to show

O O =

iz* was arbi-

that if w £ Wsiv),

then Wsiw) Ç Wsiv). 4.

The relationship

on SM.

We first

geodesic

flow

list

of the sets

some

basic

Tt on SM, M = H/D,

Ws, Wss,

properties and relate

Wu and

Wuu to the geodesic

and dynamical these

concepts

to the stable


flow

for the and unstable

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II sets.

Propositions

4.6 and

In Theorems

tivity

and topological

very likely

that

these

but this

sets,

Theorem

4.11 exhibit

mixing

ergodicity

in terms

to investigate

[10] is a good geometric

tion 4.5 contains

the basic

v £ SM such

Wssiv)

horocycle

Definition

4.1.

(0) C\ U /0

of this

sets.

It seems in terms

4.16 is a corollary

of

is a generalization

The discussion

discussion.

of

on pp. 248-250

of the sets

Wss.

The existence

in SM is equivalent

of

Proposi-

of a vector

to the existence

of a tran-

terminology.

we can find a sequence

fot every

of

point of SM is nonwandering.

If v £ SM, P iv) = \w £ SM: for any neighborhoods

SM of v, w respectively T

[12]).

transi-

be formulated

mixing

of the importance

is dense

in Hedlund's

that every

([ll],

idea

and unstable

Theorem

topological

description

of topological

can also

investigated.

that used by Hedlund for surfaces

that

mixing

we do not assume

used

relations.

the problems

of the stable

and metrical

has not been

4.15, in which

The method

sitive

fundamental

4.14 and 4.16 we formulate

69

n\.

t

C R, t

P iv) is the positive

0, U in

—> + °° such

prolongational

that

limit set

of

v [2]. Similarly requiring

P~iw). there /

72

one defines

in the definition

Equivalently exist

v

72

Definition

For any under

T .

/ £ R and

/

that

C R and

—> v and

4.2.

Q denote

above

the negative /

one may define

sequences

—►- oo)

Let

P~iv),

T

v

v

C_ SM such

the set of nonwandering

It is easy

P~(v)

to verify

if and only

P~(v)) t

if v £

to be \w £ SM:

—►+ °° (respectively

+

if v £ P iv) (equivalently

points and

v £ P~iv)).

in SM.

Q are closed

the following

v £ SM ate arbitrary

(1) T((-v) = -T_tv, (2) P-(- t/) = - P%),

that

—• w\.

(„72

limit set of v, by

—> - oo. w £ P iv)

P (v) (respectively

v £ SM is nonwandering

v £ SM, P iv),

prolongational

subsets

properties

of SM invariant

of the geodesic

flow.

If

then

P\-

v) = - P-(v),

(3) Q = -Q. The following Definition

For vectors

idea

4.3.

will be convenient

Let

v, w £ SM we say that

0, U in SM oí v, w respectively,

Using

the definitions

Proposition (1) some

4.4. t

t

related

following

that

of topological

mixing.

I 72 —. + or / 72 —>- oo.

to w if for any neighborhoods

fot n sufficiently

Definition

large.

4.2 we easily

obtain

Let M = H/D be a complete manifold with K < 0.

C R,

72 —

v is

such

T( (0) O U / 0

and the facts

If v, w £ SM, then

sequence

in the discussion

t ZZ Ç R be a sequence

t

72

w £ P+(v)

if and only

if v is t

related

—>+■oo.

(2)

v is

tn related

to w if and only

if w is -t

(3)

v is

tn related

to w if and only

if - w is

t


related

to v.

related

to -v.

to w for

PATRICK EBERLEIN

70 (4) x ' t

n

For each

(5) t

v £ SM and each

—» - oo {w £ SM: v is t

n

For each

exists

v is t a further 1

7i

to w\

w £ SM and each

—>- oo {v £ SM: v is t

(6)

sequence l

related

related

n

sequence

related

to w\

r

í

of s

n

'

t

t 7i —,+

72

v and T r„ v n —>w.

Proposition some

sequence 1

4.5.

t„n — C R,

If v* £ Wss(v)

(2)

If w* £W""iw)

then

(3)

// v* £Wss(v)

and

Proof.

(1)

show that

to w.

—►v and

72

T

rn

Viq, x) and

Viq

a

'

n

—* w.

to w*.

result

v* —»v* and

T

7i

s

rn

r

^72

p £ Liq,

x).

will also

of s

Since

—> q and

v is t

v 72 — C SM such

work for v*.

of v*, v and q

to

72

be given.

r 72 of s 72 and a sequence *

r

for any

it suffices

v* —>w.

of t

v

x

Let

respectively

such

—» x by Proposition

and c^ = YV(q x y then a/ (o) -> a'(0) r ) we show

72 72

that

The vectors

that

T they

on

V(p, a

n

subtend

dip, q 71) is bounded.

lim„_oo

^7- w„ = w-

limn_^

Tr v* = lim^^^ £ Wuu(w)

related

71

—> 0 as

and, by Proposi-

then

to

T

rn

Viq

'n

, x ) are tangent 71

\a

Thus

°

, a 72r 72) and

72 '

- diq,

a r )|

\a - r I —♦ 0. '7272'

to H at

following

This

proves

'

that

= Wssi-

w).

v is /

n

related

to w*.

from (1) and (2).

Let M = H/D and let v, w £ SM.

(1) If w £ P%) then Wu(w) C P+(v). (2) If w £ P~(v) then \Vsiw) Ç P'iv).


a r

v* is l

- y by (3) of the preceding

to - v by (1) and

4.6.

2.9,

n —. oo by the remark

- w* £ - V/uu{w)

(3) This follows immediately Proposition

By Proposition

v* —> v* = Z7+V(p, x) since

Ta v* = w.

related

r 72 = d(q'

a r , and 72 72'

Proposition

If v*72 = n±Vip, a r ), then lim 72 — «O T í, v*í = *r'7272

Furthermore,

to w, - w is t

Since

pr £ Liq," x) and a n r 71 —>x.

r ) and

2.1 since

lt"w*

\a 72 - r 721I —» 0. '

\d(q, a r ) - r | —> 0.

that

n

tn related

we can find a subsequence

Viq , x ) £ SH be lifts

the angle

t

is

we need only show that,

The sequence

= \dip, a 7272 r ) - diq,J a 72r 72)|' —» 0 since

- w* is

v*

72(r 72 ) —> a(oo) = x.

qn —» q we see

(2)

then

result

of t

that

, x ) —» Viq, x) and

If a = yv(9x) 1.4,

to w*.

and let a subsequence

v

to w £ SM for

to w.

By (6) of the preceding

s

n —

If a 71 = d(p, a r

lated

related

to w we can find a subsequence ^

v

1.2. tion

t

w* £ Wuuiw)

related

v* £ Wss(v)

Vip, x), that

is

v is tn related

v* C SM such

^

related

v*

for any subsequence

and a sequence Let

then

By (5) of the preceding

v* is /

related

t n —> + oo.

(1)

tz* £ Wss(v),

that

Let M = H/D, and let v £ SM be t

—>x.

related

Since

result.

v is

Finally

to w. t

n

re-

Therefore

GEODESIC FLOWS ON NEGATIVELY CURVED MANIFOLDS. II Proof. P'iv)

t

We shall

and let

prove only (2) since

w* £ WAw).

For some

related

to v since

related

to v by the preceding

72

t and

v £ P+iw*)

P~iv)

is closed

and the stable

discussed

that

Let

w* is it

since

w £

tn —* + °°, w is

T w* is

Z

Therefore,

P~

clarify

sets.

in terms

t

+ t) related

w* was arbitrary

Let

a sequence

the relationship

For this purpose

in [7] and [8] and details 4.7.

further

of the action

to v and

between

it is useful

of D on H(o°).

P , P~

to characterize

The following

idea

is

may be found there.

D be a properly

x, y £ rV(oo), not necessarily

exists

Ç R such

t £ R, T w* £ WsA^w) and

WA^w) C P~iv)

4.11 we shall

and

Definition Points

result.

t

in SM. and unstable

P

For some

or tz;* £ P~iv).

In Proposition

the sets

v £ P+iw).

the proof of (1) is similar.

sequence

71

discontinuous

distinct,

r/j Ç D such that

group of isometries

are dual relative

p —>x and

of H.

to D if there

~ p —»y for any point

p £ H. Note that

for any point

for any point 2.1.

Hence

r/j q —>x for any point

Dual points point.

q £ H, di(f> p, yi w*„(oo). "n

of a geodesic

Xp^nP'

of V(oû)) = ^ y

„(00).

P'(v)

Ç WAv).

If t/ 6 Q then v £ P'(v)

Therefore

P'iv)

Hence