Maximal ergodic theorems and applications to ... - Springer Link

ISRAEL JOURNAL OF MATHEMATICS 139 (2004), 319-335

MAXIMAL ERGODIC THEOREMS AND APPLICATIONS TO RIEMANNIAN GEOMETRY BY S E R G I O .'~IENDON~A AND D E T A N G Z H O U *

Instituto de Matemdtica, Universidade Federal Fluminense (UFF) Rua Mdrio Santos Braga SIN. Valonguinho, 2~.020-140 Niterdi, R J Brazil e-mail: s_mendon, ca(@hotmail.com, zhou@im,pa.br The first author dedicates this paper to his parents Josd Martiniano and Zoraide ABSTRACT We prove new ergodic t h e o r e m s in t h e context of infinite ergodic theory, a n d give some applications to R i e m a n n i a n a n d K/ihler manifolds witho u t c o n j u g a t e points. O n e of t h e consequences of t h e s e ideas is t h a t a complete manifold w i t h o u t c o n j u g a t e points has nonpositive integral of t h e i n f i m u m of Ricci curvatures, whenever this integral makes sense. We also show t h a t a complete K/ihler manifold with n o n n e g a t i v e holomorphic c u r v a t u r e is flat if it has no c o n j u g a t e points.

O.

Introduction

Infinite ergodic theory is the study of measure-preserving transformations of infinite measure spaces. A class of very natural examples is that of null-recurrent Markov chains (resp. their shifts) such as the symmetric coin-tossing random walk on the integers. There is a great variety of ergodic behavior infinite measurepreserving transformations can exhibit, and they have undergone some intense research within the last twenty years, much of which is associated with the name of Aaronson ([Aa]). In his book Aaronson studied the standard a-finite measure spaces and non-singular measure preserving transformations. This paper will provide another class of natural examples in the category of infinite ergodic theory. We prove new maximal ergodic theorems, which include *

Both authors are partially supported by CNPq, Brazil. Received January 28, 2002 319

320

S. MENDON~A AND D. ZHOU

Isr. J. Math.

some known geometric results and have some new geometric consequences. We will restrict the discussion on manifolds even though some of the results can be generalized to more general cases. Let N be a manifold equipped with the a-algebra/3 of Borel sets, a flow T¢, t E R, and a Tt-invariant measure p. Let g: N --+ R be a measurable function. We say that a measurable flmction g has w e l l - d e f i n e d i n t e g r a l if either the positive or the negative part of g is integrable on M. We start with the statement of the classical Maximal Ergodic Theorem (see [Pt], here we use "inf" instead of the usual "sup" for later convenience in applications).

Let g be a measurable function, with welldefined integral on N, and Z C N be a Tt-invariant Bore1 subset. Set MAXIMAL ERGODIC THEOREM:

E[g]=

{

wee

I inf

s>o

(

}

g(rtwldt < O .

Then fEM gdp < O. Our main ergodic result is the following new maximal ergodic theorem. THEOREM 0.1: Let f be a measurable function with well-defined integral on

N and Z C N be a Tt-invariant Borel set. Consider the following Tt-invariant subset, E(f) =

{

w E Z I liminf

L+

f ( T t w ) d t < O, liminf

r

}

f ( T t w ) d t < +oc ,

where Is denotes the interval [0, s] ifO < s, and [s, 0] if s < O. Then rE(l) f d# tl. Then, for all c > O, we

have ~l = l i m i n f t ~ + ~ f:l f(s)ds < +oo. Now we are in position to prove T h e o r e m 0.2.

Proof of Theorem 0.2: First we consider the hypothesis (a) in T h e o r e m 0.2. Suppose by contradiction t h a t rE(I)fdp. = 0 and t h a t there exists w E E ( f ) with f ( w ) ¢ 0. Since f is continuous, and N has few recurrent orbits, we can assume without loss of generality that w is in the interior of E ( f ) , and t h a t 7w is not constant. So there exists a neighbourhood W of w of the form [0, ¢] x B, where B is an open disk and W is given by the T h e o r e m of the Tubular Flow. We can assume also t h a t f ¢ 0 on W, and t h a t e is the time t h a t each connected c o m p o n e n t of an orbit remains on W. Let /) be the Borel subset of points x c W, such t h a t at least one of the pieces "/x[t>o, %It 0. Let E be the set of points x e /) satisfying %,(t) ~ W for sufficiently large t > 0. We have It(E) > 0 or p , ( / ) \ E ) > 0. W i t h o u t loss of generality we assume t h a t It(E) > 0 (the other possibility can be treated similarly). Given z E E, let ~/~ be the last connected c o m p o n e n t of ~.~ which enters in W and let %~, %3. . . . be the preceding components. If x is in %J we set j(a') = j. So we define g: E ( f ) + R given by g(x) =

max{i(x, I ) , - 1 } 2J(x)e

if x E E,

g(x) = 0 if x e E ( f ) \ E .

It is not difficult to see that g is a measurable function.

In fact, the flmetion

~: E x R 2 --+ R given by ~(x, u, v) = j::' f(Tt:r)dt is measurable, hence i(x, f ) is measurable. By the continuity of the flow Tt, for y in a small neighbourhood of x we have j(y) > j(x), so the function j is semi-contimmus, hence it is measurable.

326

s. MENDON~A AND D. ZHOU

Isr. J. Math.

Note also that

f g(Ttx)dt J

= max{i(x, f ) , - 1 } 2J '

hence 0 > f+~ g(Tta')dt >_max{i(x, f ) , - 1 } . So we define f = f-g. By considering the cases i(x, f ) _< - 1 and 0 > i(x, f) > - 1 we still have i(x,f) < O, i+(x,f) < +cx~ and i_(x,f) < + ~ , for almost all x E E(f). Then we have rE(f) fdp, < 0 by Theorem 0.1. On the other hand, we have

This contradiction proves Theorem 0.2 under the hypothesis (a). Now we assume hypothesis (b) of Theorem 0.2. By integration we arrive at inequality (1.1). Fix ~ > 0. First we rewrite the inequality (1.1) in the following fornl: (1.2) a , ( t 2 ) - x ( t l ) + a e

r

xs(t)dt+

dtl

For ahnost all w E

E(f)

fl (f(t)+(1-e)ax2(t))dt O, such that 2 = x o 7~ satisfies the Riecati inequality: 2' + a~:2 + f o 7

< 0

( respectively .~ - a~:2 + f o ? >_ 0). Then f - O. Proof of Lemma 2.1: Given ally filtration of M by bounded Borel sets Di, we have a corresponding filtration SDi of S]~./. By the Fubini Theorem we have (2.5)

/D S / d v i

l Wn--1

f~

fdtt. Di

So if fSM f d p is well defined on S M it follows easily fi'om (2.5) that the integral of S I on M is also well defined. Lemma 2.1 follows from Theorems 0.1 and 0.2.

|

Proof of Theorem 2.2: We can assume that there exists a well-defined vector field X = J ~ / o n S M which is parallel along geodesics. We note that the fimetion x = (UX, X } is mea~surable since U is lneasurable by [Gn]. It is well known t)y a result of Berger ([B]) that n+l SM -- 2(2n - 1) SH,

Vol. 139, 2004

MAXIMAL ERGODIC THEOREMS

331

where n is the complex dimension of M. So Theorem 2.2 is a direct consequence Lemma 2.1 together with Proposition 2.1, item (a). Note that it is well known that. I((w, Jw) =- 0 implies that M is flat,.

|

Proof of Corollaw 2.1: It follows directly from Theorem 0.2 and from the wellknown fact that. H -- 0 implies that M is flat,.

|

Now we prepare the proof of Theorem 2.1. When we have f: M --+ N, we can define f: S M + R by f(p, v) = f(p). In this case we have S f = f . LEMMA 2.2: Let f, f be as above. If f has well-defined integral on M then f has well-defined integral on SM.

Proof:

We have that the negative or the positive part of f is integrable. For example, let us assume that the negative part of f is integrable. Consider a filtration Di of ~I by con:pacts sets Di. Note that. ( f _ ) - = ( f ) _ . So we have

fDi f-d~/7~"/DiS((f_)-) d~'7- ,~l;,z~ 1 /~Di(])-d'p'--~~Alf -(1~7" This implies that ( f ) _ is integrable. So we conclude that the integral of f is well defined on SM. |

Proof of Theorem 2.1: Consider R: M -+ IR given by R(p) = infves{p} Ric(v). If we assmne that R has well-defined integral on M we conclude by Lemnm 2.2 t h a t / ) has well-defined integral on S.hI. If .hi does not contain conjugate points, then for all w E S M Proposition 2.1 (b), (d) imply that i(u,, ft) 0 on the interior of B and a - 0 on the b o u n d a r y given by f ( 7 ( t ) , x ) =

Isr. J. Math.

OB. Consider f : N ~ R,

a(x)sin(~rt/s), for ( 7 ( t ) , x ) E W, and f = 0 outside W.

Fix an orbit or: R -~ N passing infinitely m a n y times in b o t h directions t h r o u g h W. Set u = a(0) = (7(0),xo). Consider sequences uk -+ - c ~ , v k -~ +cx~, such t h a t a(uk) is of the form (7(-e),xh~), x~, E B, and a(vk) is of the form (7(~/2), Yk), Yk E B. Thus we have

,,k f(Ttu)dt = k

I

TM-~'~

f(Ttu)dt +

~' 'uk

f(Ttu)dt = 3~/2

f(Tt'u)dt < O. k--3¢/2

i(u, f ) < 0. It is not difficult to see t h a t i+ ( a ( - z ) , ]') _< 0 and i - ( a ( - e ) , f )