Mathematics, Northeastern University, Boston, MA
Statistics, Stanford University, Stanford CA
ABSTRACT. Tivo results are established ,o.n.cerniiig the ability of a set
to cover any ele ' m ent of a class of sets t~nder Ihe.acion of an c riyodic flow.
The classes considered are th class of couj .,*able sets and the class of sets
measure zero.
Rtsunit. Deux r6sultats sont d~mon'tr~s concernant )a possibilit6 pour
un ensemble de recouvrir, sous Paction d'un flot ergodique, tous les 6Mments
c
d'une classe Wensembles. Les classes cons:d~.r~es sont ]a classe des ensembles d6nombrables et ]a classe des ensembles de mesure z6ro.
1. INTRODIXTION
One of the earliest results in ergodic thwny is Poinear~'s *recurrence
theorem sv'hich states that if a measure pieserving transformation on a finite
space has no non trivial invariant sets flien e iery set of positive measure
flits almost every point infinitely often under Ilic action of the transfor ,
mation. The first result of this paper a to Poincar6S,
theorern by Illat if 1 l); , llow on a prob
Armaks do 1'1t.,stlitit Herri Pc).,*i.,, ari Seel ion B Voll. NV. n, 1 1979.
34 M. H. ELLIS AND 3. M. STEELE
ability space has no non triiial invariant sets then there exists a set of measure
U
,~ro which completely contains each countable set. infinitely often rider' fhe action of the flow.
To isolate the difference between 'this result and PoincarCs theorem we note that the representation theorem of Ambro se [11 easil
Y implies there is a set of measure zero ,,,hich will catch any element of 0, but a more elaborate procedure is needed to catch every co,, ,n.able subset. Also we' note that under the action of a set of d;screte trap sform at ions { Ti : j c Z } one can easily show that no set other than 0 is capable of catching every countable set.
Our second theorem shows that no set of less than full measure is able to completely catch each set of ineasure Zero, hence the countable.set in the result mentioned above cannot be substan t i ally enlarged. An essential step in the proof of this second result is a lemma ,,,hi.ch states that given
any positive real. numbers { aj for which 3~i < 1 there is a closed
'I
set F c: [0, 1] of measure zero such that F rt 1, for any collection
U
of intervals Ii } with lengths { ai
We will call a collection of measure preserving., transformations
T, : 1 c R a ineasurable' is, on the complete p, ot2ibility space (Q, ~~, p)
J10
prov ided (1) T, is a bimeasurable measure presc n,ir,2 transformation of QI
onto f? for each t c R, (2) T,,+, = T, o T, and (3) ]M* (co, i) : co c T,M
is measurable in the product space n x !R for each M c 9. A flow
T, : t oR 1 is ergodic provided that p(A) > 0 implies that UTA is of. ten
full measure. With these preliminaries, we now have our main results
2. CATCHING THEO.R..EMS.
T IEOREM 1. Suppose T, is a measurable ergodic flow.on a complete probability space (Q, 9, p). There is a set A of m casure zero such that for any countable C there is a i t(C) such t~at C c T,A. In fact, for any such.C, the set S t : C c TIA is dense in 2.
Proof. To prove this result we will use Rudolph's representation theorem [6] assharpened by Ki en el [51. Let { T,') be an ergodic measurable iow on the coii.,pjete probability (0, 9, p). FLI 111 Cri Ilet p, q be two.po.sitive
Arnales de ]']c:. rl '. ut jicnri poittca,~ 5..,ction B
CAfCHING S.MALL SETS C',;DER FLOWS 35
teal num bers with plq irrational. Then there exists a finite measure space. ~B, &*, v), an ergodic, measure presen ing, invertible map S from B to B and a set D in 9* with the following property. Let
f2' = (D x [0, p)) u (D' x [0, q)),
a' be therestriction to Q' of the completion of the product of 9* with the L6besaue measurable sets, and letu'be the restriction to a* of the completed, Product of v with Lebeseue measure. Let T, be the measurable flow on (Q', 9% p') satisf~k?ing
+ x c= D and r < p r
(~x, r + t p) xeD and t ~~P r
T;(x, r) (x, r + 1) x c D' and t < q r
((Sx, + i . q) xeD' and t,?: q r
for all (x, r) c fl' and 0 :9 t :5 min (p, q). There is a set N c= wit~ p(N) 0,
T,(N) = N for all i c 5q, and a b 1i nz asu, ab'je rneasure prescriIilg bijedtion
Q N Q' such that T,(w) for all t e R and
co e 0 N.
In words, T, IS is the flow on (f2', 9't p') which sends each point (x, r) in D x [0, p) (respectively, D' x [0,,q)), Lpv,. ard at unit velocity until the second coordinate reaches p (respectively q), at which instant the point
. z
((jumps)) to (Sx, 0) and continues iiio~ inLupvard; (D is a measu.r.c .pre.,;crvlng
isomorphism from (Q N, to (!Q', which carries T, to T'
ty t'
Let ( Q', 9% T,'}, N and (D be as above. For each positive integer n
let 0. be a dense open subset of (0, 1) i,.,bos, measure is less than 2', and
let
and rq~1e0J
Q, {(x,r):xc D and rp
For each positive integer n. p'(Q.)
Furthermore, for each (..r, r) c Q' and positis c integer n one has that
(x, r) e T;(Q~)
is a dense open subset of R. The Baire Caterogy Theorem now implie's. that for each countable subset C of 0',
A(C) (X. r) c T' JQ
nn~
n 9 x.r ~~C
1 3979.
M. 2. kt LIS AND J. %I. STEELL
is dense in R. We then see that i e A(C) if and only if C c T,' nQ.).
C'01
Tbus A = NuO 1 catches each countable subset of Q at a dense
nQ.)
( n 0.~ 1 1
set of times, and p(A) = 0.
Before proving our second theorem we establish the ]cmma, mentioned in the 'introduction.
LEMMA. Let aj 1` 1 be a countable set of positive reals Ibs which
~oci = oc < 1. There is then a closed set F of measure zero, F c= 10, 1),
CO
such that F 4 Uli for any collection of open intervals { Ii } satisfying
aj.
Proof. We will first construct closed subsets F;. c [0, 1) such that
Fk + 1 Fk, t?i(Fk) 0, and such !hat F, c~ Uli for any { Ii ) with
n2(1 j) a,
We.can assume the cci are monotone dcc ,e ,s ine. and use the fact that
aj < 1 to choose ard increasing sequence of po siltive intc ers r, r2, satisfying
+
4
(2.1) 2 ai + + 21 1 ai
for some e > 0. Sequentially choose positive nj 'i= 1 to satisfy
(2.2) ni > (2eoc,,) '
and
(23) > (2
feT all 1 ~t 1 .
Let F,
U U11n, jlpz, + 1/2nj. Next F, is defined by partilloning
J=0
7
0
CATCRING SMALL S~TS USDER FL9W3
37
each interval of F, into 2n2. equal intervals andlettin F, be the set formed by taking every other one of these smaller intervals (including endpoints). Similarly F, is obtained by partitioning each in~eri~al of F,~, into 2ni equal intervals and taking every other one (inclueing endpoints) as an interval of Fl. We have F,,, c F, and ni(F) = 2~1. We need to show
cc
that for all 1, F, I: UIi when ai.
i=1
Fix 1 and define a measure of the efficiency of covering Fi with an interval 1
.oflength x by
eff (x) = sup { in(l c) Fl);1xm(F1) : 1 is an interval of length x
Inequality (2.2) was imposed precisely to guarantee that
(2.4)
eff (X) < 1 + £ if X 2~: a,,.
Similarly, (2,3) insures that eff (x) < 2 if x ~~ 7, eff (x) < 22 if x ~t a".,, and generally
(2.5) eff (x) < 2k 1 if
rk p
2 :!~ k :5 1 + 1.
Suppose now { Ii is any collection of open in' tern als with aj ?77(1
Then (2.1), (2.4) and (2.5) imply
172
(( UI) n F,)
i=1
j efr(7i) +
< + Z)7 i +
<
i=1 +r,
r2
2.ai+ 2 2.ai + ?7i(Fg)
i= 1 +r,
This last inequality shows F, t U li, as claimed.
CO
To complete the proof we let F = nFi. Since ?7i(Fj,.) = 2 & we see
i=1
7?z'F) = 0. Now for any coflection of open intervals Ii ith m(l) = a,
c
cc
U ii 0. Since the Gk are nested and compact
,ve have Gk Fk n
VOL XV, nt 1 19791.
f
33 3.1. H. ELLIS NI.
C
we also have F n Ii 0 and the proof of the lemmia is complete.
U
In the next result we use Rudolph's represe ntaltion theorem to create circumstance~ where the preceding leninna can be ej'iecti , cly. applied.
THEd~ Em 2. Given any A of less than full measure there is an E of measure zero such that E is not contaiDed in TA for any t c R.
Proof. As in Theorem 1 we consider the flow T; built under the two.~ step (p, q) function, where D and D' denote the parts of the base B which lie under the heights p and q respectively. Let A' be the image of A N under the isomorphism (D.
The product measurability of A' together with the fact that A does not have full rnc" ure imply there is an 0 a < 1 such that p'(A') = Qp'(n') and one of the followin must hold.
(1) There is an x c D such that in t : (x, t)E.k', 0 :5 t < p or
(2) There is an x c D' such that m { t Cx, i)~A', 0 :5 t < q 1 .5 aq.'
We can suppose without loss of generality that (1) occurs. Next we take a collection of open intervals { Ii }'i=, such that
t)rA', 0 < i < p
Uli
cc
and for which 2(1) a' < P. Setting qc, 17?(1i) for i 1, 2, and
.Cto cc , (p a'),13 we apply the preceding I; mma to obtain a closed F t : 0 t~ 1 < p } Of MCiSUTC Zero Y,',hich cannot be covered by any collection of intervals with len ths { a ,, a07 ~(1, ~~21 ... }. We claim that the set i) : teF 1 is tot contained in any elerpent of the class
To pro\~e the clairn we note by the defirinion of a flow under a function
t 1 hat the image of A' n { (x, r) : 0 < r < p 1 .ii ieler T, is just a translate
of A' n { (x, r) t5 r < p } up the filber (x, r) by an amount t except
fot the bottom (x, r) : 0 < r < 1 ind the irnage of the top
A' n p 1 :5 r < p
Hoin B
1 0 1
CA7CHING SNIALL SETS UNDER FLOWS 39
The fact that a system of inlervals of lentch { 2 1, OCO, a, CC2
cannot cover F then implies that (X, 1) : t&F is not cau ght by any elerinent of
T,A' t (p a')13
To complete the proof we let U T~.., t) : taF where
S (p a113. Since (x, t) t&F cp TA' for t s ive have
T~ { (x, t) : iFF } q: T'.j,,A' for Iti
This says T,A' for all t c~ R. For the las, step we let E = (D 'f. Since (D preserves measure p(E) = 0 and since (TA) n E = (T,(A 1\~) n E for all 1 c R the theorem is established.
3. FURTIXER REYLARKS
Rudol.ph*s representation theorem 151, [6] played a key role in the'conccptualization and proof of the results given but one should note that at the expense of greater complexity one need enly appeal to the represen'iation theorems of An, brose [11 or Ambrose a ,"d Ka'k'ulani [2]. The assumption of ergodicity made in our results can be weakened to aperiodicity since the j epresentation theorem remains ,alid. This is mentioned in Rudolph [61 and the extension is described in detail in R ren gel [5]. For simplicity of exposition we have omitted the full discussion of this rnole t.ichriical hypothesis.
Since we have dealt here only with fie, ,,vs, we should note that there are some related results for transformations. It is proved in Steele [7) that for an ergodic T on a Lebesgue probability space there is for any e > 0 an A with p(A) < e which satisfies the condition:
For any finile F theye, is a j = j(F) suefi that F VA.
One can easily show that this result is best possible in 'the senses that neither can A be taken to be of measure 0 nor can F be allowed to be countable.
This result provided one of the motivations of the pre sent paper, and it has also been extended in a quite different direction in Ellis [31.
Finally, we note that a mi)ch"earlier contrIbution to covering with sets
of measure zero ,~as made by, Erdos apd [4], but their work
(cr((jrs Eucl "d(~.n ralP.tr il in Tr(.c!i:3e
mations.
Vol. )CV, n, 1 1979.
,h .1
10
4,
40 M. H. ELLIS AND J. NI. STEELE
REFERENCES
[1) W. AmBRos£, Representation of ergodic flows, Ann. of ~fath., 11, ser. 42, 1941, p. 723739.
121 W. AmBROSE, S. Y Am r.ANi, Structure and connnulty of Measurable flows, Duke Math. J., t. 9, 19422, p. 25 42.
fif5,§. ELL s, Catching and rnissing finite sets. Can. BulL Vath. (to appear), 1978.
[4] P. IRD65, S. KA,,1LrA~l, On a ;erfect set, Colloa, ~ fath., t. 4, 957, p. 195 196.
[5] U. KRENGEL, On Rudolph's representation of aperiodic flows, Ann. Inst. HenrI Poincard, sect. B.. iol. XII, nO 4, 1976, p. 319 338.
16] D. RUDOLPH, A tiio valued step coding for orgy~'ic flows, Afath. Z.. t. 1 50, 1976't p. 201 220.
(7] M. STEELE, Co,,eriiig finite sets by ergodic irnages, Can. Bull. Math., t. 21, 1979, A P. 85 92.
le 4 septembre 19,7S).
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