Proceedings of the International Congress of Mathematicians
Vancouver, 1974
On Generators in Ergodic Theory
W. Krieger
We are concerned with measure preserving transformations T of a Lebesgue
measure space {X, &9 fj) where fj,{X) = 1. Consider afinitepartition &> = {Pi, • • •, Pn)
or a countably infinite partition 0> = {Ph •••) of X. Such a partition & is said to
be a generator of T if
3 = V TW (mod pi).
More generally a sub-^-algebra sé of ^ is called a generator for Tif
^ = V 7 W (mod a).
For a survey of the theory of generators see U, Krengel's 1971 Prague conference
address [23].
Generators have been of use in entropy theory. The entropy of the partition &> is
defined by
h{&)= -
EpiPJlog/iiPJ.
m
The mean entropy of the partition &> with respect to the measure preserving
transformation T is defined by
h{&>, T) = lim L h( V T-*0>\
A-oo k
\l£i^k
J
and the entropy of Tis then defined by h{T) = sup h{&>, T) where the supremum is
taken over allfinitepartitions. The Kolmogoroff-Sinai theorem states that h{T) =
h{&>, T) if 0> is afinitegenerator for T. As a consequence one has that h{T) <nifT
has a generator of size n. In the converse direction it is known that every ergodic
measure preserving transformation T with entropy h{T) < oo has a generator with
© 1975, Canadian Mathematical Congress
303
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W. KRIEGER
no more than eÄ(TJ + 1 elements [24], [25]. A simpler proof of this result has been
given by M. Denker [4]. Denker observed further thatfinitegenerators are dense in
the set 77 of all partitions with mean entropy h{T) < oo if 77 is given the entropy
metric \0>,£\ = 2h{0> v J) - h{0>) - h{â\ ^ , J2e 77 [5].
A partition & is said to be a strong generator for T if <% = V ÏŒN T~* &> (mod /a).
Finite strong generators (in fact strong generators of size 2) exist if and only if
h{T) = 0. Recall that by a theorem of Parry and Rohlin every aperiodic transformation T has a countable strong generator. 0> is a strong generator for T if and
only if its remote past equals ^ ,
Si = f) V T-*9 (mod p).
;eJV fèj
Partitions with such a property have been called deterministic. Define £P as bilaterally deterministic if
0 = f] V T-*0> (mod p).
D. Ornstein and B. Weiss have shown that every ergodic measure preserving transformation with finite entropy has a bilaterally deterministic finite partition [33].
This answers also a question that was raised by O. E. Lanford and D. Ruelle in
connection with X-automorphisms [31].
Consider the shift space Qn = Hï(=z {1, •••, n} over the alphabet {1, •••,«}. Qn is
given the product topology of the discrete topologies. The «-shift S on Qn, {Sx){ =
Xf-i, ieZ, x — {xt)t<=z e Qn is a homeomorphism of Qn. Let now &> = (PÌ9 •••, Pn)
be a generator for the measure preserving transformation T. One obtains a mapping of X into Qn if one sets
{Ux)i = m, if T{x e Pm,
U/W^H,
i e Z , f, ^-a.a. x e X.
Setting for a Borei set A c Qn, Ufx{A) — fx{U~lA), one transports the measure
fi from X to Qn, and in this way one has produced a shift-invariant measure
v = U/u on ß„. As a consequence of the generation property of 0> the systems
{X9 fi, T) and {Qn9 y, iS) are isomorphic in the sense of measure theory. For the case
of the shift thefinitegenerator theorem provides therefore an answer to the following question : Does a given homeomorphism 0 of a compact metric space E have
an invariant probability measure v such that the system {E, y, 0) is isomorphic to
a given system {X,fi, T)l The notion of topological entropy furnishes a necessary
condition for such an imbedding. Topological entropy is defined as follows [1]:
For an open cover <$ of E let h{<£) be the logarithm of the minimal cardinality of
a subcover of ^. Then set, for a homeomorphism 0 of E9
^•«-sK^*)
and define the topological entropy of 0 by h{0) = sup h{<£9 0) where the supremum
is taken over all open covers. The «-shift has topological entropy log n. One has for
all (^-invariant probability measures fi that h{fi, 0) ^ h{0) [12]. In fact, one has
h{0) = sup h{fi, 0) where the supremum is taken over all 0-invariant probability
GENERATORS IN ERGODIC THEORY
305
measures fi [10]. This notion of topological entropy is an invitation to define the
topological analog of a generator. This was done by H. Keynes and J. Robertson [20]
as follows : An open cover ^ is a topological generator of the homeomorphism 0 if
for all (C(/)),.eZ G ^ z the s e t Oi^z Q 0 contains at most one point. This definition
stands the test: If # is a topological generator for 0 then h{0) = h{&9 0) [20]. 0 is
called expansive if there exists a ö > 0 such that, for all x, y e E, x ^ y, there is an
i e Z such that d{0*x, 0*y) > d where dk a metric of £*. Exactly the expansive homeomorphisms have topological generators [20]. The expansive homeomorphisms
of the Cantor discontinuum are given by the subshifts; these are the closed shiftinvariant subsets of the Qn with the shift acting on them. Topological Markoff
chains are subshifts of what is called finite type. A topological Markoff chain can,
e.g., be described as the shift acting on a set M that is given by a transition matrix
7){m, l) G {0, 1}, 1 g m, I ^ «, where M = f]iŒZ {xeQn: 7}{xi9 xi+i) = 1}. One
can extend the finite generator theorem from the «-shift to aperiodic topological
Markoff chains [28] : For all aperiodic topological Markoff chains M and for all
ergodic measure preserving transformations T such that h{M) > h{T) there exists a
shift-invariant probability measure v on M such that the systems {X, fx9 T) and
(M, y, S) are isomorphic in the sense of measure theory. In this context one can consider minimal expansive homeomorphisms instead of ergodic measure preserving
transformations. Recall that a homeomorphism is called minimal if all of its orbits
are dense. One has [28] : If C is a minimal subshift and if Mis a topological Markoff
chain such that h{C) < h{M) then C is topologically conjugate to the shift acting
on a closed invariant subset of M.
Consider the 77-shift {Qn, S) and let M be the set of ergodic »^-invariant probability Borei measures on Qn with support Qn. Denote by #f the group of homeomorphisms of Qn that commute with *S*. j f acts on J( by fi -> U/a {fie Jt)9 Ue 34?.
The following theorem was proved by A. Kuntz [30] : Let fi, ve M, h{\>) ^ h{fi), and
let / e N, e > 0. Then there exists a t / e / such that for all cylinder sets Z{a) =
{xeQn: Xi = ai9 1 g / g /}, ae {1, •••,«} / , one has \fi{Z{a)) - Uv{Z{a)) \ < e>
a e{l, •••, n}1. This is also a statement about generators since every ordered generator of size n yields a shift-invariant measure on Qn. An approximation result of this
kind wasfirstobtained in [24, §3]. The present proofs of the isomorphism theorem
for Bernoulli systems [32], [35], [36] apply in their initial stage such an approximation (for partitions, not necessarily for generators). In their later stage they use an
approximation with respect to a different topology that is given by the rf-metric.
A homeomorphism is said to be strictly ergodic if it is minimal and if it has a
unique invariant probability measure. Representing an ergodic measure preserving
transformation T a s a strictly ergodic homeomorphism amounts to finding a
generator of T with the appropriate properties. It was proved by R. Jewett that
every weakly mixing measure preserving transformation has a representation as a
strictly ergodic homeomorphism of the Cantor discontinuum [16]. The possibility
of such a representation for all ergodic measure preserving transformations was
shown in [26]. This proof used the finite generator theorem. A proof that did not
use thefinitegenerator theorem was then given by G. Hansel and J. P. Raoult [14].
306
W. KRIEGER
For the case at an ergodic measure preserving transformation T withfiniteentropy
h{T) one has that T can be represented as a strictly ergodic expansive homemorphism of the Cantor discontinuum, more precisely, as a strictly ergodic subshift in
a shift space over [eh -f 1] symbols [26]. Denker has given a simpler proof of this
[3], This line of investigation was continued by G. Hansel who considered measure
preserving transformations that are not necessarily ergodic [13]. He proved that
every measure preserving transformation can be imbedded in a homeomorphism
all of whose orbit closures are strictly ergodic.
There exist minimal homeomorphisms that are not strictly ergodic (see [34]).
Indeed, E. Effros and F. Hahn [9] have constructed a minimal homeomorphism
with more than countably many ergodic invariant probability measures. Their
example is distal and hence has entropy zero [19]. I. P. Kornfel'd [21] has shown
that one can have this situation also with positive entropy, and T. N. T. Goodman
[11] has produced such examples in the shift space. These examples suggest the
possibility of imbedding (not necessarily ergodic) measure preserving transformations into minimal homeomorphisms. Ch. Grillenberger has recently shown that
such an imbedding is always possible.
Attempts have been made to develop this theory for groups other than Z [2],
[18], [28]. Let us consider the case of a countably infinite group ^ that acts ergodically and freely on (Z, (%9 fi) by measure preserving transformations Tg9 g e ^.The
action is called free if p,{x e X: Tgx = x} = 0 for all g e & that are not equal to the
unit e of &. Call a sequence ^{k) c &,keN, offinitesets a summing sequence if
Urn | &{k) I"1 \^{k)Ag^F{k) | = 0,
g e <&.
Using a summing sequence ^{k), keN, one attempts for a partition 0> the definition
h{&>, &) = lim |#"(*) H hl V TgeÄ
and one sets then h{&) = sup h{&>, &) where the supremum is taken over all finite
partitions. We know that we obtain in this way an entropy theory with the familiar
features including, e.g., a finite generator theorem, for a class of groups (e.g., for
solvable groups), provided that we can prove an analog of the tower theorem
[27]. The tower theorem asserts for an ergodic measure preserving transformation
T that for all / e TV and e > 0 there exists an F a Zsuch that Ifi{F) > 1 - e and
F fl T{F = 0 , 0 < i < J. An analog statement for the action Tg, g e &, would be
as follows: For some summing sequence ^{k)9 each ^{k) containing e9 one can
findforallfcGJVrandalU>OanFc:JirsuchthatÄ:ia(iO> 1 - e a n d i ^ fl TgF=0,
g e #"(&), g =£ e. In most cases proving such a statement is equivalent to proving
the hyperfiniteness of {Tg : g e &}. Let us recall here the notion of hyperfiniteness
that is due to H. A. Dye [7]: The action {Tg:g e &} is called hyperfinite if there
exists a measure preserving transformation T whose orbits are a.e. the same as the
orbits of the action:
{Tgx:ge&} = {T'xiieZ}
f.a.a.xGZ.
There are results available on hyperfiniteness in the papers of H. A. Dye [7], [8],
GENERATORS IN ERGODIC THEORY
307
E.g., abelian groups are hyperfinite. On grounds of these results we have the tower
theorem for a significant class of groups, and hence we have entropy theory for a
significant class of groups. The opinion that hyperfiniteness would enter into the
entropy theory of groups was expressed by A. Stepin [37] and A. Veräik, who has
announced further results on hyperfiniteness [38].
Investigations on generators for one-parameter flows of measure preserving
transformations have also been carried out, e.g., the representation of an ergodic
one-parameter flow of measure preserving transformations as a strictly ergodic
flow was achieved by K. Jacobs [15] in the weakly mixing case and by M. Denker
and E. Eberlein [6] in the general case. For more information on the results for
flows we refer to U. Krengel's survey [23].
The situation changes completely if one drops the assumption that a finite
measure is preserved. As a matter of fact, as was shown by A. Kuntz [29], if one has
a group <g of nonsingular transformations acting on {X, 08, /u) such that no probability measure that is absolutely continuous with respect to y, is preserved by ^,
then there exists a set A c X such that {TA: Te &} is dense in ^ . For previous
results on single nonsingular transformations see [22] and [17]. For more information on this topic we refer again to U. Krengel's survey [23].
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