Entropy dimensions and a class of constructive examples∗
Sébastien Ferenczi
Institut de Mathématiques de Luminy
CNRS - UMR 6206
Case 907, 163 av. de Luminy
F13288 Marseille Cedex 9 (France)
and Fédération de Recherche des Unités de Mathématiques de Marseille
CNRS - FR 2291
[email protected]
Kyewon Koh Park
Department of Mathematics
Ajou University
Suwon 442-729
Korea
[email protected]
December 9, 2004
Abstract
Motivated by the study of actions of ZZ2 and more general groups, and their noncocompact subgroup actions, we investigate entropy-type invariants for deterministic
systems. In particular, we define a new isomorphism invariant, the entropy dimension,
and look at its behaviour on examples. We also look at other natural notions suitable
for processes.
AMS subject classification: 37A35, 37A15.
Keywords: Ergodic theory, entropy, examples.
Abbreviated title: Entropy dimensions.
∗
The authors were supported in part by a joint CNRS/KOSEF Cooperative Research Grant between
Korea and France. The second author was also supported in part by grant BK 21. The second author would
like to thank Korea Institute for Advanced Study for the pleasant stay while this work was completed.
1
Let (X, B, µ, σ, P ) be a process, where σ denotes an action of a group G, and P =
{P0 , . . . , Pk−1 } denotes a (finite, measurable) partition of X. In the study of a general group
action, subgroup actions play an important role: if a G-action has positive entropy, it is not
hard to see that every non-cocompact subgroup action has infinite entropy (see for example
[3]). In the case of a Z 2 -action generated by two commuting maps, say T and S, if either
h(T ) or h(S) is finite, the entropy of the Z 2 -action is 0. Hence it is increasingly important
to study systems of entropy zero, as they may give rise to interesting subgroup actions, and
to classify them up to measure-theoretic isomorphism. One way to achieve this goal is to
look at the amount of determinism in the system, in a more precise way that is given by
the mere knowledge of the entropy. Several refinements of the notion of entropy have been
introduced by various authors, such as the slow entropy [5], the measure-theoretic complexity
[4], the entropy convergence rates [1]; following a suggestion of J. Milnor, we propose a new
notion, the entropy dimension; though it seems most promising for actions of groups like Z p ,
for simplicity we develop here the basic definitions and examples in the case of Z -actions.
1
Growth rates and names
A first tentative way to define an entropy dimension would be to define
H
D (P ) = sup{0 ≤ α ≤ 1; lim sup
1
H(∨n0 T −g P ) > 0}.
nα
This can be generalized to Z k by taking a joint on a suitable part Gn instead of the interval
[0, n], and letting α vary from 0 to k. However, this does not define an isomorphism invariant,
H
as the following proposition implies that supP D (P ) = 1 :
H
Proposition 1 For any given P with D (P ) < 1 and any δ > 0, there exists P̃ such that
H
H
|P − P̃ | < δ, and D (P̃ ) > D (P ).
Proof
H
Let α0 = D (P ). We choose α0 < α < 1. We build a Rokhlin stack of height n1 such that
nα−1
≤ 2−L for a very large L. We may ensure that the distribution of the columns on the
1
1 −1
base level B0 of the stack is the same as the distribution of ∨ni=0
T i P . We divide each column
α
into 2n1 subcolumns of equal measure and change the partition P into P̃ 1 on the first nα1
levels from the bottom, so that each subcolumn has a different P̃ 1 -[0, nα1 )-name. For x and
y in B0 , their P̃ 1 -[0, n1 )-names may agree if their P̃ 1 -[nα1 , n1 )-names are the same, so the
α
number of different P̃ 1 -[0, n1 )-names may be smaller than 2n1 times the number of columns.
α
α
However, there are at least 2n1 different P̃ 1 -[0, n1 )-names, each one of measure at most 2−n1 .
Hence
α
1 −1
2nX
1
1
1
1
n1 −1 i 1
H(∨
T
P̃
)
≥
log nα ± i=0
α
α
nα
n1
n1 i=0 2 1
2 1
= log 2 ± 1
where comes from the error set. Also
nα
1 −1
1
1 X 2X
λ
λ
n1 −1 i 1
H(∨
T
P̃
)
≤
log nα ± i=0
α
α
nα
n1
n1 λ i=0 2 1
2 1
=
X
X
1
(−
λ
log
λ
+
nα1 log 2) ± ) = log 2 ± α
n1
λ
λ
where λ denotes the measure of a column and we sum over all columns.
We note that |P − P̃ 1 | < 2−L . Let E1 denote the nα1 levels where P and P̃ 1 may differ.
We repeat this for Rokhlin stacks of height nk where nkα−1 ≤ 2−L−k for k = 2, 3, . . .. In the
k-th Rokhlin stack, we choose nαk many levels in each column and change P̃ k−1 to P̃ k on
α
these levels so that there are at least 2nk many different names for each column. we choose
k
k−1
these levels so that their union Ek is disjoint from ∪k−1
| < 2−L−k , and
i=1 Ei . Thus |P̃ − P̃
H
we can define P̃ = lim P̃ k . And we have D (P̃ ) ≥ α. Note also that |P − P̃ | < 2−L+1 , thus
P̃ can be chosen arbitrarily close to P . And since each P̃k is measurable with respect to
the σ-algebra generated by P , so is P̃ ; if P̃ generates a factor σ-algebra, we can modify it
further so that it generates the whole σ-algebra. ♣
Remark
It is possible to define D using lower instead of upper limits. Note that if α = 1 the
construction of P̃ is not possible.
For a point x in X, the P -name of x is the sequence P (x) where Pi (x) = l whenever
σ i (x) is in Pl ; we denote by P[0,n) (x) the sequence P0 (x) . . . Pn−1 (x). Between P[0,n) (x) and
P[0,n) (y), there is the natural Hamming distance, counting the ration of different coordinates
in the names: for two sequences a = (a1 , ...ak ) and b = (b1 , ...bk ) over a finite alphabet, we
recall that
1
d(a, b) = #{i; ai 6= bi }.
k
We can define a complexity dimension for a process by
1
log #{different P − [0, n) − names} > 0},
nα
1
D0 (P ) = sup{0 ≤ α ≤ 1; lim inf α log #{different P − [0, n) − names} > 0}.
n
However it is easy to see, as in the previous case, that this is not an isomorphism invariant.
Hence, instead of counting names, we should use the number of d-balls around names.
D0 (P ) = sup{0 ≤ α ≤ 1; lim sup
2
Entropy dimensions and subgroup actions
Definition 2 For a point x ∈ X, we define
B(x, n, ) = {y ∈ X; d(P[0,n) (x), P[0,n) (y)) < }.
And let K(n, ) be the smallest number K such that there exists a subset of X of measure at
least 1 − covered by at most K balls B(x, n, ). Then
D(P, ) = sup{0 ≤ α ≤ 1; lim sup
2
1
log K(n, ) > 0},
nα
D(P ) = lim D(P, ),
→0
D = sup d(P ).
P
Similarly
1
log K(n, ) > 0},
nα
D(P ) = lim D(P, ),
D(P, ) = sup{0 ≤ α ≤ 1; lim inf
→0
D = sup d(P ).
P
We call D, resp. D, the upper, resp. lower, entropy dimension of the system (X, B, µ, σ). If
D = D, we just call it the entropy dimension and denote it by D.
Note that for a Z -action, the entropy dimension may be 1 while the entropy is 0.
It is a straightforward consequence of our definition, proved by the same proof as Corollary 1 in [4], that
D(P ) = D
when P is a generating partition.
We want to investigate the relation between the entropy dimension and the entropy of
subgroup actions, particularly in the case of Z 2 : if one of the directions has positive entropy,
then K(n, ) grows at least at the rate of ecn and the lower entropy dimension is at least
one. Hence, if D < 1, then h(v) = 0 for every direction v, and, moreover, the cone entropy
[2] has the property that hc (v) = h(v) = 0. The converse is not true: Katok and Thouvenot
[5] provide an example where the upper entropy dimension is arbitrarily close to 2 while the
directional entropy is 0 for almost all directions; note that in this example the upper and
lower entropy dimensisons do not agree.
We recall that there exists an example in [7] where h(σ (1,0) ) > 0 while all the remaining
directional entropies (including the irrational directions) are 0; this Z 2 -action has clearly
entropy dimension equal to 1. In the well-known ewample of Ledrappier ([6]), the entropy
dimension is 1 and every directional entropy is positive. By making a direct product of
countably many copies of that example, we can build a Z 2 -action whose entropy dimension
is 1 and every direction has infinite entropy, because of the following lemma, which holds
also for countable products:
Lemma 3
D(σ × τ ) = max(D(σ), D(τ )).
Proof
The entropy dimension of σ × τ may be computed by taking only partitions of the form
P × Q. But then for these partitions B((x, y), n, 2) contains B(x, n, ) × B(y, n, ) (respectively for P and Q) and is included in B(x, n, 2) × B(y, n, 2), which yields the result. ♣
3
3
Examples of entropy dimensions
We define inductively a family of blocks Bn,i , 1 ≤ i ≤ bn , in the following way; given two
sequences of positive integers en and rn :
• b0 = k, B0,i = i − 1, 1 ≤ i ≤ k,
• bn+1 = (bn )en ,
0
• the Bn,i
, 1 ≤ i ≤ bn+1 are all the possible concatenations of en blocks Bn,i ,
• for each 1 ≤ i ≤ bn+1 , Bn+1,i is a concatenation of rn blocks Bn,i .
0
Let hn be the length of the Bn,i , h0n be the length of the Bn,i
.
We can thus define a topological system as the shift on the set of sequences {xn , n ∈ ZZ}
such that for evry s < t there exists n and i such that xs . . . xt is a subword of Bn,i . We put
an invariant measure on it by giving to each block Bn,i the measure b1n . We denote by P the
natural partition in k sets given by the zero coordinate.
The above construction is well known to ergodic theory specialists, and a generalization of
it to Z 2 -actions is used in [5]; however, even its one-dimensional version can yield new types
of counter-examples. This system will be referred in the sequel as the standard example.
Proposition 4 There is a choice of parameters such that the standard example satisfies
D = 1,
D = 0.
Proof
Lower limit.
For any , K(hn+1 , ) is smaller than the total number of P − [0, hn+1 )-names. The possible names of length hn+1 are all the Wn+1 (a, i, j) where, for 0 ≤ a ≤ hn+1 , 1 ≤ i ≤ bn+1 ,
1 ≤ j ≤ bn+1 , Wn+1 (a, i, j) is the suffix of length a of Bn+1,i followed by the prefix of
length hn+1 − a of Bn+1,j . Hence their numbers is at most hn+1 b2n+1 , with bn+1 as above and
hn+1 = e0 . . . en−1 r0 . . . rn . If, e0 , . . . ,en , r0 , . . . , rn−1 being fixed, we choose rn large enough,
we shall have log K(hn+1 , ) ≤ (hn+1 )δn for any given sequence δn .
Upper limit
¯
Let L0n () be the number of -d-balls
than can be made with blocks Bn0 . Note that, on
an alphabet of k letters, for a! given word w of length m, the number of words w0 with
m
d(w, w0 ) < is at most
k m ≤ k mg() for some g() → 0 when → 0. In the above
m
construction, the number of different blocks Bn0 is bn+1 = k e0 ...en . As in every of these blocks
0
the repetitions occur exactly at the same places, for
a given word Bn,i
, the number of words
!
e0 . . . en
0
0
0
Bn,j
with d(Bn,i
, Bn,j
) < is at most
k e0 ...en . Hence
e0 . . . en L0n () ≥ k e0 ...en −g() .
4
As all different blocks are given the same measure, we have
K(h0n , ) ≥ (1 − )Ln ().
As h0n = e0 . . . en−1 r0 . . . rn−1 , if, e0 , . . . ,en−1 , r0 , . . . , rn−1 being fixed, we choose en large
enough, we shall have log K(h0n , ) ≥ (h0n )1−δn for any given sequence δn .♣
Proposition 5 For any 0 < α < 1, there is a choice of parameters such that the standard
example satisfies D = α.
Proof
We make the proof for α = 21 . We define a sequence ln by choosing a very large l1 , then
1
1
2(n−1)
2(n−1)
ln = [ln−1
][ln−1
]ln−1 ; then, in the standard construction, starting from the two 0-blocks
1
√
2(n−1)
0 and 1, we put e0 = r0 = [ l1 ], and, for n ≥ 1, en = rn = [ln−1
]. [x] denotes the integer
part of x, but in the following computations we shall assimilate a large enough x with its
integer part.
Then, the lower limit is reached along the sequence {hn+1 = ln } and the upper limit
1+
1
along the sequence h0n = {ln−12(n−1) }.
Lower limit
As in the second part of the proof of the last proposition, K(hn+1 , is at least (1−) times the
¯
number Ln+1 () of -d-balls
than can be made with blocks Bn+1 . Because of the repetitions,
and the computation in the last proposition
Ln+1 () ≥ L0n () ≥ 2e0 ...en −g() .
So we have only to compute
1
lim √ log(2e0 ...en ) =
n→+∞
ln
q q
q
1
1
1
√ √
n−1
1
l1 l1 l22 l33 ... ln−1
lim √ log 2
=
n→+∞
ln
1 q
ln log 2 = log 2.
lim √
n→+∞
ln
Hence D ≥ 12 .
Upper limit
As in the first part of the proof of the last proposition, K(h0n , ) is smaller than the total
number of P − [0, h0n )-names, and this is at most b2n+1 h0n . We take some b > 21 ;
lim sup
n→+∞
1
log b2n+1 h0n =
0
b
(hn )
2 lim sup
n→+∞
1
log bn+1 =
(h0n )b
5
2 lim sup
n→+∞
√ √
l 1 l1
1
1+ 1
(ln−12(n−1) )b
log 2
r
2 lim sup
n→+∞
q q
1
l33 ...
q
1
n−1
ln−1
=
1
1+
ln−12(n−1)
1+
1
l22
1
= 0.
(ln−12(n−1) )b
Hence D ≤ 12 , which gives what we claimed.
α
1−α
n−1
n−1
The general case (for a given α) follows with the same proof, by taking ln = ln−1 [ln−1
][ln−1
],
α
1−α
n−1
n−1
en = [ln−1
], rn = [ln−1
].♣
The above examples can be generalized to Z 2 -actions; by alternating repetitions and independent stacking, we can build an example whose entropy dimension is any given 0 ≤ α ≤ 2.
In [4], where the rate of growth of K(n, ) is used to define the so-called measure-theoretic
complexity, it is asked whether this growth rate can be unbounded but smalller than O(n)
(its topological version for symbolic systems, the symbolic complexity has to be bounded if
it is smaller than n). Our class of examples allows to answer this question; note that the
proofs are slightly more involved as we are dealing with sub-exponnetial growths:
Proposition 6 For any given function φ growing to infinity with n, there is a choice of
parameters such that the standard example satisfies, for every fixed small enough,
K(n, ) → +∞
with n, but
K(n, ) ≤ φ(n)
for all n.
Proof
Upper bounds
¯
We give upper bounds for K 0 ≥ k, where K 0 (n, ) is the smallest number of -d-balls
of
names of length n necessary to cover a proportion of the space of measure 1. We look at K 0
at the end of its times of maximal growth, namely K 0 (h0n , ). The possible words of length
h0n are all the Wn0 (a, i, j) where, for 0 ≤ a ≤ h0n − 1, 1 ≤ i ≤ bn+1 , 1 ≤ j ≤ bn+1 , Wn0 (a, i, j)
0
0
is the suffix of length a of Bn,i
followed by the prefix of length h0n − a of Bn,j
. Each one of
0
¯
these words is at a (d) distance at most of some Wn (as , is , js ) for 1 ≤ s ≤ K 0 (h0n , ).
We look now at words of length hn+1 ; they are all the Wn (a, i, j) where, for 0 ≤ a ≤
hn+1 − 1, 1 ≤ i ≤ bn+1 , 1 ≤ j ≤ bn+1 , Wn (a, i, j) is the suffix of length a of Bn+1,i followed
by the prefix of length hn + 1 − a of Bn+1,j . Hence for 0 ≤ t ≤ rn − 1, 0 ≤ a ≤ hn+1 − 1,
1 ≤ i ≤ bn+1 , 1 ≤ j ≤ bn+1 ,
Wn (a + th0n , i, j) = W (a, i, i)t W (a, i, j)W (a, j, j)rn −t−1 .
6
Each one of these will be at a distance at most + r1n of some W (as , is , is )t+1 W (as , js , js )rn −t−1 ;
0
0
and, for fixed s, W (as , is , is )t+1 W (as , js , js )rn −t−1 and W (as , is , is )t +1 W (as , js , js )rn −t −1 are
0
|
at a distance at most |t−t
. Hence, for a given sequence vn , we have
rn
K 0 (hn+1 , (1 +
K 0 (h0n , )2
1
+ vn )) ≤
.
rn
vn
Then, during the stage of independent stacking, a straightforward computation gives that
K 0 (h0n+1 , ) ≤ K 0 (hn+1 , )en .
P
1 < +∞; we choose any sequence vn such that
If we fix the sequence en , and suppose
rn
P
vn < +∞; then, if we choose rn large enough in terms of K 0 (h0n , ), h0n , and en , we get
that K 0 (h0n+1 , 2) is smaller than φ(h0n+1 ), and this is true a fortiori for other values.
Lower bounds
¯
We shall show that K(n, ) → +∞ with n. For this, let Ln () be the number of -d-balls
¯
than can be made with blocks Bn , and L0n () be the number of -d-balls
that can be made
0
with blocks Bn . During the repetition stage, we have
Ln () ≥ L0n−1 ().
¯
Then, during the independent stage, we start from L = Ln () blocks which are -d-separated;
en
we call them Bn,s1 , ... , Bn,sL . Then, if en is a multiple of L, the 2L blocks Bn,si , 1 ≤ i ≤ L,
en
en
en
en
en
1
L
L
L
L
L
¯
and Bn,s
i Bn,si+1 ...Bn,sL Bn,s1 ...Bn,si−1 , 1 ≤ i ≤ L, are (1 − L )-d-separated.
Thus whenever en is large compared to Ln () we have
Ln+1 ((1 −
1
)) ≥ 2Ln ()
Ln ()
and hence Ln ( 2 ) tends to infinity with n; and, because of the structure of the names and
the fact that each block Bn,i has the same measure for fixed n, we get that K(hn , ) tends
to infinity with n. ♣
Remarks
To make our examples weakly mixing, it is enough to place a spacer between two consecutive
blocks at each repetition stage.
It is easy to see that all our examples satisfy a form of Shannon-McMillan-Breiman
theorem (indeed, all atoms have the same measure); in a forthcoming paper, we shall give
examples which do not satisfy it.
References
[1] F. BLUME: Possible rates of entropy convergence, Ergodic Theory Dynam. Systems 17
(1997), no. 1, 45–70.
7
[2] R. BURTON, K. K. PARK: Spatial determinism for a Z 2 -action, preprint.
[3] J.-P. CONZE: Entropie d’un groupe abélien de transformations, (in French), Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 11–30.
[4] S. FERENCZI: Measure-theoretic complexity of ergodic systems, Israe̋l J. Math. 100
(1997), 189-207.
[5] A. KATOK, J.-P. THOUVENOT: Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Ann. Inst. H. Poincaré Probab.
Statist. 33 (1997), no. 3, 323–338.
[6] F. LEDRAPPIER: Un champ markovien peut être d’entropie nulle et mélangeant, (in
French), C. R. Acad. Sci. Paris Sr. A-B 287 (1978), no. 7, A561–A563.
[7] K. K. PARK: On directional entropy functions, Israe̋l J. Math. 113 (1999), 243–267.
8