Entropy dimensions and a class of constructive examples∗

Motivated by the study of actions of Z 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.

Let (X, B, µ, σ, P ) be a process, where σ denotes an action of a group G, and P = {P 0 , . . . , P k−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 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 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 Z p , for simplicity we develop here the basic definitions and examples in the case of Z Z-actions.

Growth rates and names
A first tentative way to define an entropy dimension would be to define This can be generalized to Z Z k by taking a joint on a suitable part G n instead of the interval [0, n], and letting α vary from 0 to k. However, this does not define an isomorphism invariant, as the following proposition implies that sup P D H (P ) = 1 : Proposition 1 For any given P with D H (P ) < 1 and any δ > 0, there existsP such that |P −P | < δ, and D H (P ) > D H (P ).

Proof
Let α 0 = D H (P ). We choose α 0 < α < 1. We build a Rokhlin stack of height n 1 such that n α−1 1 ≤ 2 −L for a very large L. We may ensure that the distribution of the columns on the base level B 0 of the stack is the same as the distribution of ∨ n 1 −1 i=0 T i P . We divide each column into 2 n α 1 subcolumns of equal measure and change the partition P intoP 1 on the first n α 1 levels from the bottom, so that each subcolumn has a differentP 1 -[0, n α 1 )-name. For x and y in B 0 , theirP 1 -[0, n 1 )-names may agree if theirP 1 -[n α 1 , n 1 )-names are the same, so the number of differentP 1 -[0, n 1 )-names may be smaller than 2 n α 1 times the number of columns. However, there are at least 2 n α 1 differentP 1 -[0, n 1 )-names, each one of measure at most 2 −n α 1 .
where comes from the error set. Also where λ denotes the measure of a column and we sum over all columns. We note that |P −P 1 | < 2 −L . Let E 1 denote the n α 1 levels where P andP 1 may differ. We repeat this for Rokhlin stacks of height n k where n α−1 k ≤ 2 −L−k for k = 2, 3, . . .. In the k-th Rokhlin stack, we choose n α k many levels in each column and changeP k−1 toP k on these levels so that there are at least 2 n α k many different names for each column. we choose these levels so that their union E k is disjoint from ∪ k−1 i=1 E i . Thus |P k −P k−1 | < 2 −L−k , and we can defineP = limP k . And we have D H (P ) ≥ α. Note also that |P −P | < 2 −L+1 , thus P can be chosen arbitrarily close to P . And since eachP k is measurable with respect to the σ-algebra generated by P , so isP ; ifP 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 ofP is not possible. For , there is the natural Hamming distance, counting the ration of different coordinates in the names: for two sequences a = (a 1 , ...a k ) and We can define a complexity dimension for a process by 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.

Entropy dimensions and subgroup actions
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 Similarly 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 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 Z 2 : if one of the directions has positive entropy, then K(n, ) grows at least at the rate of e cn 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 h c (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 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 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:

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. ♣

Examples of entropy dimensions
We define inductively a family of blocks B n,i , 1 ≤ i ≤ b n , in the following way; given two sequences of positive integers e n and r n : • the B n,i , 1 ≤ i ≤ b n+1 are all the possible concatenations of e n blocks B n,i , Let h n be the length of the B n,i , h n be the length of the B n,i .
We can thus define a topological system as the shift on the set of sequences {x n , n ∈ Z Z} such that for evry s < t there exists n and i such that x s . . . x t is a subword of B n,i . We put an invariant measure on it by giving to each block B n,i the measure 1 bn . 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 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.

Upper limit
Let L n ( ) be the number of -d-balls than can be made with blocks B n . Note that, on an alphabet of k letters, for a given word w of length m, the number of words w with d(w, w ) < is at most m m k m ≤ k mg( ) for some g( ) → 0 when → 0. In the above construction, the number of different blocks B n is b n+1 = k e 0 ...en . As in every of these blocks the repetitions occur exactly at the same places, for a given word B n,i , the number of words As all different blocks are given the same measure, we have As h n = e 0 . . . e n−1 r 0 . . . r n−1 , if, e 0 , . . . ,e n−1 , r 0 , . . . , r n−1 being fixed, we choose e n large enough, we shall have log K(h n , ) ≥ (h n ) 1−δn for any given sequence δ n .♣ 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 = 1 2 . We define a sequence l n by choosing a very large l 1 , then l n = [l Hence D ≥ 1 2 .
Upper limit As in the first part of the proof of the last proposition, K(h n , ) is smaller than the total number of P − [0, h n )-names, and this is at most b 2 n+1 h n . We take some b > 1 2 ; The above examples can be generalized to Z 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, for all n.

Proof
Upper bounds We give upper bounds for K ≥ k, where K (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 at the end of its times of maximal growth, namely K (h n , ). The possible words of length h n are all the W n (a, i, j) where, for 0 ≤ a ≤ h n − 1, is the suffix of length a of B n,i followed by the prefix of length h n − a of B n,j . Each one of these words is at a (d) distance at most of some W n (a s , i s , j s ) for 1 ≤ s ≤ K (h n , ).
We look now at words of length h n+1 ; they are all the W n (a, i, j) where, , j) is the suffix of length a of B n+1,i followed by the prefix of length hn + 1 − a of B n+1,j . Hence for 0 ≤ t ≤ r n − 1, 0 ≤ a ≤ h n+1 − 1, Each one of these will be at a distance at most + 1 rn of some W (a s , i s , i s ) t+1 W (a s , j s , j s ) rn−t−1 ; and, for fixed s, W (a s , i s , i s ) t+1 W (a s , j s , j s ) rn−t−1 and W (a s , i s , i s ) t +1 W (a s , j s , j s ) rn−t −1 are at a distance at most |t−t | rn . Hence, for a given sequence v n , we have Then, during the stage of independent stacking, a straightforward computation gives that If we fix the sequence e n , and suppose 1 rn < +∞; we choose any sequence v n such that v n < +∞; then, if we choose r n large enough in terms of K (h n , ), h n , and e n , we get that K (h n+1 , 2 ) is smaller than φ(h n+1 ), and this is true a fortiori for other values.

Lower bounds
We shall show that K(n, ) → +∞ with n. For this, let L n ( ) be the number of -d-balls than can be made with blocks B n , and L n ( ) be the number of -d-balls that can be made with blocks B n . During the repetition stage, we have L n ( ) ≥ L n−1 ( ).
Then, during the independent stage, we start from L = L n ( ) blocks which are -d-separated; we call them B n,s 1 , ... , B n,s L . Then, if e n is a multiple of L, the 2L blocks B en n,s i , 1 ≤ i ≤ L, and B Thus whenever e n is large compared to L n ( ) we have L n+1 ( (1 − 1 L n ( ) )) ≥ 2L n ( ) and hence L n ( 2 ) tends to infinity with n; and, because of the structure of the names and the fact that each block B n,i has the same measure for fixed n, we get that K(h n , ) 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.