Realizing subexponential entropy growth rates by cutting and stacking

We show that for any concave positive function $f$ defined on $[0,\infty)$ with $\lim_{x\rightarrow\infty}f(x)/x=0$ there exists 
a rank one system $(X_f,T_f)$ such that $\limsup_{n\rightarrow\infty} H(\alpha_0^{n-1})/f(n)\ge 1$ for all nontrivial partitions $\alpha$ of $X_f$ into two sets and that there is one partition $\alpha$ of $X_f$ into two sets for which the limit superior of $H(\alpha_0^{n-1})/f(n)$ is equal to one whenever the condition $\lim_{x\rightarrow\infty}\ln x/f(x)=0$ is satisfied. Furthermore, for each system $(X_f,T_f)$ we also identify the minimal entropy growth rate in the limit inferior.

1. Introduction. In order to enable the reader to place the present paper's subject matter in a broader conceptual context, we will briefly review to begin with some prior results concerning entropy growth rates. To do so, we need to introduce the pertinent standard notation: if (X, B, µ) is a probability space, T : X → X a measure-preserving transformation and α a finite partition of X, then the n-th refinement of α under T is denoted by α n−1 always exists (see [10], p.240). Furthermore, the supremum taken over all finite partitions α, is an isomorphism invariant that can be viewed to be a measure for the maximal exponential rate at which the elements of a partition α become dispersed in the space X under the action of T . Put differently, h(T ) is a measure for the increase in disorder under the repeated application of T and is said to be the entropy of T . As it turns out, however, within the group G of measure-preserving transformations on a Lebesgue space X the set of positive-entropy transformations is of 3436 FRANK BLUME first category with respect to the weak topology on G (see [9], p.103), that is, with respect to the topology that is generated by basis sets of the form {S ∈ G | µ(T (B 1 ) S(B 1 )), . . . , µ(T (B n ) S(B n )) < ε}, where ε > 0, T ∈ G, and B 1 , . . . , B n ∈ B. In other words, in this well defined topological sense, almost every measure-preserving transformation that we may happen to construct or encounter has entropy zero because the set of zero-entropy transformations is residual. Consequently, entropy as an isomorphism invariant is useful only for a very thin set of systems, and the question that therefore naturally arises is whether there are entropy-type invariants that can be used to distinguish zero-entropy systems. The most obvious way to define such an invariant is to replace n in the denominator in the defining equation (1) by a sublinear rate a n . So for a given s = (a n ) n∈N , with lim n→∞ a n n = 0, we may perhaps attempt to define Unfortunately, though, according to Theorem 2.6 in [7], it is the case that h s (T ) = ∞ universally for any aperiodic T and any positive increasing sequence s = (a n ) n∈N that satisfies (2). That is to say, the invariant h s (T ) as a generalization of h(T ) is altogether useless. As it turns out, however, the matter is different if we do not aim to find the maximal growth rate by taking the supremum over the partitions α but rather focus our attention on determining lower-bound growth rates that cannot be undercut. The following general result that establishes the existence of such lower-bound rates was proven in [6]: 1.1. Theorem. Let (X, T ) be an aperiodic measure-preserving system and assume that g is a positive monotone increasing function defined on [0, ∞) which satisfies the condition In the given generality this statement cannot be improved as was shown in [7]. More precisely, for each sufficiently regular function g : [0, ∞) → R with lim n→∞ g(x)/x = 0 and ∞ 1 g(x)/x 2 dx = ∞ there exists a weakly mixing system (X g , T g ) such that (g(log 2 n)) n∈N is the fastest growing sequence (up to trivial adjustments) for which lim sup n→∞ H(α n−1 0 )/g(log 2 n) ≥ 1/8 for all nontrivial partitions α of X g into two sets. Furthermore, in [7] we also constructed weakly mixing rank one systems for which the strongest lower-bound growth rate (in the limit superior) was given by (log 2 n) n∈N .
The next three theorems show that improvements of Theorem 1.1 are possible if the requirement that T be aperiodic is replaced by the stronger requirement that T be completely ergodic, rank-one, or rank-one mixing, respectively. The relevant proofs can be found in [2], [3], and [7].

Theorem.
If (X, T ) is completely ergodic (i.e., T n is ergodic for all n ∈ Z), then there exists a positive concave function g defined on [0, ∞) with for all partitions α of X into two sets of positive measure.
1.3. Theorem. Let (X, T ) be a rank-one system and assume that g is a positive monotone increasing function defined on [0, ∞) which satisfies the condition for all partitions α of X into two sets of positive measure.
Furthermore, the nearly universal existence of nontrivial lower-bound growth rates in the limit inferior was established in [2] where we proved the following general theorem: 1.5. Theorem. If (X, T ) is completely ergodic, then there exists a positive monotone increasing sequence (a n ) n∈N with lim n→∞ a n = ∞ such that for all partitions α of X into two sets of positive measure.
Some additional theorems concerning the estimation of minimal entropy growth rates for interval-exchange transformations and the relation of these rates to the average computational complexity of dynamical-system trajectories can be found in [1] and [4], but more important with respect to the topic of the present paper are the topological results established in [5]. Setting ) a n > 0 for all finite nontrivial partitions α , these latter results show that EI(s) is of first category in G whenever lim sup s = lim sup n→∞ a n = ∞ and that ES(s) is residual whenever the growth rate of s is sublinear.
Given the topological abundance of the transformations that make up the sets ES(s), the question naturally arises whether for any limsup-rate s that falls between the lower limit set by Theorem 1.1 and the upper limit given by the positive-entropy rate (n) n∈N there exists a corresponding transformation that realizes this rate. Since we already know that the range of rates from the lower limit set by Theorem 1.1 to the rate (log 2 (n)) n∈N can be realized by the systems (X g , T g ) (as explained above), the purpose of the present paper is to demonstrate that the remaining range from (log 2 n) n∈N to (n) n∈N can be realized as well. Moreover, for any system that realizes a rate in this latter range we will exhibit as well a minimal entropy growth rate in the limit inferior.
Remark. One of the problems in determining minimal entropy growth rateseither in the limit inferior or limit superior-is that these minimal rates cannot be computed by considering only generating partitions. Consequently, it is worth mentioning that an alternative counting-type entropy measure for zero-entropy systems that does allow for this reduction to generating partitions was introduced by Katok and Thouvenot in [8].
In order to construct, for a given positive concave function f that satisfies the conditions lim x→∞ f (x)/x = 0 = lim x→∞ ln(x)/f (x), a measure-preserving transformation T f on an interval J f that realizes the limsup-rate s = (f (n)) n∈N , we will employ a rank-one cutting-and-stacking procedure that produces a sequence of towers τ i whose union equals J f . That is to say, we will construct a tower τ i from a given tower τ i−1 by cutting τ i−1 into a large number of vertical subtowers, placing on top of each of them either one or no spacers according to an essentially random pattern, and then stacking up these resulting subtowers with the spacers added. As we will see, the number of vertical subtowers depends on the asymptotic behavior of f : the faster f diverges to ∞ the larger this number will be. However, before we can begin to discuss this construction in any more detail, we need to list several simple facts and definitions. Proof. Since h is increasing on [0, 1/e] and decreasing on [1/e, 1] and since there can be at most two sets in β whose measure is greater than 1/e, it follows that H F (β) ≤ H(β) + 2h(1/e) < H(β) + 2, as dessired.

Construction of the transformations.
To begin with, we assume that f : R → R is a concave monotone increasing function that is strictly positive on [0, ∞) and satisfies the following growth-rate conditions: Remark. For the proof of the fact that the sequence (f (n)) n∈N represents a minimal entropy growth rate of the system (J f , T f ) assumption (4) can be replaced with the weaker condition lim x→∞ f (x) = ∞. The stronger condition (4) is needed only afterwards, when we show that the rate given by f is optimal.
In order to control the placement of the spacers, in the cutting-and-stacking procedure described in the Introduction, we enumerate, for a given n ∈ N, the elements of {0, 1} n in the natural way by considering them to be binary expansions of natural numbers: The concatenation of these 01-names-with s(0) at the beginning and the end-we denote by u n , that is, u n = (u n (0), . . . , u n (n2 n + n − 1)) := (s(0), s(1), . . . , s(2 n − 1), s(0)).
Proof. Using elementary (but slightly tedious) combinatorial arguments, it is easy to see that for all k ∈ {0, . . . , n − 1}, and this immediately implies the statement of the lemma. Now let l 0 := 1, x 0 := 1, and τ 0 := ([0, 1)) (so τ 0 is a tower consisting of only one level), and assume that l i−1 , x i−1 , and τ i−1 have been defined in such a way that . Using assumption (3), it follows that is well defined. Next we cut τ i−1 into n i 2 ni + n i − 1 vertical subtowers of equal width µ(I i−1 (0))/(n i 2 ni + n i − 1) and place a spacer on top of those σ i (k) for which u ni (k) = 1. The resulting towers-with the spacers added-we denote by η i (0), . . . , η i (n i 2 n i + n i − 2) and define Moreover, the bottom intervals of the towers η i (k) we denote by K i (k) and by l i we denote the length of τ i .
Illustration of the construction: For convenience we arrange the spacers in such a way that for some x i > x i−1 . Then and therefore, Thus we may define where m denotes Lebesgue measure on R. If we denote by B f the σ-algebra of all Lebesgue measurable subsets of J f , then (J f , B f , µ f ) is a probability space and the preceeding tower construction induces a measure-preserving rank one transformation T f : J f → J f in the obvious way via the defining equation

FRANK BLUME
3. Identifying similar 01-names. Our purpose in this section is to prove the central technical result of this paper (Lemma 3.9) concerning the similarity of certain 01-names. To begin with, we need to establish several simple facts that are mostly analytic in character.
3.1. Theorem. If α is a nontrivial partition of J f into two sets, then Proof. Since f is concave on R, it follows that f is continuous, and therefore, the assumed strict positivity of f on (0, ∞) in conjunction with (3) implies that the equation f (x)/x = 1/n has a solution z n ≥ 1 for all sufficiently large values n ∈ N. If it were not the case that lim n→∞ z n = ∞, then the sequence (z n ) would contain a bounded subsequence which in turn would contain a convergent subsequence (z n k ) k∈N . Denoting by z the limit of this latter subsequence, it would be the case that z ≥ 1 > 0, and the continuity of f would thus allow us to infer that f (z)/z = lim k→∞ f (z n k )/z n k = lim k→∞ 1/n k = 0 in contradiction to the assumed strict positivity of f on (0, ∞). Having thus shown that lim n→∞ z n = ∞, we may apply (4) (or the weaker assumption mentioned above) to conclude that Consequently, for a given K ∈ N we can find an L ∈ N such that z n /n > K for all n ≥ L. Since f is concave, we further find that for all x ∈ [0, z n ] and therefore, in particular, for all x ∈ [0, nK] whenever n ≥ L (because if n ≥ L then nK < z n ). So for all n ≥ L it is the case that Setting x := mn, the statement of the lemma follows as desired.

Corollary.
For the values n i defined in (5), it is the case that Proof. Let K ∈ N and let L ∈ N be given as in Lemma 3.2. Since lim i→∞ l i = ∞, we have l i−1 ≥ L for all sufficiently large i, and therefore, Lemma 3.2 in conjunction with the definition of n i in (5) implies that n i ≥ K for all sufficiently large i.
Let E ∈ B f and ε > 0 such that 0 < µ f (E) < 1 and Since (J f , T f ) is a rank one system, it is in particular ergodic, and, using the ergodic theorem, we can therefore find a K E ∈ N and a set X E ⊂ J f such that for all x ∈ X E and n ≥ K E . Moreover, we can choose K E ∈ N so large that for all i ≥ K E we have Combining (6), (7), and (9), we fiind that According to Lemma 1.13, we can therefore find a D i ⊂ I i−1 (0) (recall that I i−1 (0) is the bottom-level interval of τ i−1 ) with and li−1 (x) for all x ∈ I i−1 (0) (by Definitions 1.11 and 1.12). Now let δ > 0 such that and Setting we may apply the Lebesgue density theorem, to infer that there is an L E ∈ N such that for all i ≥ L E it is the case that Furthermore, according to Corollary 3.3, we can choose L E so large that for all i ≥ L E we have 1 and Using Lemma 1.13 in conjunction with (14), we may conclude that for any i ≥ L E there exists a set C i ⊂ I i−1 (0) such that and for all x ∈ C i . To proceed we set N E := max{K E , L E }, and for all i ≥ N E we define B i := C i ∩ D i . Then, according to (10), (12), and (19), we have

Now let
we may apply Lemma 3.4a to infer that #Ω i n i 2 ni + n i − 1 < ε 2 .
Now we define

Using (21) and Lemma 3.5 (with
and therefore, (15) implies that Since FRANK BLUME for all k ∈ Ψ i , it follows that for any k ∈ Ψ i there are integers m k ∈ N and j k (1), . . . , j k (m k ) ∈ N such that To proceed we artificially define, for any k ∈ Ψ i , the tower and assign to it the transformation T ρi(k) as specified in Definition 1.10. Setting further for all k ∈ Ψ i , the definitions of Ω i and ρ i (k) imply that for all k ∈ Ψ i . Introducing the definition for all k ∈ Ψ i and x ∈ K i (j k (1)), we may use (24) and Lemma 1.13 to infer that there exists a set B i (k) ⊂ K i (j k (1)) such that for all x ∈ B i (k) and all k ∈ Ψ i . Now let A i (k) be the projection of B i (k) onto K i (k), that is, for all k ∈ Ψ i and let and Given these definitions, we notice that is the vertical subtower of η i (k) generated by A i (k). Moreover, (by (25) and the definition of A i (k)) and therefore, (9) and (17)).
3.8. Definition. For any x ∈ L i we denote by k x the unique integer in Ψ i for which there is a unique m ∈ {0, . . . , ). Using this integer k x , we define Proof. Our first goal is to show that the assumption implies that r i (x) and r i (y) are [(1 − 11ε)n i ]-similar whenever x ∈ A i (k x ) and y ∈ A i (k y ). To do so, we will use r i (x) and r i (y) to construct a finite sequence of pairs (a 0 , b 0 ), . . . , (a l−1 , b l−1 ) ∈ {0, . . . , n i − 3} 2 . Setting (a 0 , b 0 ) := (0, 0), we assume that (a j−1 , b j−1 ) ∈ {0, . . . , n i − 5} 2 has been defined in a such a way that the sums satisfy one of the following two pairs of inequalities (which are both trivially satisfied for j − 1 = 0): (Note: For the definition of S a (j − 1) and S b (j − 1) we agree that −1 m=0 · · · = 0.) To proceed, we will define a j and b j in dependence on which of the two pairs of inequalities-(29) or (30)-is satisfied. If (29) is satisfied, then we set a j := a j−1 +1 and b j := b j−1 + 1 in case that and otherwise, if this inequality is violated, we set a j := a j−1 + 1 and b j : Similarly, if (30) is satisfied, then we set a j := a j−1 + 1 and b j : and otherwise, if this inequality is violated, we set a j := a j−1 + 2 and b j := b j−1 + 1. For clarity we wish to point out that there is no ambiguity in the case where (29) and (30) are valid simutaneously. For in this case we evidently have and therefore, a j := a j−1 + 1 and b j := b j−1 + 1 because both (31) and (32) are satisfied. Furthermmore, in order to explain why the pair (a j , b j ), defined in the manner just described, satisfies either we will assume w.l.o.g. that (29) is satisfied for (a j−1 , b j−1 ). Given this assumption, we consider first the case that (31) is satisfied. In that case we have a j = a j−1 + 1 and b j = b j−1 + 1 and (31) is therefore equivalent to the second inequality in (33). Thus (33) is satisfied whenever Consequently, we only need to consider the case in which this latter inequality is violated, that is, the case in which Given that we assumed (29) to be satisfied, this strict inequality can be valid only and u ni (k x + a j−1 ) = 0. Since these three conditions together imply that (34) is satisfied, we only need to examine the remaining case where (31) is violated and (a j , b j ) = (a j−1 + 1, b j−1 + 2). Here we find that and therefore, (34) is satisfied. Having thus established the validity of (33) or (34), we define Using the notation that we introduced in the context of defining the towers ρ i (k), we may infer that and therefore, #Φ x > n i − 5εn i (by (15)).
Similarly, we find that #Φ y > n i − 5εn i .

FRANK BLUME
To proceed, we denote by Φ the set of all pairs (a j , b j ) for which the following conditions are satisfied: and It is easy to see that the number of pairs (a j , b j ) satisfying (37)  that is, Using (35), (36), and (18), we may thus conclude that Next we claim that for all (a j , b j ) ∈ Φ. To give a proof by contradiction, we assume that there is a pair (a j , b j ) ∈ Φ for which u ni (k x + a j ) = u ni (k y + b j ).
or equivalently Thus we may assume w.l.o.g. that Since (a j , b j ) satisfies either (33) or (34), we will assume w.l.o.g. that (33) is satisfied, because the alternative case, where (34) is valid, is completely analogous. Setting we apply (33) to infer that and, by implication, that 1 3 Since (a j , b j ) ∈ Φ, we have k x + a j ∈ Φ x and k y + b j ∈ Φ y , and therefore, (41) implies that and Since B i = C i ∩ D i , we may apply (20) to conclude that 1 Hence because the second inequality in (43)  and s E li−1 (T f β+li−1 y), respectively. Thus, using (45), we may deduce in a completely analogous fashion that Setting we may combine (46) and (47) with (43) to infer that #∆ Similarly, for the set we combine (44) and (45) with (11) to infer that #Γ < 2ε 4 l i−1 and that #Γ γ < 6ε 4 < 1 2 (by (43) and (6)).
Moreover, according to (13), (16) and (43), we have Consequently, we may apply Lemma 3.6 in conjunction with (48), (12), and (6) to the set Combining (49) and (50) yields and therefore, r i (x 0 ) and r i (y 0 ) are [(1 − 11ε)n i ]-similar, and, by implication, r i (x) and r i (y) are ([(1 − 11ε)n i ] − 1)-similar. To complete the proof, we apply (15) to infer that 4. Minimal entropy growth rates. In this concluding section we will show, by means of Lemma 3.9, how the sequence (f (n)) n∈N defines a minimal entropy growth rate for the system (J f , T f ) in the limit superior as well as-in somewhat altered form-in the limit inferior. for all r ∈ {0, 1} ni , we find that According to Stirling's estimate, we can find an N ∈ N such that for all n ≥ N we have 1 2 √ 2πn n n e n ≤ n! ≤ 3 2 √ 2πn n n e n . Let i ∈ N such that Then a few simple calculations using Stirling's estimate show that Since x −x is increasing for x ≤ 1/e and decreasing for x ≥ 1/e, we can use (6) and (15) to conclude that Combining this estimate with (53), yields