Exponential Decay of Lebesgue Numbers

We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.


Motivation
Entropy, which measures complexity of a dynamical system, has various definitions in both topological and measure-theoretical contexts. Most of these definitions are closely related to each other. Given a partition on a measure space, the famous Shannon-McMillan-Breiman Theorem asserts that for almost every point the cell covering it, generated under dynamics, decays in measure with the asymptotic exponential rate equal to the entropy. It is natural to consider analogous objects in topological dynamics. Instead of measurable partitions, the classical definition of topological entropy due to Adler, Konheim and McAndrew involves open covers, which also generate cells under dynamics. We would not like to speak of any invariant measure as in many cases they may be scarce or pathologic, offering us very little information about the local geometric structure. Diameters of cells are also useless since usually the image of a cell may spread to the whole space. Finally we arrive at Lebesgue number. It measures how large a ball around every point is contained in some cell. It is a global characteristic but exhibits local facts, in which sense catches some idea of Shannon-McMillan-Breiman Theorem. We also notice that the results we obtained provides a good upper estimate of topological entropy which is computable with reasonable effort.

Preliminaries on Lebesgue number
First we briefly discuss some preliminaries on Lebesgue number and open covers. Some of those can be found in any textbook of elementary topology. For the rest, as well as other facts we discuss in succeeding sections without proof, one can refer to, for example, [5].
The basic result we shall use is the following Lebesgue Covering Lemma. Proof. If X ∈ U then the theorem is trivial.
Then δ(U, x) is a continuous function on X taking strictly positive values. Since X is compact, the function attains its minimum value on X. So is the Lebesgue number of the open cover.
Remark. Another widely used formulation of Lebesgue Covering Lemma states that there isδ > 0 (the largest one) such that every set of diameter less thanδ is contained in some element of U. It is easy to see δ ≤δ ≤ 2δ. This guarantees that Definition 3.1 is not affected if Lebesgue number is defined this way.
We have some simple facts on Lebesgue numbers.

Lemma 2.2.
For two open covers U and V, we say U is finer than V, denoted by

Lemma 2.4. For any two open covers U and V, let
Then Proof. On one hand, U V is finer than U and V. By Proposition 2.2 On the other hand, for every x, there are U x ∈ U and V x ∈ V such that Corollary 2.6. If U ≻ V, then for every n, we have f −n (U) ≻ f −n (V) and U n f ≻ V n f , hence δ(f −n (U)) ≤ δ(f −n (V)) and δ n (f, U) ≤ δ n (f, V).

Decay of Lebesgue numbers
Now we turn to the asymptotic decay of Lebesgue numbers.
Definition 3.1. Let U be an open cover of X. We set From now on we use h * L to denote either h + L or h − L , when the argument works for both cases. We note that these numbers possess some properties analogous to entropy.
. If in addition f is a homeomorphism, then the first equality also holds for n < 0. Taking limit we obtain h * . Applying Corollary 2.5, we also have: (We intentionally replace f 1−n by f −n in the limits.) In fact, we have: Remark. The analogous result is not necessarily true for h − L .
The proposition is a corollary of the following lemma. We also note that Lebesgue number is always bounded by the diameter of the space. Lemma 3.6. Let a n ≥ K be real numbers, uniformly bounded from below. Let Then lim sup n→∞ a n n = lim sup n→∞ b n n Proof. By definition, a n ≤ b n . So lim sup n→∞ a n n ≤ lim sup n→∞ b n n .
Note that {b n } is a non-decreasing sequence. If there is N such that for all Otherwise, the set J = {n j |b nj −1 < b nj } has infinitely many elements. If n j ∈ J then we must have Remark. This proposition implies that in Definition 3.1 the supremums may be taken over all open covers (not necessarily finite).

Corollary 3.8. For every sequence of open covers
Proof. By assumption, every element of ∞ n=−∞ f n (U) contains at most one point. U is a generator. By [5,Theorem 5.21], for every ǫ > 0 there is .
Proof. We only show the first inequality. Proof of the other one is similar. Let For any open cover U of diameter less than d(x 0 , y 0 ), y 0 is not covered by any can not cover any point of f −n (y 0 ). This implies that Apply Proposition 3.7 and let ǫ → 0, then we obtain h − L (f ) ≥ λ. Remark. The inequalities may be strict. See Example 6.7.

Lebesgue numbers, entropy and dimensions
In this section we investigate the relations between decay of Lebesgue numbers, topological entropy and sorts of dimensions. We consider the three definitions of topological entropy: one using open covers, oene using separated sets, and one closely related to Hausdorff dimension. Each of them has something to do with Lebesgue numbers. Proof. Let W = {W j }. Then for each j, there is x j ∈ W j such that x j / ∈ W k for all k = j. Otherwise W − {W j } is a sub-cover of U with smaller cardinality.
As every δ(W)-ball is covered by some element of W, it can cover at most one element in {x j } 1≤j≤|W| . So the minimal number of δ(W)-balls needed to cover X is no less than |W|.
where the supremum is taken over all finite open covers.
If h(f, U) = 0 then the theorem is trivial since ∆ n is non-increasing.
Fix a small ǫ > 0. For every N 0 > 0, by definition of the upper box dimension, there is γ 0 > 0 and N 1 > N 0 such that ∆ n < γ 0 for all n > N 1 , and  Recall for ǫ > 0, E ⊂ X is said to be an (n, ǫ)-separated set if d n f (x, y) > ǫ for distinct points x, y ∈ E. Let s n (f, ǫ) = max |E|, where the maximum is taken over all (n, ǫ)-separated sets. Then Proof. Let E be an (n, ǫ)-separated set of cardinality s n (f, ǫ). If distinct points x, y ∈ E are covered by the same δ n -ball, then the ball is covered by some element So d n f (x, y) < ǫ, which contradicts the fact that x and y are (n, ǫ)-separated. So each δ n -ball can cover at most one point in E. N (δ n ) > s n (f, ǫ).
Fix a small θ > 0. For every N 0 > 0, by definition of the upper box dimension, there is γ 0 > 0 and N 1 > N 0 such that δ n < γ 0 for all n > N 1 , and This is true for every U with diameter less than ǫ and every θ > 0. Let ǫ → 0 and apply Proposition 3.7.  and for any real number λ,

Hausdorff dimension and
Then there is a number h U (f ) such that is ∞ for λ < h U (f ) and 0 for λ > h U (f ). As showed in [1], we have The classical Hausdorff measure is defined as We know the Hausdorff dimension of X is a number dim H (X) such that µ λ (X) = ∞ for λ < dim H (X) and µ λ (X) = 0 for λ > dim H (X).
Then there is n 0 such that for every n > n 0 , (1) is satisfied for n ≤ − log(diam(B))/K + 1.

Lipschitz maps
We have shown that h − L and h + L provide upper estimates of topological entropy. Now we show that these numbers are bounded for Lipschitz maps. Recall that a continous map f is Lipschitz with constant L(f ) > 0 if for every x, y ∈ X, d(f (x), f (y)) ≤ L(f ) · d(x, y). Here we assume that L(f ) to be the smallest one among such numbers. Proof. Let L = max{L(f ), 0}. For every x ∈ X and every y ∈ B(x, δ(U) · L −(n−1) ), for j = 0, 1, . . . , n − 1. This implies that In particular, Remark. We note (thanks to Anatole Katok) long before Bowen's definition of topological entropy was introduced, the weaker result involving the box dimension (see, e.g.[3, Theorem 3.2.9]) had been proved by Kushnirenko:

Then for every finite open cover
Remark. If f is Lipschitz (L(f ) < ∞), then the sequence {log L(f n )} 1≤ n≤∞ is sub-additive. In this case In general, h − L (f ), h + L (f ) and l(f ) may be different from each other. Some examples will be discussed in the next section. Proof. Let H be a bi-Lipschitz conjugacy between f on X and g on Y .
For a finite open cover U of X and every x ∈ X, there is U ∈ U such that As H is a homeomorphism, this implies Moreover, H is a conjugacy, Replace U by g −n (H(U)) in (3), then Taking the upper limit we have h * L (f ) ≥ h * L (g). The other direction is the same.
Remark. h * L depends on the metric chosen and is not a topological invariant. By the above theorem each of them is the same for strong equivalent metrics (C 1 d(x, y) < d ′ (x, y) < C 2 d(x, y)). Box dimension and Hausdorff dimension also depend on the metric. However, the inequalities we obtained holds for any metric, making entropy, a topological invariant, bounded by geometric numbers.

Examples
To finish this paper, we put here several examples. Proof. Take any finite open cover U with diameter less than 1/10. Then 1/2 / ∈ U for every U ∈ U covering 0.
This example shows that the numbers h * L (f ) may be unbounded even if h(f ) = 0. It also shows that h *  x, x = 2 −2 k for some integer k or x = 0; 2x, otherwise.
It is easy to check that f is continuous.
On the other hand, denote by If U is an open cover of X, then there is an element U 0 of U and N 0 > 0 such that U 0 ⊃ X N0 . Let k 0 be an integer such that 2 k0−1 ≤ N 0 < 2 k0 . Then for every n, is still an open cover, we have which is independent of n. So by Proposition 3.5, h + L (f ) = 0. Example 6.4. Fix a ≥ b > 1. Consider the compact space with the induced metric and topology from R. Define f on X by x, x = 2 −2 k for some integer k or x = 0; min{y ∈ X a,b : y > x}, otherwise.
It is easy to check that f is a homeomorphism. Similar argument as Example 6.3 shows that l(f ) = log a and h * L (f ) = log b. This example also shows that h * L (f ) is far from a topological invariant since all these functions are topologically conjugate for arbitrary values of a and b.
Then l(f ) = log a and h * L (f ) = 0. This together with Example 6.4 imply that for every l ≥ 0, the collection of possible values Moreover, if we consider g = (f, Id) on X a × [0, 1], then we have dim H (X) = 1, h(g) = 0, l(g) = log a and h * L (g) = 0. This shows that Corollary 4.9 can be a strictly better estimate than Corollary 5.3 (also the results in [2][4]).