MEAN TOPOLOGICAL DIMENSION

ELON LINDENSTRAUSS AND BENJAMIN WEISS

Abstract. In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.

1. Introduction

One of the basic invariants of a dynamical system (X, T) is its topological entropy. This quantifies to what extent nearby points diverge as the system evolves. For the shift on {1, 2, …, k}^{ℤ}, the topological entropy is log k and thus gives a dynamical interpretation of the cardinality of the set of states. For the shift K^{ℤ}, where K is an infinite compact space, this invariant is always +∞, and thus gives no information about K other than the fact that it is infinite. Recently, M. Gromov suggested a definition of a new dynamical invariant, the mean dimension, that would recover in the above example, for a nice K, its topological dimension. Our purpose in this paper is to explore several ways of defining the mean dimension and to relate this new invariant to some other problems in topological dynamics.

A classical theorem of Beboutov states that any real flow (X, T_{t}) whose fixed point set can be imbedded in ℝ can be imbedded in the space of continuous functions on ℝ, with the natural action of ℝ (see Kakutani (1968)). An open problem for many years (see Auslander (1988)) has been to decide whether every minimal system (X, T) is imbeddable in ([0, 1]^{ℤ}, σ), where σ is the shift operation. As we will see, a necessary condition for such an imbedding is that the mean dimension of (X, T) be at most 1. Since it is not hard to see that there are minimal systems with any value for the mean dimension, it follows that, in general, the ℤ-analogue of Beboutov’s theorem fails. In a sequel to this paper by the first author (Lindenstrauss (1998)), it will be shown that conversely, any extension of a minimal system, with mean dimension at most Cd (where C is some constant), can be imbedded in the coordinate shift on the Hilbert cube ([0, 1]^{d})^{ℤ}.

Date: December 6, 1998.

Key words and phrases. dimension, mean dimension, entropy, imbeddings.

The initial examples encountered in topological dynamics of systems with a unique invariant measure were equicontinuous systems and their isometric extensions. These examples fostered the belief that these systems had special metric properties. This belief was dispelled with the discovery of Jewett that any weakly mixing system has a uniquely ergodic model. Since then, there have been no results indicating that uniquely ergodic systems have special dynamical properties. We will relate zero mean dimension to a dynamical version of being totally disconnected, and it will then turn out that any uniquely ergodic system has mean dimension zero.

In the usual approach to topological dimension, one focuses on the degree to which open covers by sets of small diameter overlap without paying any attention to the cardinality of these covers. However, as Pontryagin and Schnirelmann (1932) have shown, one can also define the topological dimension using only cardinalities of open covers. This suggests that the mean dimension should be directly related to the topological entropy of a system, and indeed we shall establish such a relationship which will imply, in particular, that any system with finite topological entropy has zero mean dimension.

While we have written this paper for the usual system (X, T) (i.e. a ℤ-action), all of the results given here can be established for discrete amenable groups, in particular ℤ^{d}. The reader familiar with this extension will have no difficulty in extending all proofs and definitions to the amenable case, with the exception of a useful subadditivity lemma for which we include a complete proof. This lemma apparently has been known by ergodic theorists, but has not been published before.

Overview: In §2 we review some well known facts related to dimension theory, define the mean dimension and deduce some simple properties of it. In §3 we compute the mean dimension for some examples, and indicate how to construct minimal systems with an arbitrary value for the mean dimension. In §4 we define an analogue of Minkowski dimension, the metric mean dimension, which is very closely related to the topological entropy, and investigate the connection between this and the mean dimension. In §5 we discuss the small boundary property ( the dynamical version of being dynamically totally disconnected), and show that a system that has this property has mean dimension zero. Finally in the appendix we prove the result alluded to above concerning subadditive functions defined on finite subsets of amenable groups. This last result is all that is needed for the knowledgeable reader to extend the results and definitions we present to actions of discrete amenable groups.

This paper is part of the first named author’s PhD thesis, prepared under the guidance of the second named author.

2. Preliminaries and basic properties of mean dimension

Let X be a compact metric space, α a finite open cover of X. We will say that a cover β refines α (β ≻ α), if every member of β is a subset of some member of α.

Definition 2.1. If α is an open cover of X we shall denote by

ord(α) = max_{x∈X} ∑_{U∈α} 1_{U}(x) − 1,

and by

D(α) = min_{β≻α} ord(β),

where β runs over all finite open covers of X refining α.

D(α) has a very important property which we shall use heavily — it is subadditive, i.e. D(α ∨ β) ≤ D(α) + D(β) (for open covers α and β of X, their join α ∨ β is defined to be the open cover whose elements are U ∩ V for all U ∈ α, V ∈ β). To verify this we shall give another characterization of D(α) using dimension. We recall the definition of the (cover) dimension for compact metric spaces: X has dimension ≤ D if for every open cover α of X there is a refinement β ≻ α with ord(β) ≤ D — i.e. if for every open cover α, D(α) ≤ D.

Definition 2.2. A continuous map f: X → Y will be called α-compatible if it is possible to find a finite open cover of f(X), β, such that f^{−1}(β) ≻ α. We will use the notation f ≻ α to denote that f is α-compatible.

Proposition 2.3. If X is compact, f: X → Y a continuous function such that for every y ∈ Y, f^{−1}(y) is a subset of some U ∈ α, then f is α-compatible.

Proof. Without loss of generality we may assume f(X) = Y. If α = {U_{1}, … U_{n}}, we define for i = 1, …,n,

V_{i} = {y ∈ Y : f^{−1}(y) ⊂ U_{i}}.

By our assumption, ⋃_{i=1}^{n} V_{i} = Y, and clearly f^{−1}(V_{i}) ⊂ U_{i}. Thus, it only remains to show the V_{i}’s are open. Assume to the contrary that there is some y ∈ V_{i} such that there is a sequence y_{n} → y with y_{n} ∉ V_{i}, i.e.

f^{−1}(y_{n}) ∩ (X \ U_{i}) ≠ ∅.

Let x_{n} ∈ f^{−1}(y_{n}) ∩ (X \ U_{i}). The x_{n} have a subsequence that converge to some x ∈ X. As f is continuous,

f(x) = lim f(x_{n_{k}}) = lim y_{n_{k}} = y

and so x ∈ f^{−1}(y) ⊂ U_{i}. On the other hand, for every n, x_{n} ∈ X \ U_{i} which implies that

x = lim x_{n_{k}} ∈ X \ U_{i}

a contradiction.

We can now state the following characterization of D(α):

Proposition 2.4. If α is an open cover of X, then

D(α) ≤ k

iff there is an α-compatible continuous function f: X → K where K has topological dimension k.

Proof. Suppose there exists such an f: X → K, and a cover β of f(X) ⊂ K such that f^{−1}(β) ≻ α. By k-dimensionality of K, and thus of f(X), the cover β has a subcover γ with ord(γ) ≤ k. And so

f^{−1}(γ) ≻ f^{−1}(β) ≻ α.

As ord(f^{−1}(γ)) = ord(γ) ≤ k we are done.

Conversely, suppose γ ≻ α, with ord γ ≤ k. We shall construct a γ compatible map from X to a k-dimensional simplicial complex |C|.

Indeed, let {w_{U}}_{U∈γ} be a partition of unity subordinate to γ, i.e. functions such that for any x ∈ X

∑_{U∈γ} w_{U}(x) = 1,

and supp(w_{U}) ⊂ U. The vertices of C correspond to the elements of γ, the s-dimensional simplices to all families {U_{0}, …, U_{s}} with ∩_{i=0}^{s}U_{i} ≠ ∅. The points in each simplex {U_{0}, …, U_{s}} can be parameterized as

∑_{i=0}^{s} a_{i}[U_{i}] a_{i} ≥ 0 for every i, and ∑_{i=0}^{s} a_{i} = 1.

We now map every point x of X into the simplex {[U]: x ∈ U ∈ γ} of |C| by

x ↦ ∑_{x∈U∈γ} w_{U}(x)[U].

This is clearly continuous. f is γ compatible (and hence α compatible) since for every y ∈ |C|, f^{−1}(y) is a subset of every U that corresponds to a vertex of the least dimensional simplex of |C| containing y.

Corollary 2.5. D is sub-additive, i.e. if α and β are finite open covers of X

D(α ∨ β) ≤ D(α) + D(β).

Proof. If f: X → A is α-compatible, with dim A = D(α), and g: X → B is β-compatible, with dim B = D(β), then h: X → A × B with h: x ↦ (f(x), g(x)) is α ∨ β compatible. By the well known properties of dimension, dim A × B ≤ dim A + dim B, and hence we have an α ∨ β-compatible mapping from X to a space of dimension ≤ D(α) + D(β).

If α is an open cover of X, we will use the notation α_{a}^{b} for b > a ∈ ℤ to denote the open cover

α_{a}^{b} = T^{−a}α ∨ T^{−a−1}α ∨ ⋯ ∨ T^{−b}α

Definition 2.6. If (X, T)is a dynamical system, then the mean dimension of (X, T), denoted by mdim(X, T) (or mdim(X) if T is understood), is defined by:

mdim(X, T) = sup_{α} lim_{n→∞} D(α_{0}^{n−1}) ; n,

where α runs over all finite open covers of X.

Remarks:

• The fact that the limit above exists is a consequence of the subadditivity of D, and is true also for the general case of amenable group actions (see appendix). In this case, the limit over n is replaced by taking the limit on subsets of the group that become more and more invariant.

• It is clear from the definition that if X′ is a T -invariant closed subset of X then

mdim X′ ≤ mdim X.

However, unlike topological entropy, mdim of a factor may be greater than the mdim of the original system. The same is, of course, true for the usual topological dimension.

• Recall that the topological dimension can be defined by

dim X = sup_{α} D(α)

where α runs over all open covers of X. It follows that if X has finite topological dimension, then for any α and n

D(α_{0}^{n−1}) < dim X,

and hence mdim(X) = 0.

• In the definition of mdim one can replace sup_{α} by

sup_{k} lim_{n→∞} D(α(k)_{0}^{n−1}) ; n

where α(k) is any sequence of open covers such that the maximal diameter of an element of α(k) tends to zero as k ⟶ ∞.

We now prove some easy results about mdim, showing that this new invariant is well behaved.

Proposition 2.7. For any dynamical system (X, T),

mdim(X, T^{n}) = n mdim(X, T)

Proof. Let α be any open cover of X. Then

lim_{k→∞} D(α ∨ T^{−n}α ∨ ⋯ ∨ T^{−(k−1)n}α) ; k ≤ n lim_{k→∞} D(α_{0}^{kn−1}) ; kn
≤ n mdim(X, T)

and so mdim(X, T^{n}) ≤ n mdim(X, T). On the other hand,

α_{0}^{kn−1} = α_{0}^{n−1} ∨ T^{−n}α_{0}^{n−1} ∨ ⋯ ∨ T^{−(k−1)n}α_{0}^{n−1}

and so

mdim(X, T) = sup_{α} lim_{k→∞} D(α_{0}^{kn−1}) ; kn ≤ mdim(X, T^{n}) ; n.

Proposition 2.8. Let (X_{i}, T_{i}), be a sequence of dynamical systems, for 1 ≤ i < R, R ∈ ℕ ∪ {∞}. Then

mdim(X_{1} × X_{2} × ⋯, T_{1} × T_{2} × ⋯) ≤ ∑_{i<R} mdim(X_{i}, T_{i}).

Proof. Let α be any open cover of X_{1} × X_{2} × · · ·. There is an N < R and open covers β(j) of X_{j} such that

β = ⋁_{j=1}^{N} π_{j}^{−1}(β(j)) ≻ α,

where π_{j} is the projection onto the jth coordinate. By subadditivity of D(·),

D(β_{0}^{n}) ≤ ∑_{j=1}^{N} D(β(j)_{0}^{n})

and so

lim_{n→∞} D(β_{0}^{n−1}) ; n ≤ ∑_{j=1}^{N} lim_{n→∞} D(β(j)_{0}^{n−1}) ; n ≤ ∑_{j=1}^{N} mdim(X_{j}).

3. Some examples

We begin with calculating the mean dimension of some natural dynamical systems.

Proposition 3.1. Let K be finite dimensional and compact, and let σ be the shift on K^{ℤ}. Then

mdim(K^{ℤ}, σ) ≤ dim K.

Proof. Let α be any open cover of K^{ℤ}. For every a < b, let π_{a}^{b}: K^{ℤ} → K^{b−a+1} be the projection of an element of K^{ℤ} to its coordinates with indices in the range a … b. As the topology on K^{ℤ} is the Tychonoff topology, there is a refinement α′ of α, and an N such that every U ∈ α′ is of the form (π_{−N}^{N})^{−1} (V) for some open V ⊂ K^{2N+1}.

Thus to show mdim(K^{ℤ}) ≤ dim K it suffices to show that for α′ of the above form,

lim_{n→∞} D(α′_{0}^{n−1}) ; n ≤ dim K.

However, α′_{0}^{n−1} is actually the inverse image (via π_{−N}^{N+n−1}) of a cover of K^{2N+1+n}. Since K^{2N+1+n} has dimension at most (2N+1+n) dim K, there is a cover β̃ that refines π_{−N}^{N+n−1}(α′) with ord(β̃) ≤ (2N + 1 + n) dim K. As

(π_{−N}^{N+n−1})^{−1} (β̃) ≻ α′_{0}^{n−1}

we see that

D(α′_{0}^{n−1}) ; n ≤ (2N + 1 + n) dim K ; n ⟶ dim K as n ⟶ ∞.

There are compact subsets K of ℝ^{d} such that dim K × K < 2 dim K, and so the above result cannot be improved in general. However for nice K we have indeed that mdim K^{ℤ} = dim K. We will need the following dimension theoretic lemma:

Lemma 3.2. If α is an open cover of [0, 1]^{d} such that no U ∈ α intersects two opposing faces of the cube, then ord(α) ≥ d.

Proof. Like most results of this type, this is merely a reformulation of the Brouwer Fixed Point Theorem. Indeed, suppose ord(α) < d. We can use this to construct a continuous map F from [0, 1]^{d} to |C|, a d − 1 (or lower) dimensional simplicial complex with a vertex [U] corresponding to each U ∈ α and simplices corresponding to all families {U_{0}, …, U_{s}} with ⋂_{i=0}^{s} U_{i} ≠ ∅ (just like we did in the proof of Proposition 2.4).

Now define G: |C| → [0, 1]^{d} as follows: first map the vertices of C by G: [U] ↦ (e_{0}(U), …, e_{d−1}(U)) where

e_{k}(U) = \begin{cases}1 & if U intersects the face x_{k} = 0 of [0, 1]^{d} \\ 0 & otherwise, \end{cases}

then extend this to the simplices of |C| in an affine way. G ∘ F: [0, 1]^{d} → [0, 1]^{d} maps the d dimensional cube to a finite number of at most d − 1 dimensional simplices, so its image cannot contain (0, 1)^{d}. Let

p ∈ (0, 1)^{d} \ Image(G ∘ F),

and take H to be the projection that sends an x ∈ [0, 1]^{d} \ {p} along the ray emanating from p and passing through x to the boundary of [0, 1]^{d}. The map H ∘ G ∘ F maps the d dimensional cube to its boundary, and maps each face of the cube to the opposite face, so this map has no fixed points neither inside the cube nor on its boundary, in contradiction to the Brouwer Fixed Point Theorem.

We now prove that if K = [0, 1]^{d} then mdim K^{ℤ} = dim K. With an eye to a construction of a minimal system that is not imbeddable in ([0, 1]^{ℤ}, σ) we prove a slightly more general statement. If I = {⋯ i_{0} < i_{1} < i_{2} ⋯ } is a finite or infinite subset of ℤ , we take π_{I} to be the projection

π_{I}: (…, x_{0}, x_{1}, …) → (…, x_{i_{0}}, x_{i_{1}}, …)

(π_{I} is a function from K^{ℤ} to K^{|I|}, K^{ℕ} or K^{ℤ} depending on I).

Proposition 3.3. Let X be a shift invariant closed subset of ([0, 1]^{d})^{ℤ} such that there is an infinite set of indices I ⊂ ℕ and an x̄ ∈ X such that

1. I has upper density θ, i.e.

θ = lim_{n→∞} |I ∩ {0, …, n − 1}| ; n;

2. Any x ∈ ([0, 1]^{d})^{ℤ} with

π_{ℤ\I}(x) = π_{ℤ\I}(x̄)

is in X.

then

mdim(X) ≥ θd.

In particular, mdim (([0, 1]^{d})^{ℤ}, σ) = d

Proof. Let α′ be a cover of [0, 1]^{d} such that no element of α′ intersects two opposing faces of the cube [0, 1]^{d}, and let α = π_{1}^{−1}(α′). α is an open cover of ([0, 1]^{d})^{ℤ}, and since X ⊂ ([0, 1]^{d})^{ℤ} it induces an open cover α̃ of X.

We now estimate D(α̃_{0}^{n−1}). Let I(n) = I ∩ {0, …, n − 1}. For any β ≻ D(α̃_{0}^{n−1}) the collection of sets

β′ = {π_{I(n)} ({x ∈ U: π_{ℤ\I}(x) = π_{ℤ\I}(x̄)}) : U ∈ β}

is an open cover of ([0, 1]^{d})^{|I(n)|}, and clearly no element of β′ can intersect two opposing faces of the cube ([0, 1]^{d})^{|I(n)|}. Thus ord(β′) ≥ d|I(n)|, and this also means that ord(β) ≥ d|I(n)|. Since β was an arbitrary refinement of α̃_{0}^{n−1} we see that D(α̃_{0}^{n−1}) ≥ d|I(n)|, hence

mdim(X) ≥ lim_{n→∞} D(α̃_{0}^{n−1}) ; n ≥ lim_{n→∞} d|I(n)| ; n = θd.

Remark: In a very similar way, one can see that if X ⊂ ([0, 1]^{ℕ})^{ℤ}, and for a set I ⊂ ℕ of indices with positive upper density and x̄ ∈ X we have that any x with π_{ℤ\I}(x) = π_{ℤ\I}(x̄) is in X then mdim(X) = ∞.

Since mean dimension is a topological invariant of (X, T), and the mean dimension of a closed invariant subset of a dynamical system is at most the mean dimension of the full system, we get:

Corollary 3.4. A necessary condition for (X, T) to be imbeddable in the dynamical system (([0, 1]^{d})^{ℤ}, σ) is that mdim X ≤ d.

The standard methods of constructing minimal subshifts yield the existence of minimal systems with mean dimension r for all r ∈ R^{+} ∪ {∞}. Any such example with mean dimension greater than 1 will not be imbeddable in ([0, 1]^{Z}, σ); any example with infinite mean dimension will not be imbeddable in any (([0, 1]^{d})^{ℤ}, σ). We remark that it is very easy to show that any dynamical system can be imbedded in (([0, 1]^{ℕ})^{ℤ}, σ) — indeed, any compact metric set X can be imbedded in [0, 1]^{ℕ} (say using the map ϕ), and thus

Φ: x ⟼ (…, ϕ(T^{−1}x), ϕ(x), ϕ(T x), …)

is an imbedding of (X, T) in (([0, 1]^{ℕ})^{ℤ}, σ). For completeness, we show in detail how to construct a minimal shift invariant closed subset of ([0, 1]^{2})^{ℤ} with mdim(X) > 1.

Proposition 3.5. There is a shift invariant closed subset X of ([0, 1]^{2})^{ℤ} such that (X, σ) is minimal and mdim(X) > 1.

Proof. We consider ([0, 1]^{2})^{ℤ} as bi-infinite words from the (rather large) alphabet D = [0, 1]^{2}. As a metric on this space we take

d(x, y) = ∑_{i=−∞}^{∞} 2^{−i}∥x − y∥

where ∥·∥ is any norm on ℝ^{2} (for example the Euclidian norm). It will be convenient to use for a < b the notation

x_{a}^{b} = (x_{a}, x_{a+1}, …, x_{b})

(we will use this notation also for x ∈ D^{N} for finite N; in this case we begin to index the coordinates of x from 0).

We will build a decreasing sequence X_{n} of closed shift invariant subsets of D^{ℤ}, and a decreasing sequence of subsets I_{n} ⊂ ℕ such that

1. The orbit of every point x ∈ X_{n} comes within distance C2^{−n} of every y ∈ X_{n}.

2. There is an x̄(n) ∈ X_{n} such that any y ∈ D^{ℤ} with π_{ℤ\I_{n}}(x̄(n)) = π_{ℤ\I_{n}}(y) is also in X_{n}.

3. π_{ℤ\I_{n}}(x̄(m)) = π_{ℤ\I_{n}}(x̄(n)) for any m ≥ n.

4. I = ∩_{n}I_{n} has upper density greater than .9.

From (1), X = ∩_{n}X_{n} is minimal. From (2) and (3), if x̄ is any limit point of the set {x̄(n)} then for all n, any y with π_{ℤ\I_{n}}(x̄) = π_{ℤ\I_{n}}(y) is in X_{n}, hence any y with π_{ℤ\I}(x̄) = π_{ℤ\I}(y) is in X. Finally, from (4) and Proposition 3.3,

mdim(X) ≥ 1.8.

So, it remains to construct the X_{n}, x̄(n) and I_{n}. The X_{n} will be of block type: there will be an L_{n} ∈ ℕ and B_{n} ⊂ D^{L_{n}} such that

X_{n} = {(… x_{−1}, x_{0}, x_{1}, …): ∃k ∈ ℤ s.t. ∀a ∈ ℤ, x_{k+aL_{n}}^{k+aL_{n}+L_{n}−1} ∈ B_{n}} .

Such a (nonempty) system of block type is always topologically transitive i.e. there is an x̃ ∈ X_{n} such that its orbit is dense in X_{n}. We take x̄(n) to be any point of X_{n} such that

x_{aL_{n}}^{aL_{n}+L_{n}−1} ∈ B_{n} for all a ∈ ℤ

We start from L_{1} = 1, B_{1} = D, X_{1} = D^{ℤ}, and I_{1} = ℕ. We proceed by induction; so assume we have defined L_{n}, B_{n} and I_{n}. Take x̃ to a point in X_{n} with dense orbit. By replacing x̃ by its shift if necessary, we can assume x_{aL_{n}}^{aL_{n}+L_{n}−1} ∈ B_{n} for all a ∈ ℤ. We take l_{n+1} to be an integer divisible by 2L_{n} such that for any y ∈ X_{n} there is a

k ∈ {−l_{n+1} + 10n, …, l_{n+1} − 10n}

such that d(y, σ^{k}x̃) < 2^{−n}. We define L_{n+1} = 2^{n+10}l_{n+1}, and B_{n+1} by

B_{n+1} = {b ∈ D^{L_{n+1}} : ∀a, b_{aL_{n}}^{aL_{n}+L_{n}−1} ∈ B_{n} and b_{L_{n+1}−2l_{n+1}}^{L_{n+1}−1} = x̃_{−l_{n+1}}^{l_{n+1}−1}}.

Finally, we define

I_{n+1} = I_{n} ∩ {k ≡ 0, …, L_{n+1} − 2l_{n+1} − 1 (mod L_{n+1})}.

We leave to readers the verification that X_{n}, x̄(n) and I_{n} so defined have the required properties.

We have already calculated the mean dimension of [0, 1]^{ℤ}; now we wish to consider factors of this system. In Lindenstrauss (1995) a relatively simple argument was used to show that all factors of [0, 1]^{ℤ} have infinite topological entropy. We use a similar argument to show that all the factors of [0, 1]^{ℤ} have strictly positive mean dimension.

Theorem 3.6. If (Y, S) is any non trivial factor of ([0, 1]^{ℤ}, σ) then

mdim(Y) > 0.

Proof. Let ϕ: [0, 1]^{ℤ} → Y be the factor map from ([0, 1]^{ℤ}, σ) to (Y, S). Let α = {U_{0}, U_{1}} be an open cover of Y such that neither U_{0} nor U_{1} is dense in Y. We shall prove that

lim_{n→∞} D(α_{0}^{n−1}) ; n > 0.

Indeed, for any β that refines α_{0}^{n−1}, ϕ^{−1}(β) refines ϕ^{−1}(α_{0}^{n−1}) and

ord(ϕ^{−1}(β)) = ord(β).

Thus it suffices to prove that for any open cover α̃ = {V_{1}, V_{2}} of [0, 1]^{ℤ}, such that for i = 0, 1 cl V_{i} ≠ [0, 1]^{ℤ},

lim_{n→∞} D(α̃_{0}^{n−1}) ; n > 0.  (3.1)

As the V_{i}’s are not dense, there are p_{i} ∈ V_{i}, N, and ϵ such that if ∥π_{−N}^{N}x − π_{−N}^{N}p_{i}∥_{∞} < ϵ then x ∉ V_{1−i}. By replacing V_{i} with

{x ∈ [0, 1]^{ℤ}: ∥π_{−N}^{N}x − π_{−N}^{N}p_{1−i}∥_{∞} > ϵ}

if necessary, we can assume the V_{i} are cylindrical sets that depend only on the coordinates −N … N. Find some line ℓ ⊂ [0, 1]^{2N+1} such that ℓ ⊄ cl(π_{−N}^{N}V_{i}) for i = 0 and 1. Limiting our attention to the subset

X′ = {x ∈ [0, 1]^{ℤ}: ∀k π_{−N+(2N+1)k}^{N+(2N+1)k}(x) ∈ ℓ}

we can use similar arguments as in the proof that mdim([0, 1]^{ℤ}) = 1 to see that

D(α̂ ∨ σ^{−(2N+1)}α̂ ∨ ⋯ ∨ σ^{−(k−1)(2N+1)}α̂) = k,

where α̂ is the restriction of the cover α̃ to X′. It follows that for all k

D(α̃_{0}^{k(2N+1)−1}) ; k(2N + 1) ≥ 1 ; 2N + 1,

and thus mdim(Y) ≥ 1 ; 2N+1 > 0.

It is also possible (with some effort) to find a minimal dynamical system with all its factors having strictly positive mean dimension — the example in Lindenstrauss (1995) of a minimal system with no finite entropy factors also has no zero mean dimensional factors. A similar example can be found in Glasner and Maon (1984).

We conclude this section with a simple example of a system with zero topological dimension (and hence also zero mean dimension) with [0, 1]^{ℤ} as a factor: Let C be Cantor’s set. C can be mapped (say by the map ϕ) onto [0, 1]. Then (…, x_{0}, x_{1}, …) ↦ (…, ϕ(x_{0}), ϕ(x_{1}), …) is a factor mapping (C^{ℤ}, σ) ⟶ ([0, 1]^{ℤ}, σ). This example shows that unlike entropy, factor maps can definitely increase mean dimension.

4. The metric mean dimension

In this section we present another definition of mean dimension, and investigate the connection between this new mean dimension, which we will call the metric mean dimension, and mdim(X). In particular, we will learn something new about mdim(X) — if h_{top}(X) is finite, mdim(X) = 0.

The metric mean dimension depends on the metric d, and can be thought as a mean Minkowski dimension.

For an open cover α, define the mesh of α according to the metric d by

mesh(α, d) = max_{U∈α} diam(U).

Definition 4.1. Let X be a dynamical system, d(·, ·) a metric on X. Define

d_{N}(x, y) = max_{0≤n<N} d(T^{n}x, T^{n}y).

Set

S(X, ϵ, d) = lim_{n→∞} inf_{mesh(α,d_{n})<ϵ} log |α| ; n  (4.1)

S is monotone nondecreasing as ϵ → 0, and we wish to measure just how fast it increases. We define the metric mean dimension of X (for the given metric d), mdim_{M} (X, d), as

mdim_{M} (X, d) = lim_{ϵ→0} S(X, ϵ, d) ; | log ϵ|.  (4.2)

To get a topological invariant we shall set

mdim_{M} (X) = inf_{d metric on X} mdim_{M} (X, d)

Notice that if mesh(α, d_{n}) < ϵ and mesh(β, d_{m}) < ϵ then

mesh(α ∨ T^{−n}β, d_{n+m}) < ϵ.

Thus inf_{mesh(α,d_{n})<ϵ} log |α| is a subadditive function of n, and the limit in (4.1) can be replaced by an infimum.

Recall how the topological entropy is defined for a dynamical system X. Let X be a dynamical system, d(·, ·) a metric on X. A set S ⊂ X is called (n, ϵ, d)-spanning if for every x ∈ X there is a y ∈ S so that for all 0 ≤ k < n,

d(T^{k}x, T^{k}y) < ϵ

Set

A(X, n, ϵ, d) = min{|S|: S ⊂ X is (n, ϵ, d)-spanning}.

The topological entropy is defined in terms of these as

S′(X, ϵ, d) = lim_{n→∞} log A(X, n, ϵ, d) ; n;

h_{top}(X) = lim_{ϵ→0} S′(X, ϵ, d).  (4.3)

Notice that

S′(X, ϵ, d) ≥ S(X, 2ϵ, d) ≥ S′(X, 2ϵ, d)

Indeed, the first inequality is a consequence of the fact that if S is (n, ϵ, d)- spanning, the d_{n} balls of radius ϵ cover X — and these are |S| sets of d_{n} diameter smaller than 2ϵ. The second inequality is a consequence of the fact that if α is a cover with mesh(α, d_{n}) < ϵ and chose one point from each element of α to form a set S, then S is (n, ϵ, d) spanning. Thus (4.2) is equivalent to

mdim_{M} (X, d) = lim_{ϵ→0} S′(X, ϵ, d) ; | log ϵ|

And in particular the metric mean dimension will be non zero only if h_{top}(X) = ∞.

The rest of this section is dedicated to showing that the mean dimension of X is not larger than the metric mean dimension for all metrics d on X.

Theorem 4.2. For any dynamical system (X, T), and any metric d compatible with the topology on X,

mdim(X) ≤ mdim_{M} (X, d).

As mdim(X) = sup_{α} lim D(α_{0}^{n−1})/n, it is enough to prove that for any finite open cover α of X,

lim_{n→∞} D(α_{0}^{n−1}) ; n ≤ mdim_{M} (X, d).

We can refine α to be of the form

α = {U_{1}, V_{1}} ∨ {U_{2}, V_{2}} ∨ ⋯ ∨ {U_{r}, V_{r}}.

where for every i, {U_{i}, V_{i}} is a two element open cover of X.

Set w_{i}: X → [0, 1] by

w_{i}(x) = d(x, X \ V_{i}) ; d(x, X \ V_{i}) + d(x, X \ U_{i})

Notice that w_{i} is Lipschitz, and

U_{i} = w_{i}^{−1}[0, 1)
V_{i} = w_{i}^{−1}(0, 1].  (4.4)

Let C_{L} be a bound on the Lipschitz constant of all w_{i}. For any N define F (N, ·): X → [0, 1]^{rN} by

F (N, x) = (w_{1}(x), w_{2}(x), …, w_{r}(x),
w_{1}(T x), w_{2}(T x), …, w_{r}(T x),
⋮
w_{1}(T^{N}x), w_{2}(T^{N}x), …, w_{r}(T^{N}x)).

From (4.4) we see that F (N, ·) ≻ α_{0}^{N−1}. As usual, if S ⊂ {1, …, rN}, then F (N, x)_{S} ∈ [0, 1]^{|S|} is the projection of F (N, x) to the coordinates in the index set S. We need the following two lemmas:

Lemma 4.3. Let ϵ > 0, D = mdim_{M} (X, d). If N is larger than some N(ϵ), there is a ξ ∈ (0, 1)^{rN} such that for any |S| ≥ (D + ϵ)N,

ξ_{S} ∉ F (N, X)_{S}

Proof. Let

δ < (2^{r}(2C_{L})^{2D})^{−2/ϵ}

be such that

S(X, δ, d) ; | log δ| ≤ mdim_{M} (X, d) + ϵ/4

If N is large enough (depending on ϵ and δ), X can be covered by δ^{−(D+ϵ/2)N} balls of the form

B_{X}(x, N, δ) = {x′ ∈ X: d(T^{k}x, T^{k}x′) < δ for all 0 ≤ k < N}

Since C_{L} is a Lipschitz constant for all w_{i}

F (N, B_{X}(x, N, δ)) ⊂ {a ∈ [0, 1]^{rN}: ∥F (N, x) − a∥_{∞} < C_{L}δ}

the set F (N, X) can be covered by δ^{−(D+ϵ/2)N} balls in the ∥·∥_{∞} norm of radius C_{L}δ. Enumerate these balls as B(k), k = 1, …, K = δ^{−(D+ϵ/2)N}.

Choose ξ with uniform probability in [0, 1]^{rN}. For any S

P (ξ_{S} ∈ F (N, X)_{S}) ≤ ∑_{k=1}^{K} P (ξ_{S} ∈ B(k)|_{S}) ≤ δ^{−(D+ϵ/2)N}(2C_{L}δ)^{|S|}

Hence

P (∃S: |S| ≥ (D + ϵ)N and ξ_{S} ∈ F (N, X)_{S}) ≤

≤ ∑_{|S|≥(D+ϵ)N} P (ξ_{S} ∈ F (N, K)_{S})

≤ (# of such S) × δ^{−(D+ϵ/2)N}(2C_{L}δ)^{(D+ϵ)N}

≤ 2^{rN}((2C_{L})^{2D}δ^{ϵ/2})^{N} ≪ 1,

and so, with high probability, a random ξ will satisfy the requirements.

Lemma 4.4. If π: F (N, X) → [0, 1]^{rN} satisfies for both a = 0 and 1, and all ξ ∈ [0, 1]^{rN},

{1 ≤ k ≤ rN: ξ_{k} = a} ⊂ {1 ≤ k ≤ rN: π(ξ)|_{k} = a}

Then π ∘ F (N, ·) is compatible with α_{0}^{N−1}

Proof. Indeed, if ξ ∈ [0, 1]^{rn}, define for 0 ≤ j < N and 1 ≤ i < r,

W_{i,j} = \begin{cases}T^{−j}U_{i} & if ξ_{jr+i} = 0, \\ T^{−j}V_{i} & otherwise. \end{cases}

It is easy to see that

π^{−1}(ξ) ⊂ ⋂_{{1≤i<r \atop 0≤j<N}} W_{i,j} ∈ α_{0}^{N−1}.

Proof of Theorem 4.2: Let ϵ > 0. Find ξ̄, N as in Lemma 4.3. Let

Φ = {ξ ∈ [0, 1]^{rN}: ξ_{k} = ξ̄_{k} for more than (D + ϵ)N indexes k}

thus F (N, X) ⊂ [0, 1]^{rN} \ Φ. For brevity we shall denote the latter set as Φ^{C}. We shall construct a continuous retraction π of Φ^{C} onto the ⌊(D + ϵ)N⌋ + 1 skeleton of the cube I = [0, 1]^{rN}. For m = 1, 2, …, we denote by J_{m} the set

J_{m} = {ξ ∈ I: ξ_{i} ∈ {0, 1} for at least m indices 1 ≤ i ≤ rN}.

Since ξ̄is in the interior of I, one can define π_{1} : I\{ξ̄} → J_{1} by mapping each ξ to the intersection of the ray starting at ξ̄ and passing through ξ and J_{1}. For each of the (rN − 1)-dimensional cubes I^{l} that comprise J_{1}, we can define a retraction on I^{l} in a similar fashion using as a center the projection of ξ̄ onto I^{l}. This will define a continuous retraction π_{2} of Φ^{C} into J_{2}. As long as there is some intersection of Φ with the cubes in J_{m} this process can be continued, thus we get finally a continuous projection π of Φ^{C} onto J_{m_{0}}, with

m_{0} + ⌊(D + ϵ)N⌋ + 1 = n.

Clearly, π satisfies the conditions of Lemma 4.4. Thus π ∘ F (N, ·) ≻ α_{0}^{N−1}. Also, as F (N, ·) ⊂ Φ^{C},

π ∘ F (N, ·) ⊂ J_{m_{0}},

the latter having topological dimension ⌊(D + ϵ)N⌋ + 1.

To summarize: we have constructed an α_{0}^{N−1} compatible function from X to a space of topological dimension ≤ (D + ϵ)N + 1, and so

D(α_{0}^{N−1}) ≤ (D + ϵ)N + 1

As ϵ > 0 can be made as small as we like, α as refined as we like, we see that indeed

mdim(X) ≤ D.

It is plausible that equality actually holds in Theorem 4.2. In general this is still open, but Lindenstrauss (1998) contains a proof of this for ℤ-actions, if in addition (X, T) is an extension of a minimal system.

5. The Small Boundary Property

Definition 5.1. Let (X, T) be a dynamical system, and E a subset of X. We define the orbit capacity of the set E to be

ocap(E) = lim_{n→∞} sup_{x∈X} ∑_{i=0}^{n−1} 1_{E}(T^{i}x) ; n.

A set E ⊂ X will be called small if ocap(E) = 0.

We remark that the limit above exists since

sup_{x∈X} ∑_{i=0}^{n+m−1} 1_{E}(T^{i}x) ≤ sup_{x∈X} ∑_{i=0}^{n−1} 1_{E}(T^{i}x) + sup_{x∈X} ∑_{i=0}^{m−1} 1_{E}(T^{i}x)

Definition 5.2. A dynamical system (X, T) has the small-boundary property (SBP) if every point x ∈ X and every open U ∋ x there is a neighborhood V ⊂ U of x with small boundary.

The notion of SBP can be thought of as an analogue of the usual definition of zero dimensional space: X is zero dimensional if for every point x ∈ X and every open U ∋ x there is a neighborhood V ⊂ U of x with empty boundary.

The definition of small sets was introduced in Shub and Weiss (1991), where it was shown that if X is uniquely ergodic (or even if it has less than 2^{ℵ_{0}} ergodic T -invariant measures) then X has the SBP. Indeed, first we note that for any closed set E that is not small there is some invariant probability measure μ such that μ(E) > 0 — let x_{k} be a sequence such that

1 ; N_{k} ∑_{i=0}^{N_{k}−1} 1_{E}(T^{i}x_{k}) ≥ c > 0

then any weak^{∗} limit point of

1 ; N_{k} ∑_{i=0}^{N_{k}−1} δ_{T^{i}x_{k}}

is invariant and satisfies μ(E) ≥ c. Conversely, if E is small and μ an invariant measure

μ(E) = 1 ; n ∫ ∑_{i=0}^{n−1} 1_{E}(T^{i}x)dμ(x) → 0.

Now, if X is uniquely ergodic, and μ the unique invariant measure, then for any x ∈ X, the set

{r ∈ ℝ^{+} : {y: d(x, y) = r} is not small} =

= {r ∈ ℝ^{+} : μ{y: d(x, y) = r} ≠ 0}

is countable, and so there are many arbitrarily small balls around x with small boundary, and X has the SBP.

The following proposition is an immediate consequence of the definitions:

Proposition 5.3. If (X, T) has the SBP, then for every open cover α of X and every ϵ there is a subordinate partition of unity ϕ_{j}: X → [0, 1] (j = 0 … |α|) such that

∑_{j=1}^{|α|} ϕ_{j}(x) ≡ 1

supp(ϕ_{j}) ⊂ U for some U ∈ α, and j = 1, …, |α|.

ocap(⋃_{j=1}^{|α|} ϕ_{j}^{−1}(0, 1)) < ϵ  (5.1)

Proof. First find (using the SBP) a cover of X by open sets with small boundary that refines α, and by taking unions of these sets it is possible to find a refinement α′ ≻ α such that there is a one-to-one correspondence between the elements U_{j} of α and U′_{j} of α′ with U′_{j} ⊂ U_{j}.

Let N satisfy

1 ; N ∑_{i=0}^{N−1} 1_{bd}_{U′_{j}}(T^{i}x) < ϵ ; |α|, for all x ∈ X and j.

and δ be small enough so that for all j

1 ; N ∑_{i=0}^{N−1} 1_{B(bd U′_{j},δ)}(T^{i}x) < ϵ ; |α|.

where B(bd U′_{j}, δ) is set of all points of distance smaller than δ from bd U′_{j}. We also take δ to be small enough so that B(bd U′_{j}, δ) ⊂ U_{j} for all j. Now take

ψ_{j}(x) = \begin{cases}1 & if x ∈ U′_{j}, \\ max(0, 1 − δ^{−1}d(x, bd U′_{j})) & otherwise. \end{cases}

We define the functions ϕ_{i}(x) as follows:

ϕ_{1}(x) = ψ_{1}(x)
ϕ_{2}(x) = min(ψ_{2}(x), 1 − ϕ_{1}(x))
ϕ_{3}(x) = min(ψ_{3}(x), 1 − ϕ_{1}(x) − ϕ_{2}(x))
⋮

These clearly satisfy the required conditions.

Theorem 5.4. Every space with the SBP has mean dimension zero.

Proof: Let (X, T) have the SBP, α a cover of X. Take ϵ > 0, Construct, according to the above proposition, an α subordinate partition of unity which obeys (5.1). Let |α| = k, A = ⋃_{i=1}^{k} ϕ_{j}^{−1}(0, 1). Assume N is large enough so that

1 ; N ∑_{i=0}^{N−1} 1_{A}(T^{−i}x) < ϵfor all x ∈ X  (5.2)

Define Φ: (X, T) → ℝ^{k} by

x ↦ (ϕ_{1}(x), …, ϕ_{k}(x))

Define the map f_{N}: X → ℝ^{kN} by

f_{N}(x) = (Φ(x), Φ(T x), …, Φ(T^{N−1}x)).

We claim that f_{N}(X) is a subset of a finite number of ϵkN dimensional affine subspaces of ℝ^{kN}.

Indeed, let e_{j}^{i}, i = 1 … N, j = 1 … k be the standard base of ℝ^{kN}. Define for every I = {i_{1}, i_{2}, …, i_{N′}}, N′ < ϵN, and every ξ ∈ {0, 1}^{kN}

C(I, ξ) = span{e_{j}^{i}: i ∈ I, 1 ≤ j ≤ k} + ξ.

Then by (5.2),

f_{N}(X) ⊂ ⋃_{|I|<ϵN, ξ} C(I, ξ)

It is easy to see that f_{N} is α_{0}^{N−1}-compatible. By Proposition 2.4 we see that

D(α_{0}^{N−1}) < ϵkN

and so (X, T) has zero mean dimension.

Note that in general the converse to this theorem may be false. Indeed, consider the dynamical system X = ℝ/ℤ × ℝ/ℤ and T: (x, y) ↦ (x, y + x mod 1). X is finite dimensional, hence mdim X = 0, but no small neighborhood of any point of the form (0, y) has small boundary, since this boundary must contain at least two points fixed by T. There are, however, cases where the converse holds (Lindenstrauss (1998)), and presumably a better definition of the SBP that takes into account the periodic points should be equivalent to mdim X = 0. See Lindenstrauss (1995) for a modification of the SBP that holds for finite dimensional dynamical system.

6. Appendix: Subadditive functions on amenable groups.

Let G be a discrete (countable) amenable group. According to Følner’s characterization of such groups for any finite K ⊂ G and any δ > 0 there is a finite set F that is (K, δ)-invariant in the sense that

|{f ∈ F: Kf ⊂ F }| ≥ (1 − δ)|F |

If ϕ is a nonnegative monotone subadditive function defined on the finite subsets of G that is right translation invariant, i.e.

ϕ(Ag) = ϕ(A) for all g ∈ G, A ⊂ G.

then as sets F become more and more invariant ϕ(A)/|A| converges to a limit. More precisely, there is a finite limit b such that for all ϵ > 0, there is a finite K ⊂ G and δ > 0 such that for any F that is (K, δ) invariant we have

|ϕ(F)/|F | − b| < ϵ

The classical application of this fact is in defining the entropy of a Gindexed stationary process (Ornstein and Weiss (1987)). Under a stronger condition than simple sub-additivity, a general result of the type is proved in Ollagnier (1985). For the record we present here a proof of the general fact.

Theorem 6.1. If G is amenable and ϕ is defined on the finite subsets of G and satisfies:

1. 0 ≤ ϕ(A) < +∞ and ϕ(∅) = 0.

2. ϕ(A) ≤ ϕ(B) for all A ⊂ B.

3. ϕ(Ag) = ϕ(A) for all A ⊂ G and g ∈ G.

4. ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B) if A ∩ B = ∅.

then 1 ; |A|ϕ(A) converges to a limit as the set A becomes more and more invariant.

It is worth pointing out that there is some novelty in the theorem even in the case G = ℤ. A special case of the theorem for G = ℤ is a well known calculus exercise — proving that for any subadditive sequence 0 ≤ a_{n}, (i.e. one satisfying: a_{n+m} ≤ a_{n} + a_{m}) the limit lim a_{n} ; n exists.

Since the proof of the theorem is based on the proof of this exercise let us briefly recall it. Set

z = inf_{n→∞} a_{n} ; n .

If ϵ > 0 is given and N chosen so that a_{N} ; N ≤ z + ϵ. Then for all n > N, writing n = qN + r with r < N we see that

a_{n} ≤ a_{qN} + a_{r} ≤ qa_{N} + a_{r}

which upon division by n gives

a_{n} ; n ≤ qN ; n · a_{N} ; N + a_{r} ; n ≤ qN ; n (z + ϵ) + a_{r} ; n

so that lim a_{n} ; n ≤ z + ϵ which shows that lim a_{n} ; n equals z.

Such a sequence corresponds to a translation invariant subadditive function defined not on all finite subsets of ℤ but only on intervals. The key property of intervals used in the above proof is that any interval tiles, via disjoint translations, any large interval, up to a small error. For general finite sets in ℤ this is no longer true, so that taking an infimum over all finite sets does not appear to work. What is needed is a substitute for this tiling property, and we shall use the machinery of quasi-tiling by approximately invariant sets, as developed in Ornstein and Weiss (1987). Here are the notions that are necessary to formulate the results that we need:

Definition 6.2. A collection of finite sets {E_{j}} is said to be ϵ-disjoint if there are subsets Ê_{j} ⊂ E_{j} satisfying:

• Ê_{j} ∩ Ê_{i} = ∅ for all j ≠ i,

• |Ê_{j}|/|E_{j}| ≥ 1 − ϵ for all j.

Definition 6.3. A collection of sets {T_{i}: 1 ≤ i ≤ M} in a group G is said to ϵ-quasi tile the group if there is a finite K ⊂ G, η > 0, and translates {T_{i}c_{ij}}_{i,j} of the T_{i} that are ϵ-disjoint, such that for any F that is (K, η)-invariant one has

|F ∩ ⋃_{ij} T_{i}c_{ij}|/|F | ≥ 1 − ϵ.

The following theorem is in Ornstein and Weiss (1987), I.2:

Theorem 6.4. For any amenable group G, given ϵ > 0, there is an M that depends only on ϵ, such that if

e ∈ T_{1} ⊂ T_{2} ⊂ ⋯ ⊂ T_{M} ⊂ G

satisfy that T_{i+1} is (T_{i}T_{i}^{−1}, η_{i})-invariant, i = 1, 2, … M − 1 (η_{i} depends on i and T_{i}) then the {T_{i}: 1 ≤ i ≤ M} ϵ-quasi tile G.

The formal meaning of the fact that η_{i} depends on i and T_{i} is that there is a function Ψ from ℕ × {finite subsets of G} to ℝ^{+}, such that if η_{i} ≤ Ψ(i, T_{i}) for i = 1,2, …,M − 1 then the conclusion of the theorem is valid.

Let K_{n} be an increasing sequence of finite sets whose union is G and set:

F_{n} = {F ⊂ G: F is finite and (K_{n}, 1/n) invariant}

(note that F_{1} ⊃ F_{2} ⊃ …). If ϕ(A) is a translation invariant subadditive function defined on the finite subsets of G, set

a_{n} = inf_{F ∈F_{n}} ϕ(F)/|F |

and put a_{0} = lim_{n→∞} a_{n}. We shall show that for any δ > 0, there is some n(δ) such that

|ϕ(F)/|F | − a_{0}| < δ  for all F ∈ F_{n(δ)}  (6.1)

To see this, fix some ϵ > 0 (which will be chosen later to be small, in a fashion that depends only on δ), and then let N be large enough for the conclusion of Theorem 6.4 to hold for this ϵ.

Inductively, find sets {F_{1}, …, F_{N}} that both quasi tile and also satisfy

ϕ(F_{i})/F_{i} ≤ a_{0} + δ/2  1 ≤ i ≤ N.  (6.2)

Let n(δ) be large enough so that any F ∈ F_{n(δ)} is itself ϵ-quasi tiled by {F_{1}, …, F_{N}}. We can now bound ϕ(F) by using (6.2) and the fact that {F_{i}} ϵ-quasi tile:

ϕ(F)/F ≤ (a_{0} + δ/2)(1 + ϵ) + ϵϕ({e}),

and so if ϵ is chosen to be small enough we get (6.1), thus proving Theorem 6.1.

We note that if ϕ satisfies a stronger subadditive inequality — namely

ϕ(A ∪ B) + ϕ(A ∩ B) ≤ ϕ(A) + ϕ(B)

for all A, B disjoint or not, then an easier proof can be given which does not require the ϵ-quasi tiling machinery. Such a result can be found in Ollagnier (1985), section 2.2. The measure theoretic entropy of the span of a partition under a measure preserving action satisfy this stronger inequality, as a consequence of the properties of conditional entropy, for which there is no clear analogue in the context of mean dimension.

References

[1] J. Auslander, Minimal Flows and Their Extensions, Chapter 13, pp 183–194, Amsterdam, North-Holland (1988).

[2] S. Glasner and D. Maon, An inverted tower of almost 1-1 extensions, J. D’Analyse Math., 44 (1984/85), pp 67–75.

[3] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press (1941).

[4] A. Jaworski, University of Maryland Ph.D. thesis, (1974).

[5] S. Kakutani, A proof of Bebutov’s theorem, J. of Differential Eq. 4 (1968), pp 194–201.

[6] E. Lindenstrauss, Lowering topological entropy, J. d’Analyse Math. 67 (1995), pp 231–267.

[7] E. Lindenstrauss, Mean dimension, small entropy factors and an imbedding theorem, preprint.

[8] J. Ollagnier, Ergodic Theory and Statistical Mechanics, SLN #1115, Springer-Verlag (1985).

[9] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d’Analyse Math 48(1987), pp 1– 141.

[10] L. Pontryagin and L. Schnirelmann, Sur une propriété métrique de la dimension, Ann. Math., II, Ser. 33 (1932), pp 152 – 162.

[11] M. Shub and B. Weiss, Can one always lower topological entropy?, Ergod. Th. & Dynam. Sys.11 (1991), pp 535–546.

Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

   E-mail address: elon@math.huji.ac.il, weiss@math.huji.ac.il