Lowering Topological Entropy

Elon Lindenstrauss Hebrew University Jerusalem, Israel

                           elon@math.huji.ac.il

Abstract The main result we prove in this paper is that for any finite dimensional dynamical system (with topological entropy h), and for any factor with strictly lower entropy h′, there exist an intermediate factor of entropy h′′ for every h′′ ∈ [h′, h]. Two examples, one of them minimal, show that this is not the case for infinite dimensional systems.

§1. Introduction

The problem considered here was suggested in the paper [1]. It can be stated as follows:

Problem (1.1) Given a dynamical system (X, T) with positive topological entropy, what can be said about the range of values of h_{top}(Y) as Y ranges over the factors of (X, T); in particular, when does such a dynamical system have factors with arbitrarily small entropy?

In what follows, by a dynamical system (X, T) we mean a compact metric space X, and a continuous bijection T: X → X. A factor of (X, T) is a system (Y, S) with a continuous surjection π: X → Y such that S ∘ π = π ∘ T. This surjection is called the factor transformation.

A natural dynamical system in this context is the system [0, 1]^{Z} (the infinite dimensional cube with the usual Tychonoff topology) together with the shift σ. The Hilbert cube is universal for compact metric spaces since every such space can be imbedded in it. Dynamically it is universal in the following sense:

Proposition (1.2) Any dynamical system (X,T) has a non-trivial factor imbedded in the (bi-)infinite dimensional cube.

Proof. Let f: X → [0, 1] be any continuous function. Define f̂: X → [0, 1]^{Z} by

(1.2.1)  f̂: x ↦ ( …, f ∘ T^{−1}(x), f(x), f ∘ T (x), …)

Set Y = f̂(X) ⊂ [0, 1]^{Z} and S = σ|_{Y}. It is clear that (Y, S) is a factor, with f̂ the factor map, since f̂ is continuous and f̂∘ T = σ ∘ f̂. ⊔ ⊓

In the next section we show that this system has no finite entropy factors.

The system [0, 1]^{Z} has infinite dimension. In §3 and §4 we prove that if X has finite dimension then for any factor Y′ of X with h_{top}(Y′) < h_{top}(X) we can find, for any h ∈ (h_{top}(Y′), h_{top}(X)), a factor Y of X compatible with Y′ with h_{top}(Y) = h. By compatible we mean that if ϕ_{Y,X}: X → Y and ϕ_{Y′,X}: X → Y′ are the factor transformations from X to Y and Y′ respectively then we can find a factor transformation ϕ_{Y′,Y}: Y → Y′ such that

ϕ_{Y′,Y} ∘ ϕ_{Y,X} = ϕ_{Y′,X}.

The proof of this result will be divided into two parts. In §3 we use dimension theory to prove some facts about any finite dimensional dynamical system. In §4 we show that a consequence of these facts is that we can construct the intermediate factors described above.

This theorem has a few interesting corollaries. First note that if we take Y′ to be the trivial factor consisting of a single point then the above theorem gives a complete answer to Problem (1.1) for finite dimensional systems:

For any finite dimensional system (X, T) the range of values of h_{top}(Y) as Y ranges over the factors of (X, T) is [0, h_{top}(X)].

Thus in particular there are no finite dimensional prime systems (i.e. dynamical systems with no non-trivial factors) with positive entropy.

Also note that in particular the theorem gives a connection between finite dimension and finite topological entropy: while it is very easy to construct finite dimensional dynamical system with infinite entropy, the theorem ensures us that any finite dimensional dynamical system has a factor with finite entropy.

In section §5 we will give a construction of an infinite dimensional minimal system with no finite entropy factors. An interesting open problem that remains in this context is whether there exists an (infinite dimensional) finite entropy system with (say) a positive lower bound on the topological entropy of its factors. We believe (at least if we require that the system be minimal) that no such system exists. Note that for the finite dimensional case we did not need any condition on the entropy.

After the preparation of the initial version of this manuscript, S. Glasner has shown that the existence of a minimal system with no finite entropy factors gives a positive answer to a question posed by H. Furstenberg ([3]) whether there exist a pair of minimal dynamical systems which have a common almost one-to-one extension but no common factors. With his permission we present this result in the last section.

This paper essentially contains the author’s M.Sc. thesis, conducted under the guidance of Professor Benjamin Weiss from the Hebrew University. The author is grateful to B. Weiss for introducing him to this problem, for many enlightening discussions about this and related topics, and for helping him bring this paper into what is hoped to be a readable form.

§2. Factors of the system [0, 1]^{Z}

For the one sided shift, a proof, due to J. Ashley, that the system [0, 1]^{N} has no finite entropy factors, is given in [1]. However, new ideas must be used to prove the same for the two sided shift, and in particular one must use a classical topological result: the Brouwer fixed point theorem. Let H =^{def}[0, 1]^{Z}.

Theorem (2.1) Let (Y, T) be a non trivial factor of ([0, 1]^{Z}, σ). Then there exists an N such that ([0, 1]^{Z}, σ) is a factor of (Y, T^{N}).

The following lemma is a simple consequence of the Brouwer fixed point theorem:

Lemma (2.2) Let F = (f_{1}, …, f_{n}): [0, 1]^{n} → [0, 1]^{n} be a continuous map such that for every 1 ≤ i ≤ n, f_{i}(x_{0}, …, x_{i−1}, 1, x_{i+1}, …, x_{n}) > 9 10 and f_{i}(x_{0}, …, x_{i−1}, 0, x_{i+1}, …, x_{n}) < 1 10. Then

(2.2.1)  [ 1 4, 3 4]^{n} ⊂ F ([0, 1]^{n}).

Proof. Suppose x_{0} ∈ [1/4, 3/4]^{n} but not in the image of F. For any x ∈ [0, 1]^{n} define G(x) as the point where the ray from F (x) to x_{0} intersects the boundary of [0, 1]^{n}. This point is well defined since x_{0} is not in the image of F, and clearly G so defined is continuous. But the conditions on F guarantees that any point on the boundary of the cube [0, 1]^{n} is not mapped to itself, and since the image of G is contained in the boundary of [0, 1]^{n} we see that G is a continuous function G: [0, 1]^{n} → [0, 1]^{n} with no fixed points, in violation of the Brouwer fixed point theorem. ⊔ ⊓

Proof of the Theorem.

Without loss of generality one can assume that the factor is of the form (1.2.1). That is, the factor is of the form (Y, σ|_{Y}) where Y ⊂ [0, 1]^{Z} and the factor mapping ϕ̂ is generated from the continuous function ϕ: [0, 1]^{Z} → [0, 1] by

ϕ̂: x ↦ ( …, ϕ ∘ T^{−1}(x), ϕ(x), ϕ ∘ T (x), …).

We can normalize ϕ so that 0 and 1 will both be in its image (Since [0, 1]^{Z} is connected this means that ϕ is onto the interval [0, 1]).

From the (uniform) continuity of ϕ it follows that there is an N_{0} such that for any two points in the infinite dimensional cube, x and y,

(2.1.1)  x_{k} = y_{k} for every −N_{0} ≤ k ≤ N_{0} ⟹ |ϕ(x) − ϕ(y)| < 1 10
Set N = 2N_{0} + 1, and let î: ([0, 1]^{Z}, σ^{N}) → ([0, 1]^{Z}, σ) be the map
î( …, x_{−1}, x_{0}, x_{1}, …) ↦ ( …, x_{−N}, x_{0}, x_{N}, …).

Since we assume (Y, T) to be imbedded in ([0, 1]^{Z}, σ) this map defines a map from (Y, T^{N}) to the infinite dimensional cube, and we will prove the image of Y contains the smaller infinite dimensional cube H′ =^{def} [ 1 4, 3 4]^{Z}. Since it is easy to find a factor transformation from [0, 1]^{Z} to itself such that the image of the smaller cube is [0, 1]^{Z} this proves the theorem.

We want to see that î(Y) ⊃ H′, that is that î ∘ ϕ̂(H) ⊃ H′. Since H is compact, so is î ∘ ϕ̂(H). If we will show that for every point h ∈ H′ and every n there is a point h′ ∈ î∘ ϕ̂(H) that agrees with h in the coordinates −n,…,n it will follow that h itself is in î∘ ϕ̂(H). So we need only look, for every n, at the projection π of î∘ ϕ̂(H) on that range of coordinates, and verify that they contain the projection of H′. We now have a function

g=^{def}π ∘ î∘ ϕ̂: H → [0, 1]^{2n+1}

We will define a function ψ: [0, 1]^{2n+1} → H such that g ∘ ψ: [0, 1]^{2n+1} → [0, 1]^{2n+1} will satisfy the conditions of lemma (2.2). Let p^{0},p^{1} ∈ [0, 1]^{Z} such that ϕ(p^{0}) = 0 and ϕ(p^{1}) = 1. Set

p_{k}(t) = (1 − t) · p_{k}^{0} + t · p_{k}^{1} for − ∞ ≤ k ≤ ∞,

and define ψ as follows

(2.1.2)  ψ(x_{−n}, …, x_{−1}, x_{0},x_{1}, … x_{n})=^{def} ; ( … 0, 0, 0,p_{−N_{0}}(x_{−n}), p_{−N_{0}+1}(x_{−n}), … p_{N_{0}}(x_{−n}), ; p_{−N_{0}}(x_{−n+1}), p_{−N_{0}+1}(x_{−n+1}), … p_{N_{0}}(x_{−n+1}), ; ⋮ ; p_{−N_{0}}(x_{0}), p_{−N_{0}+1}(x_{0}), … p_{0}(x_{0}), … p_{N_{0}}(x_{0}), ; ⋮ ; p_{−N_{0}}(x_{n}), p_{−N_{0}+1}(x_{n}), … p_{N_{0}}(x_{n}), 0, 0, 0, …)

(the 0^{th} coordinate is p_{0}(x_{0})).

Suppose x_{0} = 0. Then ψ( …, x_{−1}, 0, x_{1}, …) is identical to p^{0} in the coordinates −N_{0},…, N_{0}. Using (2.1.1) we see that

|ϕ ∘ ψ( …, x_{−1}, 0, x_{1}, …) − ϕ(p^{0})| < 1 10,

that is, ϕ ∘ ψ( …, x_{−1}, 0, x_{1}, …) < 1 10. In this way we see that if x_{k} = 0 then the k^{th} coordinate of g ∘ ψ is smaller than 1 10, and if x_{k} = 1 then the k^{th} coordinate is bigger than 9 10 — i.e. g ∘ ψ satisfies the conditions of lemma (2.2). Thus the image of this map contains the small cube [ 1 4, 3 4]^{2n+1}. ⊔ ⊓

Conclusion (2.2) Any factor of [0, 1]^{Z} has infinite entropy.

§3. The Existence of Open Sets with Small Boundary

If (X, T) is a dynamical system, we will denote by per(X) the set of its periodic points, and by per_{k}(X) the set of its periodic points with period ≤ k. We shall denote the metric on X by d(· , ·); and to simplify notations we shall (without loss of generality) assume that for every x, y ∈ X, d(x, y) < 1. Also, in this and the next section every measure is assumed to be a normalized Borel measure. We will denote the set of all T -invariant measures of (X, T) by M_{T}.

We wish to show that every open subset of X can be approximated (in a rather strong sense) by an open set such that the set of the non-periodic points on its boundary will be small, in the sense introduced in [1]:

Definition (3.1) A set E ⊂ X will be called (N, n)-disjoint if the intersection of any n sets from the collection

E, T (E), T^{2}(E), T^{3}(E), …, T^{N−1}(E)

is empty.

Definition (3.2) A set E ⊂ X will be called T -small (or simply small if T is understood) if for every ϵ > 0 there is an N such that E is (N, ⌊ϵN⌋)-disjoint.

Note that if a (measurable) set E is (N, n)-disjoint then no point of X can be covered by more than n sets from the collection {T^{i}(E)}_{i=0}^{N−1} and so for any μ ∈ M_{T},

μ(E) = 1 ; N ∑ μ(T^{i}(E)) ≤ n ; N μ(X) = n ; N .

In particular, if a set E is small then for any such measure μ, the measure of E is zero. For closed sets the converse also holds, but we will not use this fact. For details see [1]. Clearly, a subset of a small set is also small; it is also very easy to see that the union of two small sets is small.

In this section we will prove the following theorem:

Theorem (3.3) Let (X, T) be a finite dimensional dynamical system. Then for every open set U ⊂ X, and any open V ⊃ ∂U \ per(X), there is an open U′, U ⊂ U′ ⊂ U ∪ V, such that ∂U′ is the union of a small set and a subset of per(X).

We will see in the next section that the fact that any open set can be approximated by an open set with its boundary satisfying the conditions above is sufficient to ensure the existence of factors of the system (X, T) with arbitrary entropy.

We will prove this theorem using ideas from [2]. The following definition is also taken from [2]:

Definition (3.4) A collection B of subsets of an n-dimensional space X will be said to be in general position if for every finite sub-collection S ⊂ B with m elements, dim ∩S ≤ max(−1, n − m).

(Recall that by definition dim M = −1 iff M = ∅). The motivation for this definition is that given a collection of n − 1 dimensional subsets of an n dimensional space then generically any two will have intersection with dimension ≤ n − 2, etc. In the rest of this section, we will take n to be dim(X). Notice that if the collection {T^{k}E}_{k=0}^{r−1} is in general position then in particular, since the intersection of any n + 1 sets from this collection is empty, E is (r, n + 1)-disjoint.

We will use the following standard results in dimension theory valid for every separable metric space M. For proofs of these results see [6].

D1 (“The Subspace Theorem”) If A ⊂ B then dim A ≤ dim B. D2 (“The Countable Closed Sum Theorem”) Let {B_{i}} be a countable collection of closed

sets, with dim B_{i} ≤ k for every i. Then dim ⋃ B_{i} ≤ k.

D3 Let E be a zero dimensional subset of the space M. Then for every x ∈ M and every

open neighbourhood U of x there is a U′ ⊂ U with x ∈ U′ such that ∂U′ ∩ E = ∅. D4 If dim M > −1 there is a zero dimensional subset E of M which is the union of a

countable number of closed sets in M and such that dim M \ E = dim M − 1.

and also the following result which is a consequence of D1 and D2:

D5 Any finite or countable union of F_{σ} k-dimensional sets is an F_{σ} k-dimensional set.

To see D5 is true note that each F_{σ} set is a countable union of closed sets that by D1 have dimension ≤ k, and so the union of the original F_{σ} sets is a countable union of closed at most k-dimensional sets. By D2 this union has dimension ≤ k.

Let P_{k} = per_{k}(X), and P = per(X).

The important step towards proving the main result of this section is the following lemma, which is a modification of a lemma proved by J. Kulesza in [2]:

Lemma (3.5) Let (X, T) be an n dimensional dynamical system, and let U ⊂ X be an open set. Then for every k and every open set V ⊃ ∂U \ P, there exists an open set U′, U ⊂ U′ ⊂ U ∪ V with ∂U′ ⊂ ∂U ∪ V such that

{∂U′ \ P, T (∂U′) \ P, …, T^{k−1} (∂U′) \ P }

is in general position. (Notice that for any U′, if ∂U′ contains a periodic point with a short period the more natural collection {∂U′, …, T^{k−1} (∂U′)} will not be in general position.)

To prove lemma (3.5) we will need the following simple result:

Lemma (3.6) Let U be an open set, and let E ⊂ X be zero dimensional. Then for any open V ⊃ cl(U), there exists an open set U′ ⊃ cl(U), with U′ ⊂ V and such that ∂U′ ∩ E = ∅.

Proof.

By D3 for every x ∈ ∂U there is neighbourhood U_{x} inside V whose boundary does not intersect E. By compactness of ∂U a finite number of these, say U_{x_{1}},…, U_{x_{r}} suffice for covering ∂U. Set U′ = U ∪ ⋃_{i=1}^{r} U_{x_{r}}.

Clearly U′ ⊂ V and cl(U) ⊂ U′. Also, since the union is finite, every point x in ∂U′ is in ∂U or in some ∂U_{x_{i}}. The first possibility is impossible since the U_{x_{j}}’s cover ∂U and so every point of ∂U is an internal point of U′. Thus x is in ∂U_{x_{i}} for some i, and since this set has empty intersection with E, x ∉ E. Thus ∂U′ ∩ E = ∅. ⊔ ⊓

Proof of lemma (3.5).

We will prove the lemma by induction on k. If k = 0 we can simply take for U′ the set U itself.

Now, suppose that the lemma is true for some k ≥ 0. We will prove it is true for k +1. Without loss of generality we can assume

V ∩ P_{k+1} = ∅

for otherwise we can take V′ = V \ P_{k+1} that also satisfies the conditions of this lemma and if the lemma holds for V′ it also holds for V. Define further for any integer r,

V_{r} = {x : d(x, X \ V) ≥ 1 ; r};

clearly each V_{r} is compact and V = ⋃_{r=0}^{∞} V_{r}. Using the induction hypothesis we know that there exists an open A_{0} with U ⊂ A_{0} ⊂ U ∪ V such that ∂A_{0} ⊂ ∂U ∪ V and such that

{∂A_{0} \ P, T (∂A_{0}) \ P, …, T^{k−1} (∂A_{0}) \ P }

is in general position.

For every x ∈ ∂A_{0} ∩ V, there exists an η_{x} > 0 such that cl B(x, η_{x}) ⊂ V and such that the sets B(x, η_{x}), T B(x, η_{x}), …,T^{k}B(x, η_{x}) are disjoint (since x ∉ P_{k+1}). Let B_{x} =^{def}B(x, η_{x}/2) and B̂_{x} =^{def}B(x, η_{x}). The B_{x}’s form a cover of ∂A_{0} ∩ V, and we can find a countable sub-cover B_{i} such that

(3.7.1)  lim_{i→∞} diam(B_{i}) = 0,

and such that for any V_{l} there exists an r_{l} such that

V_{l} ∩ ⋂_{i=r_{l}}^{∞} B̂_{i} = ∅.

Indeed, for every integer r define the compact set C_{r} as

C_{r} =^{def} {x ∈ ∂A_{0} : 1 ; r ≥ d(x, X \ V) ≥ 1 ; r+1} .

We can construct a sub-cover with the required properties by taking for every r a finite number of sets of the form B_{x} with x ∈ C_{r} that cover C_{r}. Since ∂A_{0} ∩ V = ⋃_{r=1}^{∞} C_{r} we have indeed a subcover of ∂A_{0} ∩ V. Furthermore, for any x ∈ C_{r} the diameter of B_{x} is ≤ 1 ; r, and so (3.7.1) holds. Since d(V_{l}, C_{r}) > 1 ; r for r large enough, we see that the second condition imposed on the above sub-cover also holds, and so the existence of a sub-cover B_{i} with the required properties is established.

We will construct recursively, starting with A_{0}, a sequence of open sets {A_{i}}_{i=0}^{∞} such that

P1 A_{i} ⊂ A_{0} ∪ ⋃_{j=0}^{∞} B_{j} P2 A_{i} ⊃ A_{i−1} P3 ∂A_{i} ⊂ ∂U ∪ V P4 A_{i} \ B̂_{i} = A_{i−1} \ B̂_{i}

P5 A_{i} =^{def} {∂A_{i} \ P, T (∂A_{i}) \ P, …, T^{k−1} (∂A_{i}) \ P, T^{k}(∂A_{i} ∩ ⋃_{j=1}^{i} B_{j})} is in general

position.

(it is easy to see that A_{0} satisfies all these conditions for i = 0).

Given this sequence we will define U′ as ⋃_{i=0}^{∞} A_{i}. By P1, and since ⋃_{j=0}^{∞} B_{j} ⊂ V we see that

U′ ⊂ A_{0} ∪ V ⊂ (U ∪ V) ∪ V = U ∪ V.

By P2 we see that U ⊂ U′. We now claim that ∂U′ ⊂ ∂U ∪V. Indeed, let x be a boundary point of U′. Then either it is in ∂A_{i} for some i or it the limit of a sequence (x_{l})_{l=1}^{∞} with x_{l} ∈ A_{n_{l}} \A_{n_{l}−1} with n_{l} strictly increasing. Consequently, since by P4 A_{n_{l}} \A_{n_{l}−1} ⊂ B̂_{n_{l}},

x ∈ ⋂_{l=0}^{∞} cl(⋃_{l=r}^{∞} B̂_{n_{l}}) ⊂ ∂A_{0}

where the last inequality follows from the fact that the centers of the B̂_{i}’s are in ∂A_{0} and their radii tend to zero. Thus we see that

∂U′ ⊂ ∂A_{0} ∪ ⋃_{i=0}^{∞} ∂A_{i} ⊂ ∂U ∪ V.

where the second inclusion is a consequence of P3.

It remains to be verified that

(3.7.2)  {∂U′ \ P, T (∂U′) \ P, …, T^{k−1} (∂U′) \ P, T^{k} (∂U′) \ P }

is in general position. First note that (for a given l) ∂U′ ∩ V_{l} = ∂A_{r} ∩ V_{l} if r is large enough. For according to P4, A_{r} \ ⋃_{i=r+1}^{∞} B̂_{i} = A_{r′} \ ⋃_{i=r+1}^{∞} B̂_{i} for any r′ > r, and so U′ \ ⋃_{i=r+1}^{∞} B̂_{i} = A_{r} \ ⋃_{i=r+1}^{∞} B̂_{i}. According to the way the B_{i}’s were chosen, there is an r such that for every i > r, B̂_{i} ∩ V_{l+1} = ∅, and so U′ ∩ V_{l+1} = A_{r} ∩ V_{l+1}. Since V_{l} ⊂ int(V_{l+1}) we see that indeed (∂U′) ∩ V_{l} = (∂A_{r}) ∩ V_{l}. We can take r large enough so that V_{l} ⊂ ⋃_{i=1}^{r} B_{i}. Now we know that

{∂A_{r} \ P, T (∂A_{r}) \ P, …, T^{k−1} (∂A_{r}) \ P, T^{k}(∂A_{r} ∩ ⋃_{j=1}^{r} B_{j})}

is in general positions and the same is true if we replace each set of the above collection by a smaller set, and so

{∂A_{r} ∩ V_{l} \ P, T (∂A_{r} ∩ V_{l}) \ P, …, T^{k−1} (∂A_{r} ∩ V_{l}) \ P, T^{k} (∂A_{r} ∩ V_{l}) \ P }

= {∂U′ ∩ V_{l} \ P, T (∂U′ ∩ V_{l}) \ P, …, T^{k−1} (∂U′ ∩ V_{l}) \ P, T^{k} (∂U′ ∩ V_{l}) \ P }

is in general position. Since this is true for every l, and since T^{i} (∂U′ ∩ V_{l}) \ P is closed in X \ P then an immediate consequence of the Countable Closed Sum Theorem (D2) is that the collection

{∂U′ ∩ V \ P, T (∂U′ ∩ V) \ P, …, T^{k−1} (∂U′ ∩ V) \ P, T^{k} (∂U′ ∩ V) \ P }

is in general position. We have already seen that ∂U′ ⊂ ∂U ∪ V; since ∂U \ P ⊂ V we see that ∂U′ ⊂ P ∪ V and so the collection above is the same as the collection (3.7.2). Thus it only remains to be seen how to construct A_{i+1} using A_{i}.

For every finite sub-collection S ⊂ A_{i} with dim (∩S) > −1 there is according to D4 a zero-dimensional set E_{S} ⊂ ∩S, F_{σ} in ∩S, such that dim (∩S \ E_{S}) = dim (∩S) − 1 (for S = ∅, ∩S is undefined; for convenience we set it to be X). E_{S} is in fact σ-compact, for ∩S is σ-compact and an F_{σ} subset of a σ-compact space is σ-compact. Set

E = ⋃_{{S⊂A_{i} \atop 0≤j≤k}} T^{−j}E_{S}.

Since T is a homeomorphism the image of a zero dimensional σ-compact set is zero dimensional and σ-compact. Thus E is a finite union of σ-compact zero dimensional sets, and so according to D5, E itself is both σ-compact and zero dimensional. Using lemma (3.6), there is an open set W ⊃ cl (B_{i+1} ∩ A_{i}) such that cl(W) ⊂ B̂_{i+1} ∩ (U ∪ ⋃_{j=0}^{∞} B_{j}) ∩ V, with ∂W ∩ E = ∅. We claim that A_{i+1} =^{def}A_{i} ∪ W satisfies P1 – P5. That A_{i+1} satisfies P1 – P4 is trivial, and thus it only remains to be shown that P5 is satisfied.

Suppose that P5 is not satisfied, i.e. there exists a collection S ⊂ A_{i+1} of cardinality m such that dim ∩S > max(−1, n−m). Let S = {S_{1}, S_{2}, …, S_{m}}, where each S_{l} is a distinct element of A_{i+1}, i.e. either T^{j_{l}} (∂A_{i+1}) for some j_{l} < k or else T^{k} (∂A_{i+1} ∩ ⋃_{j=1}^{i+1} B_{j}) (in which case we set j_{l} = k). Notice that since the S_{l}’s are distinct, j_{l} ≠ j_{l′} for l ≠ l′.

Now ∂A_{i+1} ⊂ (∂A_{i} \ W) ∪ ∂W and so for any l such that j_{l} < k,

S_{l} ⊂ T^{j_{l}} (∂A_{i}) ∪ T^{j_{l}} (∂W) .

A similar result holds for j_{l} = k:

S_{l} = T^{r}(∂A_{i+1} ∩ ⋃_{j=1}^{i+1} B_{j})

⊂ T^{r}((∂A_{i} \ W) ∩ ⋃_{j=1}^{i+1} B_{j}) ∪ T^{r} (∂W)

= T^{r}((∂A_{i} \ W) ∩ ⋃_{j=1}^{i} B_{j}) ∪ T^{r} (∂W)

⊂ T^{r}(∂A_{i} ∩ ⋃_{j=1}^{i} B_{j}) ∪ T^{r} (∂W) ,

where the second equality holds since cl (B_{i+1} ∩ A_{i}) ⊂ W. Combining the two cases, we see that each S_{l} is a subset of the union of two sets: S_{l}^{0}, the element of A_{i} corresponding to S_{l}, and S_{l}^{1} =^{def}T^{j_{l}} (∂W). Thus ∩S is a subset of the finite union

∩S ⊂ ⋃_{α∈{0,1}^{m}} (⋂_{l=1}^{m} S_{l}^{α_{l}})

and since dim (∩S) > max(−1, n − m), at least one of these intersections must have dimension larger than max(−1, n − m). But since cl(W) ⊂ B̂_{i+1} and the T^{j}B̂_{i+1}’s for j = 0,…,k are pairwise disjoint, any α with more than two α_{l}’s 1 has ⋂_{l=1}^{m} S_{l}^{α_{l}} = ∅. For α = (0, … 0), each S_{l}^{α_{l}} is a (distinct) element of A_{i}, and since this collection is in general position we see that dim ⋂_{l=1}^{m} S_{l}^{0} ≤ max(−1, n − m). We haven’t assumed anything on the order of the S_{l}’s yet, so without loss of generality we can assume that

dim(S_{1}^{1} ∪ ⋃_{l=2}^{m} S_{l}^{0}) > max(−1, n − m).

But this too is impossible — to see this, let Ŝ = {S_{2}^{0}, …, S_{m}^{0}}. Now Ŝ ⊂ A_{i} so we know that

dim (∩Ŝ \ EŜ) ≤ dim (∩Ŝ) − 1 ≤ min(−1, n − m).

By construction, ∂W is disjoint from E and thus also from T^{−j_{1}}EŜ ⊂ E, so that S_{1}^{1} = T^{j_{1}} (∂W) ⊂ X \ EŜ. Thus

S_{1}^{1} ∩ ⋂_{l=2}^{m} S_{l}^{0} ⊂ ∩Ŝ \ EŜ,

a set of dimension ≤ min(−1, n − m) — a contradiction.

⊔ ⊓

Lemma (3.8) Suppose that A is closed and that A \ P is (k, m)-disjoint. Then there is an open set V ⊃ A \ P such that V is also (k, m)-disjoint.

Proof.

Set A′ =^{def}A \ P.

Take any m sets from the collection {T^{i}(A′)}_{i=0}^{k−1}, say A_{1}, …, A_{m}, where for every i,

A_{i} = T^{n_{i}}(A′) = T^{n_{i}}(A) \ P.

Note that cl(A_{i}) ⊂ T^{n_{i}}(A) ⊂ A_{i} ∪ P, and so

(3.8.1)  A_{1} ∩ ⋂_{i=2}^{m} cl(A_{i}) = A_{1} ∩ ⋂_{i=2}^{m}(A_{i} ∪ P) ⊂ ⋂_{i=1}^{m} A_{i} ∩ (A_{1} ∩ P) = ∅,

where the last equality is true since A′ = A \ P is (k, m)-disjoint, and since A_{1} ∩ P = ∅. Define for every 1 ≤ j ≤ k the open set

V_{j} = \{x : d(x, T^{j}(A′)) < min_{{S⊂\{1,...,k\} \atop |S|=m−1, j∉S}} max_{r∈S} d(x, T^{r}(A′))\}.

From (3.8.1) we see that for every S ⊂ {1, …, k} with |S| = m − 1 and j ∉ S, and every x ∈ T^{j}(A′), max_{r∈S} d(x, T^{r}(A′)) > 0 since x ∉ ⋂_{r∈S} cl T^{r}(A). It follows that every x ∈ T^{j}(A′) is in V_{j}, and thus for every j, T^{j}(A′) ⊂ V_{j}.

Now take any m sets form the collection {V_{j}}_{j=1}^{k}, say V_{j_{1}},…,V_{j_{m}}, and let S = {j_{2}, …, j_{m}}. Then

⋂_{l=1}^{m} V_{j_{l}} ⊂ {x : d(x, T^{j_{1}}(A′)) < max_{r∈S} d(x, T^{r}(A′))}

∩ ⋂_{r∈S}{x : d(x, T^{r}(A′)) < d(x, T^{j_{1}}(A′)} = ∅.

Define V = ⋂_{j=0}^{k} T^{−j}(V_{j}). Since T^{j}(A′) ⊂ V_{j} for every j we see that A′ = A \ P ⊂ V. And since for every j, T^{j}(V) ⊂ V_{j}, the equation above shows that V is (k, m)-disjoint. ⊔ ⊓

We are now ready to prove theorem (3.3). For the convenience of the reader we will restate it:

Theorem (3.3) Let (X, T) be a finite dimensional dynamical system. Then for every open set U ⊂ X, and any open V ⊃ ∂U \ P, there is an open U′, U ⊂ U′ ⊂ U ∪ V, such that ∂U′ is the union of a small set and a subset of P.

Proof.

We will construct recursively two sequences of open sets {U_{i}}_{i=0}^{∞} and {V_{i}}_{i=0}^{∞}, starting from U_{0} = U and V_{0} = V such that the following conditions hold:

C1 ∂U_{k} \ P ⊂ V_{k}, C2 U_{k+1} ⊂ U_{k} ∪ V_{k}, C3 (cl V_{k+1}) \ P ⊂ V_{k}, C4 V_{k} is (k, n + 1)-disjoint, C5 V_{k+1} ⊂ V_{k} and U_{k+1} ⊃ U_{k}.

Suppose that U_{i}, V_{i} have already been defined. Define U′ = ⋃_{i=0}^{∞} U_{i}. Using C2 and C5 we see that for every r,

U_{k+r} ∪ V_{k} ⊂ U_{k+r−1} ∪ V_{k+r−1} ∪ V_{k} = U_{k+r−1} ∪ V_{k} ⊂ ⋯ ⊂ U_{k} ∪ V_{k},

and thus for every k, U_{k} ⊂ U′ ⊂ U_{k} ∪ V_{k}. Taking k = 0 we see that U ⊂ U′ ⊂ V. Furthermore using this, C1 and C2 we see that for every k

∂U′ \ P ⊂ (cl(U_{k+1} ∪ V_{k+1}) \ U_{k+1}) \ P
⊂ (∂U_{k+1} \ P) ∪ (cl(V_{k+1}) \ P)
= cl(V_{k+1}) \ P ⊂ V_{k}.

Since V_{k} is (k, n + 1)-disjoint the same is true for ∂U′ \ P. Since n is fixed ( = dim(X)) and k arbitrary, we see that ∂U′ \ P is T -small.

Thus it only remains to show how to build such sequences of sets. Clearly U_{0} and V_{0} satisfy C1 and since any set is (0, n + 1)-disjoint also C4. Suppose U_{0}, …, U_{k} and V_{0}, …, V_{k} have already been defined, satisfying the conditions imposed on these sets by C1 – C5.

From C1 we know that ∂U_{k} \ P ⊂ V_{k}, and so using Lemma (3.5) we can find a set U_{k+1} such that U_{k} ⊂ U_{k+1} ⊂ U_{k} ∪ V_{k}, ∂U_{k+1} \ P ⊂ V_{k}, and such that ∂U_{k+1} \ P is (k, n + 1)-disjoint. Using the previous lemma we can find a (k, n + 1)-disjoint open set V′_{k+1} ⊃ ∂U_{k+1} \ P. Define W_{k} ⊂ V_{k} as

W_{k} = {x : d(x, ∂U_{k+1}) < d(x, X \ V_{k})} ,

and set

V_{k+1} = V′_{k+1} ∩ W_{k}.

Clearly U_{k+1} and V_{k+1} satisfy C2, C4 and C5. By definition ∂U_{k+1} \ P ⊂ V′_{k+1}. For every x ∈ ∂U_{k+1} \ P, d(x, U_{k+1}) = 0 but d(x, X \ V_{k}) > 0 (since x ∈ V_{k}). Thus ∂U_{k+1} \ P ⊂ W_{k} and thus C1 holds. To prove C3 holds, it clearly suffice to show that cl(W_{k}) \ P ⊂ V_{k}. So suppose x ∈ cl(W_{k}) \ V_{k}. Then

d(x, ∂U_{k+1}) ≤ d(x, X \ V_{k}) = 0,

and so x ∈ ∂U_{k+1} \ V_{k} ⊂ P. Since this is true for any x ∈ cl(W_{k}) \ V_{k} we see that indeed cl(W_{k}) \ P ⊂ V_{k}. ⊔ ⊓

Note that we have in fact proved that for any k the set ∂U′ \ P is (k, n + 1)-disjoint, which is a much stronger statement than merely saying that ∂U′ \ P is T -small. We will also need the following corollary to Theorem (3.3) and Lemma (3.8):

Corollary (3.10) Suppose E is closed and E \ P is (k, m)-disjoint, with V ⊃ E an open set. Then there exists an open set U′, V ⊃ U′ ⊃ E \ P such that U′ is (k, m)-disjoint and such that ∂U′ \ P is small.

Proof.

According to Lemma (3.8) there is an open, (k, m)-disjoint V′ ⊃ E \ P. Without loss of generality V′ ⊂ V for otherwise we can replace V′ by V′ ∩ V. Define the open set W as

W = {x : d(x, E) < d(x, X \ V′)} .

As in the proof of Theorem (3.3), cl(W) ⊂ V′ ∪ P. Thus we can apply Theorem (3.3)to find an open set U′ with W ⊂ U′ ⊂ V′ and such that ∂U′ \ P is small. Since U′ ⊂ V′, U′ is also (k, m)-disjoint. ⊔ ⊓

§4. The Construction of Factors With Arbitrary Entropy

In this section we will prove that if the conclusion of theorem (3.3) holds* for the dynamical system (X, T), then we can find for every ξ, ξ′ ∈ X and any η > 0 a factor (Y, S) with entropy ≤ η and such that the factor transformation maps ξ and ξ′ into distinct points in Y. From this we will deduce that if (Z, R) is any factor of the system (X, T) with h_{top}(Z) < h_{top}(X), then for any h ∈ (h_{top}(Z), h_{top}(X)), we can find a factor (Y, S)

* And thus its corollary (3.10). of (X, T) with topological entropy h, and such that if ϕ: X → Z and ψ: X → Y are the respective factor transformations then ϕ∘ψ^{−1} is well defined, and is a factor transformation Y → Z.

We first need to see how the information given on the system (X, T) by (3.3) can help to construct factors with small entropy.

For any measurable set A we can construct in a natural way a (not necessarily continuous) map m_{A}: X → {0, 1}^{Z} as follows:

m_{A}: x ↦ (…, 1_{A} ∘ T^{−1}(x), 1_{A}(x), 1_{A} ∘ T (x), …) .

Define the set S_{A} =^{def} cl m_{A}(X) ⊂ {0, 1}^{Z}. This set is shift invariant — and so together with the shift σ it is a dynamical system. We will define for any set A the entropy of A as

h(A)=^{def}h_{top}(S_{A}).

Proposition (4.1) For any two (measurable) sets A and B,

h(A ∪ B) ≤ h(A) + h(B).

Proof. Take any x ∈ X. Note that we can calculate m_{A∪B}(x) from m_{A}(x) and m_{B}(x), since

1_{A∪B}(T^{n}x) = max(1_{A}(T^{n}x), 1_{B}(T^{n}x)).

Consider the (continuous) map

f: (…, a_{−1}, a_{0}, a_{1}, …) × (…, b_{−1}, b_{0}, b_{1}, …) ⟼ (…, max(a_{−1}, b_{−1}), max(a_{0}, b_{0}), max(a_{1}, b_{1}), …)

from S_{A} × S_{B} to {0, 1}^{Z}. We claim that S_{A∪B} ⊂ f(S_{A} × S_{B}). Indeed, for any x ∈ X,

f (m_{A}(x), m_{B}(x)) = m_{A∪B}(x),

and so m_{A∪B}(X) ⊂ f(S_{A} × S_{B}). Since f is continuous, f(S_{A} × S_{B}) is closed (as the image of a compact set) and so contains cl m_{A∪B} = S_{A∪B}.

Thus S_{A∪B} is a subsystem of a factor of S_{A} × S_{B}, and so we have:

h(A ∪ B) = h_{top}(S_{A∪B}) ≤ h_{top} (f(S_{A} × S_{B})) ≤ h_{top}(S_{A} × S_{B}) = h(A) + h(B).

⊔ ⊓

Lemma (4.2) Let E be a closed set. If E \ per(X) is small then for any V ⊃ E, and any ϵ > 0, there is an open set V ⊃ U ⊃ E such that:

a. h(U) < ϵ; b. ∂U \ per(X) is small.

Proof.

Fix some sequence of integers N_{0}, N_{1}, . . .. We will use this sequence to construct an open U, and show that if the integers N_{i} are large enough this set has the required properties.

Applying Corollary (3.10), we can find an open set U_{0}, V ⊃ U_{0} ⊃ E \ P such that ∂U_{0} \ per(X) is small and such that U_{0} is (N, ⌊N/N_{0}⌋)-disjoint for some N and the given N_{0}. Using theorem (3.3), there is a V′, V ⊃ V′ ⊃ E such that ∂V′ \ P is small. For every k, let P^{k} = {x : T^{k}(x) = x}, and set

U_{k} =^{def} ⋂_{i=0}^{N_{k}} T^{ki}(V′).

Clearly U_{k} ⊃ P_{k} ∩ E. Thus {U_{k}}_{k=0}^{∞} form an open cover of the compact set E, and so a finite number of these, say U_{0}, …, U_{r}, suffice to cover E. Define U =^{def} ⋃_{i=0}^{r} U_{i}.

Note that for any k ≥ 1,

∂(U_{k} \ P) ⊂ ⋃_{i=0}^{N_{k}} T^{ki}(∂V′ \ P),

and so, since the union of a finite number of small sets is small, for every k the set ∂U_{k} \ P is small. Thus ∂U \ P, being a subset of ⋃_{i=0}^{r}(∂U_{k} \ P) is small. Also it is clear that E ⊂ U ⊂ V.

It remains to be shown that if the sequence N_{i} increases fast enough, then h(U) < ϵ. Since by the previous proposition, h(U) ≤ ∑_{i=0}^{r} h(U_{i}), this lemma will be proved if we show that if N_{k} is large enough then h(U_{k}) < ϵ2^{−k−1}.

If S ⊂ {0, 1}^{Z} is a shift invariant set, denote by W_{n} (S) the set

W_{n} (S) =^{def} {(x_{1}, x_{2}, …, x_{n}) : (…, x_{−1}, x_{0}, x_{1}, …) ∈ S} .

It is well known (and easy to deduce from the definition of topological entropy) that if S ⊂ {0, 1}^{Z}, the entropy of the system (S, σ) is

h_{top}(S) = lim_{n→∞} 1 ; n log |W_{n} (S) | = inf 1 ; n log |W_{n} (S) |,

where the second equality is true since for any n and k, |W_{kn} (S) | ≤ |W_{n} (S) |^{k}.

If some n-tuple is the first n coordinates of some point in S_{U_{0}}, it is also the first n coordinates of a point of m_{U_{0}}(X) which is dense in S_{U_{0}}. Since U_{0} is (N, ⌊N/N_{0}⌋)-disjoint, for no x ∈ X can m_{U_{0}}(x) have more than N/N_{0} ones in the first N coordinates. Thus

h(U_{0}) ≤ 1 ; N log |W_{N} (S_{U_{0}}) | ≤ 1 ; N log ( ∑_{i=0}^{⌊N/N_{0}⌋} \binom{N}{i})

which can be made arbitrarily small by taking N_{0} to be very large.

We will now consider what n-tuples can appear in W_{n} (S_{U_{k}}) for k > 0. Recall that U_{k} = ⋂_{i=0}^{N_{k}} T^{ki}(V′). Suppose now that (for some x ∈ X) 1_{U_{k}}(x) = 1 but 1_{U_{k}} (T^{−k}(x)) = 0. If this is true then this x must be in T^{ki}(V′) for i = 0, …, N_{k}, but not in T^{k(N_{k}+1)}(V′). Thus we see that this x is not in T^{ki}(U_{k}) (or, equivalently, 1_{U_{k}} (T^{−ki}(x)) = 0) for i = 1, …, N_{k}. Take n = kN_{k}, and consider for any n-tuple

(a_{1}, a_{2}, …, a_{kN_{k}}) ∈ W_{kN_{k}} (S_{U_{k}})

the k N_{k}-tuples

(a_{i}, a_{i+k}, …, a_{i+k(N_{k}−1)}) for i = 1, …, N_{k}.

We know that if for some j, a_{j} = 1 and a_{j−k} = 0 then for any l such that 1 ≤ j − lk < j we know that a_{j−lk} = 0. Similarly, if for some x, 1_{U_{k}}(x) = 1 but 1_{U_{k}}(T^{k}(x)) = 0 then for every 1 ≤ i ≤ N_{k}, 1_{U_{k}}(T^{ki}(x)) = 0. Thus if for some j, a_{j} = 1 but a_{j+k} = 0 then for any l such that j < j + kl ≤ N_{k}, a_{j+lk} = 0.

Thus the sequence (a_{i}, a_{i+k}, …, a_{i+k(N_{k}−1)}) must be of the form

(0, 0, …, 0, 1, 1, … 1, 0, 0, …, 0).

The number of N_{k}-tuples of this form is less than N_{k}^{2}, and so

|W_{kN_{k}} (S_{U_{k}})| < N_{k}^{2k},

Thus

h(U_{k}) ≤ 1 ; kN_{k} log |W_{kN_{k}} (S_{U_{k}})| ≤ 1 ; kN_{k} log(N_{k}^{2k}) = 2 ; N_{k} log(N_{k}),

and so we see that indeed if N_{k} is large enough, h(U_{k}) will be smaller than 2^{−k−1}ϵ. ⊔ ⊓

(4.3) The Construction

Let ξ, ξ′ be distinct points of X, and ϵ > 0. We will now construct a factor imbedded in ([0, 1]^{Z}, σ) and a (continuous) factor map ϕ̂: X → [0, 1]^{Z} such that ϕ̂(ξ) ≠ ϕ̂(ξ′). In the next subsection we will show that the entropy of this factor is less than ϵ.

We will first find a countable collection A of open sets such that

A1 For any U ∈ A and any δ > 0 there is a V ∈ A such that ∂U ⊂ V ⊂ B(∂U, δ). A2 ∑_{U∈A} h(U) < ϵ.

A3 There is a U ∈ A such that ξ ∈ U but ξ′ ∉ cl(U).

This collection A will be the union of a monotone increasing sequence of collections

A_{0} ⊂ A_{1} ⊂ A_{2} ⊂ · · · .

To construct these collections first note that for any x ∈ X, {x} \ per(X) is small, and thus according to Lemma (4.2) there is an open set U such that ξ ∈ U, ξ′ ∉ cl(U), the boundary of U is a union of a small set and a set of periodic points, and h(U) < ϵ/2. Define A_{1} = {U}.

Suppose A_{n} has already been defined. According to Lemma (4.2)we can find for any U ∈ A_{n} a set V_{U,n} such that

a. ∂U ⊂ V_{U,n} ⊂ B(∂U, 1 ; n)

b. ∂V_{U,n} is a union of a small set and a set of periodic points c. h(V_{U,n}) < (2^{n+1} · |A_{n}|)^{−1} ϵ

and define A_{n+1} as

A_{n+1} =^{def} {V_{U,n} : U ∈ U} ∪ A_{n}.

It is easy to see that A=^{def} ⋃_{n=1}^{∞} A_{n} has all the required properties.

Define

A_{T} = {T^{n}U : n ∈ Z and U ∈ A} .

It is easy to see that A_{T} still satisfies properties A1 and A3. Indeed, A_{T} clearly satisfies the following three properties:

B1 For any U ∈ A_{T} and any δ > 0 there is a V ∈ A_{T} such that ∂U ⊂ V ⊂ B(∂U, δ). B2 There is a U ∈ A_{T} such that ξ ∈ U but ξ′ ∉ cl(U). B3 U ∈ A_{T} iff T (U) ∈ A_{T}.

We will now define a (closed) equivalence relation ∼ on X: we will say that x ∼ y iff for every U ∈ A_{T}, 1_{U}(x) = 1_{U}(y). To show ∼ is closed we must show that if x ≁ y there are open neighbourhoods V_{x} ∋ x, V_{y} ∋ y such that for every x′ ∈ V_{x} and any y′ ∈ V_{y}, x′ ≁ y′.

Without loss of generality we can assume that for some U ∈ A_{T}, x ∈ U but y ∉ U. There are two cases:

A If y ∉ ∂U, there is a neigbourhood V_{y} ∋ y such that V_{y} ∩ U = ∅ and we can take as V_{x} the set U. Since all points of V_{y} are not in U, no point of V_{y} can be equivalent to a point of V_{x}.

B If y ∈ ∂U then there is an n such that d(x, ∂U) > 1 ; n. According to the properties of the collection A_{T} there is a set U′ ∈ A_{T} such that ∂U ⊂ U′ ⊂ B(∂U, 1 ; n), and thus y ∈ U′, x ∉ cl(U′), and Case A is applicable.

Using the well know construction of quotient space, X/∼ is a compact metric space, and let π: X → X/∼ denote the projection map from X to Y =^{def}X/∼. Recall that if d(· , ·) is a metric on X, the metric on X/∼ is defined by

ρ(x, y) = δ(π^{−1}x, π^{−1}y),

where δ(· , ·) is the Hausdorff metric on the closed subsets of X, that is

δ(A, A′) = inf{δ > 0 : A ⊂ B(A′, δ) and A′ ⊂ B(A, δ)}. We can define a homeomorphism S: Y → Y by specifying that for any x ∈ X, S ∘ π(x) = π ∘ T (x).

This map is well defined since x ∼ y iff T (x) ∼ T (y), and is easily seen to be continuous and to have a continuous inverse. Thus we have constructed a factor (Y, S) of (X, T).

(4.4) Estimating the Entropy

Recall the following definition of topological entropy: Let (Z, R) be a dynamical system and let d(· ·) be a metric on Z. A set E is said to be n,δ-separated if for every two distinct x and y in E, there is a 0 ≤ k < n such that d(T^{k}x, T^{k}y) > δ. Let S(Z; n, δ) (or S(n, δ) if Z is understood) be the maximal cardinality of an n,δ-separated set. Set

h_{δ}(Z)=^{def} lim sup 1 ; n log S(Z; n, δ),

then the topological entropy of (Z, R) is

(4.4.1)  h_{top}(Z)=^{def} lim_{δ→0} h_{δ}(Z).

We wish to show that the entropy of Y is less then ϵ. We will do so by showing that Y is not only a factor of X but also of some dynamical system Z with h_{top}(Z) < ϵ.

A is denumerable, and so we can present it as

A = {U_{1}, U_{2}, …}.

For any U_{i} we have the dynamical system (S_{U_{i}}, σ) with entropy h(U_{i}). Thus Z′ = ∏_{i=1}^{∞} S_{U_{i}}, together with the transformation σ′ = ∏_{i=1}^{∞} σ is a dynamical system with entropy at most ∑_{i=1}^{∞} h(U_{i}) < ϵ.

We have the following natural (non continuous) map ν from X to Z′ ν: x ↦ (v_{i,j})_{{1≤i<∞ \atop −∞<j<∞}} ,

where v_{i,j} = 1_{T^{j}U_{i}}(x) for every i, j. This map ν induces a map ν′ from Y = X/∼ to Z since if x ∼ y then ν(x) = ν(y). Let Z = cl ν(X) ⊂ Z′ and define the transformation R: Z → Z as R = σ′|_{Z}. Since (Z, R) is a subsystem of (Z′, σ′), h_{top}(Z) < ϵ.

To show that h_{top}(Y) < ϵ we wish to construct a continuous factor map f: Z → Y. For any i, define U_{i}^{0} = X \ U_{i} and U_{i}^{1} = cl U_{i}. Take any α = ν(x) ∈ ν(X). Then

x ∈ ⋂_{1≤i<∞_{−∞<j<∞}} T^{j}(U_{i}^{α_{i,j}}),

and so for any α ∈ Z = cl ν(X),

U_{α,N} =^{def} ⋂_{1≤i≤N_{−N≤j≤N}} T^{j}(U_{i}^{α_{i,j}}) ≠ ∅.

and thus U_{α} =^{def} ⋂_{N=1}^{∞} U_{α,N} ≠ ∅.

Proposition (4.5) diam (π(U_{α,N})) tends to zero uniformly in N.

Proof. Suppose that the proposition is false, that is there exists a δ > 0 and a sequence of α_{n} ∈ Z, an increasing sequence N_{n} and two sequences x_{n}, y_{n} ∈ U_{α_{n},N_{n}} such that

ρ(π(x_{n}), π(y_{n})) > δ.

Since Z is compact there is a subsequence of α_{n} (which w.l.o.g. we can assume to be α_{n} itself) that converges to a point of Z, say α. This means that there is a sequence N′_{n} → ∞, N′_{n} < N_{n} for every n, such that (α_{n})_{i,j} = α_{i,j} for 1 ≤ i ≤ N′_{n}, −N′_{n} ≤ j ≤ N′_{n}. Thus x_{n}, y_{n} ∈ U_{α,N′_{n}}.

X is compact, and so again we may assume that x_{n} and y_{n} converge to x ∈ U_{α} and y ∈ U_{α}, respectively. Since ρ(π(x), π(y)) ≥ δ, x ≁ y. This means that there is some T^{j} U_{i} such that x ∈ T^{j} U_{i} and y ∉ T^{j} U_{i}. Without loss of generality we can assume that

(4.5.1)  x ∈ T^{j}u_{i}  and  y ∉ cl(T^{j} U_{i}),

for otherwise there is an U_{i′} ⊃ ∂U_{i} such that x ∉ cl(T^{j} U_{i′}), and as in the proof that ∼ is closed we see that if we exchange the names of x and y and take T^{j} U_{i′} instead of T^{j} U_{i} (4.5.1) will be true.

But this is impossible, for if α_{i,j} = 0 both x and y can not be in T^{j} U_{i}; and if α_{i,j} = 1 both x and y must be in cl(T^{j} U_{i}). ⊔ ⊓

Note that in particular we have proved that π(U_{α}) consists of exactly one point for any α ∈ Z. We will define f(α) as the single point of Y in π(U_{α}). Clearly, as a direct consequence of the definitions of f and S,

f ∘ R(α) = S ∘ f(α) for every α ∈ Z.

f is onto Y since for any x ∈ X, x ∈ U_{ν(x)}, and so for any y ∈ Y, y ∈ π U_{ν′(y)}. f is also continuous — this is merely a restatement of the fact that diam π(U_{α,N}) → 0 uniformly in N. Thus f is a factor transformation from Z to Y and we are done.

We have in fact proved the following theorem:

Theorem (4.6) Let (X, T) be a finite dimensional dynamical system. Then for every two points x,x′ ∈ X and every ϵ > 0 there is a factor transformation ψ that maps the system (X, T) onto a system (Y, S) with h_{top}(Y) < ϵ and such that ψ(x) ≠ ψ(x′).

In fact, if we argue along the lines of remark

, (4.7) then we see that we can construct such a factor transfomation ψ so that ψ(x) ≠ ψ(y) for any y ∈ X.

(4.8) Constructing Intermediate Factors

We will need a small lemma before proving our main result:

Lemma (4.9) Let {(Y_{i}, S_{i})}_{i=1}^{∞} be a sequence of factors of the system (X, T), and let ψ_{i}: X → Y_{i}, i = 1, …, be the associated factor transformations. Define for every n the factor (Z_{n}, R_{n}) imbedded in

(Y_{1} × Y_{2} × ⋯ × Y_{n}, S_{1} × S_{2} × ⋯ × S_{n})

as the image of the system X under the factor mapping ψ_{1} × ψ_{2} × ⋯ × ψ_{n}. Similarly, define (Z_{∞}, R_{∞}) as the image of X under the infinite product ψ_{1} × ψ_{2} × · · ·. Then

(4.9.1).  h_{top}(Z_{∞}) = lim_{n→∞} h_{top}(Z_{n})

Proof. We will first define for each n ≤ ∞ a metric d(·, ·) on Z_{n} as

d(z, z′) = ∑_{i=1}^{n} 2^{−i}d_{Y_{i}}(y_{i}, y′_{i}),

where z = (y_{1}, …, y_{n}) and z′ = (y′_{1}, …, y′_{n}).

Since (Z_{m}, R_{m}) is a factor of (Z_{n}, R_{n}) for m < n ≤ ∞ the limit on the right-hand side of equation (4.9.1) exists and is no bigger than the left-hand side. However, in view of the metrics defined on ∏_{i=1}^{∞} Y_{i} and ∏_{i=1}^{n} Y_{i}, for every ϵ there is an N and ϵ′ such that

(4.9.2)  h_{ϵ}(Z_{∞}) ≤ h_{ϵ′}(Z_{N}).

This is true since for large enough N, the projection π_{N}: Z_{∞} → Z_{N} does not change the distances too much, i.e.

|d(x, y) − d(π_{N}(x), π_{N}(y))| < ϵ/4

for any x, y ∈ Z_{∞} = ∏_{i=1}^{∞} Y_{i}. Thus the projection of any n, ϵ-separated set to the first N coordinates will be a n, ϵ/2-separated set in Z_{N}, and so we have (4.9.2).

Assume h_{top}(Z_{∞}) < ∞ (the case h_{top}(Z_{∞}) = ∞ is almost the same and is omitted). Since h_{top}(Z_{∞}) = lim_{ϵ→0} h_{ϵ}(Z_{∞}), for any δ there exist an ϵ such that h_{top}(Z_{∞}) < h_{ϵ}(Z_{∞})+ δ. Thus

h_{top}(Z_{∞}) − δ < h_{ϵ}(Z_{∞}) ≤ h_{ϵ′}(Z_{N}) ≤ h_{top}(Z_{N}) ≤ lim_{n→∞} h_{top}(Z_{n}).

⊔ ⊓

Theorem (4.10) Let (X, T) be a finite dimensional dynamical system, and let (Y′, S′) be a factor of (X, T) such that h_{top}(Y′) < h_{top}(X) (in particular (Y′, S′) can be the trivial factor consisting of a single point). Then for any h ∈ (h_{top}(Y′), h_{top}(X)), we can find a factor (Y, S) of (X, T) with topological entropy h, and such that if ϕ: X → Y′ and ψ: X → Y are the respective factor transformations then ϕ ∘ ψ^{−1} is well defined, and is a factor transformation Y → Y′.

Proof.

Take any 0 < h < h_{top}(X) − h_{top}(Y′). Take ϵ_{1} = h ; 2. According to theorem (4.6), for every two points x ≠ x′ in X there is a factor (Y_{x,x′}^{1}, S_{x,x′}^{1}) of X with entropy ≤ ϵ_{1} such that x and x′ are mapped to different points in Y_{x,x′}^{1}. The factor transformation is continuous and so there are open sets U_{x,x′}^{1} ∋ x and V_{x,x′}^{1} ∋ x′ such that the factor transformation maps U_{x,x′}^{1} and V_{x,x′}^{1} into disjoint sets.

The sets U_{x,x′}^{1} × V_{x,x′}^{1} are an open cover of X × X \ Δ, where Δ is the diagonal in X × X. A countable number of these U_{x,x′}^{1} × V_{x,x′}^{1} suffice to cover X × X \ Δ, say U_{i}^{1} × V_{i}^{1}. Denote by (Y_{i}^{1}, S_{i}^{1}) the associate factor and by ψ_{i}^{1} (i > 1) the factor transformation. Also set (Y_{1}^{1}, S_{1}^{1}) = (Y′, S′) (the given factor of X), and ψ_{1}^{1} = ϕ.

Define using the (Y_{i}^{1}, S_{i}^{1})’s the factors (Z_{n}^{1}, R_{n}^{1}) and (Z_{∞}^{1}, R_{∞}^{1}) as in the previous lemma. The U_{i}^{1} × V_{i}^{1}’s cover X × X \ Δ and so for every x ≠ x′ in X there is an i such that ψ_{i}^{1}(x) ≠ ψ_{i}^{1}(x′). Thus the factor map X → Z_{∞}^{1} is one-on-one; by definition it is onto. Since X is compact we see that Z_{∞}^{1} is homeomorphic to X, and this homeomorphism is in fact an isomorphism between these two dynamical systems. Consequently, we see that h_{top}(Z_{∞}^{1}) = h_{top}(X) > h. Using lemma (4.9) we see that already for some finite N, the entropy h_{top}(Z_{N}^{1}) > h. Note that for any N we have

h_{top}(Z_{N+1}^{1}) ≤ h_{top}(Z_{N}^{1}) + h_{top}(Y_{N+1}^{1}) < h_{top}(Z_{N}^{1}) + ϵ_{1},

since we can imbed (Z_{N+1}^{1}, R_{N+1}^{1}) as a dynamical system in (Z_{N}^{1} × Y_{N+1}^{1}, R_{N}^{1} × S_{N+1}^{1}). If we take N_{1} to be the smallest integer such that h_{top}(Z_{N}^{1}) > h, we see that

h − ϵ_{1} < h_{top} (ψ_{1}^{1} × ψ_{2}^{1} × ⋯ × ψ_{N_{1}−1}^{1}(X)) ≤ h

(Note that since h_{top}(Y′) + ϵ_{1} < h_{top}(X), the integer N_{1} must be greater than 1).

We now take ϵ_{2} = ϵ_{1}/2 and find exactly as for ϵ_{1} a sequence of factors (Y_{i}^{2}, T_{i}^{2}) and factor transformations ψ_{i}^{2} such that for every i, h_{top}(Y_{i}^{2}) < ϵ_{2} and so that for every x ≠ x′ there is an i such that ψ_{i}^{2}(x) ≠ ψ_{i}^{2}(x′). For the same reasons as in the preceding paragraph

ψ_{1}^{1} × ⋯ × ψ_{N_{1}−1}^{1} × ψ_{1}^{2} × ψ_{2}^{2} × ⋯ (X) ≅ X

and so there is an N_{2} such that

h − ϵ_{2} < h_{top} (ψ_{1}^{1} × ⋯ × ψ_{N_{1}−1}^{1} × ψ_{1}^{2} × ⋯ × ψ_{N_{2}−1}^{2}(X)) ≤ h.

Proceeding in this way and using (4.9) we see that

h_{top} (ψ_{1}^{1} × ⋯ × ψ_{N_{1}−1}^{1} × ψ_{1}^{2} × ⋯ × ψ_{N_{2}−1}^{2} × ψ_{1}^{3} × ⋯ × ψ_{N_{3}−1}^{3} × ⋯ (X)) = h,

and so if we define ψ to be

ψ=^{def}ψ_{1}^{1} × ⋯ × ψ_{N_{1}−1}^{1} × ψ_{1}^{2} × ⋯ × ψ_{N_{2}−1}^{2} × ψ_{1}^{3} × ⋯ × ψ_{N_{3}−1}^{3} × ⋯,

Y =^{def}ψ(X) is the required factor. Indeed, since ψ_{1}^{1} = ϕ the map ϕ∘ψ^{−1}: Y → Y′ is simply a map that projects elements of Y (imbedded in Y′×Y_{2}^{1} ⋯) according to the first coordinate and so is obviously well defined and continuous. It is also onto Y′ since ϕ is.

⊔ ⊓

§5. A Construction of a Minimal Dynamical System with no Small Factors

In §2 we used the fact that each coordinate of a point in [0, 1]^{Z} can be varied continuously from 0 to 1 without having to change any other coordinate. Obviously there can’t be any minimal system with this property, but we will build a minimal system with enough flexibility that we will be able to use reasoning similar to that in §2 to prove this system has no finite entropy factors. Our system will be imbedded in [0, 1]^{Z}, and we will use the following metric on this space:

d(x, y)=^{def} ∑_{−∞<k<∞} 2^{−|k|}|x_{k} − y_{k}|.

We will define inductively a monotone decreasing sequence of subsystems (i.e. closed and shift invariant subset) [0, 1]^{Z} ⊃ X_{0} ⊃ X_{1} ⊃ … ⊃ X_{n} … such that:

a. No X_{n} is empty. b. For any two points x, y ∈ X_{n} there is a k such that d(y, σ^{k}x) < a_{n}, where (a_{k})_{k=0}^{∞} is

a sequence tending to zero as k → ∞.

Lemma (5.1) If X_{n} satisfy conditions a and b above then X =^{def} ⋂_{n=0}^{∞} X_{n} is minimal.

Proof. Since [0, 1]^{Z} is compact ⋂ X_{n} is nonempty. Let x ∈ X. We must show that {σ^{k}x} = X. Take any y ∈ X. For every n, both x and y are in X_{n}; thus there is a k such that d(y, σ^{k}x) < a_{n}. Since a_{n} → 0 this proves that y is in the closure of {σ^{k}x}. ⊔ ⊓

The following terminology will be convenient for the construction of the sequence X_{n}, and in proving X has no small factors. We will look at finite sequences of real numbers in the interval [0,1] as words from the (infinite) alphabet [0, 1] (the letters). By an infinite word we simply mean an element of [0, 1]^{Z}. If w is a word we will designate its length by l(w). If x ∈ [0, 1]^{Z}, a and b in Z ∪ {−∞, ∞} define x|_{a}^{b} = (x_{a}, x_{a+1}, … x_{b−1}). We will also use the notation w|_{a}^{b} for words, where the first letter of a word has index 1 (if a < 1 or

b > l(w) we set w|_{a}^{b} =^{def}w|_{max(a,1)}^{min(b,l(w)+1)}).

A word w′ is said to be a substring of a (finite or infinite) word w (or w′ ⪯ w) if there exist integers t_{1},t_{2} such that w′ = w|_{t_{1}}^{t_{2}}. If X ⊂ [0, 1]^{Z} we define the words of X or W (X) to be:

W (X) = {w | ∃x ∈ X w ⪯ x},

W_{l} (X) to be the words of X of length l, W_{≤l} (X) those of length ≤ l. And finally, if w,w′ are two words then w · w′ is their concatenation.

Let w_{0} be the length 1 word ‘0’, t_{0} = 100. Set

X_{0} = { x ∈ [0, 1]^{Z} | ∀t w_{0} ⪯ x|_{t}^{t+t_{0}} } .

For every k set L_{k} = {0, 1 ; 2^{k} , …, 1 − 1 ; 2^{k} }^{Z} ⊂ [0, 1]^{Z}, with L_{∞} = ∪_{k=0}^{∞}L_{k}. We will continue inductively to define X_{k},w_{k} and t_{k}, with the following properties:

1. w_{k} ∈ W (X_{k−1} ∩ L_{k})

2. Every word w ∈ W_{t_{k−1}} (X_{k−1} ∩ L_{k}) is a substring of w_{k}.

3. t_{k} = 50 · 2^{k}l(w_{k})

4. X_{k} = { x ∈ X_{k−1} | ∀t w_{k} ⪯ x|_{t}^{t+t_{k}}}

If we will define how to get w_{k} with the above properties given X_{k−1} we will be done with the definition. Set W = W_{3t_{k−1}} (X_{k−1} ∩ L_{k}). W is obviously a finite set. By property 4, for k − 1, each w ∈ W has a w_{k−1} as a substring of its first (and also last) t_{k−1} letters. For every such w remove at most t_{k−1} letters from its beginning and its end until we have a word that begins and ends with the substring w_{k−1}, and let W′ be the set of all these smaller words. Take w_{k} to be the concatenation of all the w′ ∈ W′.

Since each w′ ∈ W (X_{k−1}), and they all begin and end with a w_{k−1}, w_{k} ∈ W (X_{k−1}) and so w_{k} ∈ W (X_{k−1} ∩ L_{k}) which is property 1. Also, since we have not tampered with the middle t_{k−1} letters of each w ∈ W we have property 2. Properties 3 and 4 are merely definitions for t_{k} and X_{k} respectively.

In terms of the w_{k} and t_{k} we see that

(5.1.1)  X_{n} = { x ∈ [0, 1]^{Z} | ∀t, k ≤ n w_{k} ⪯ x|_{t}^{t+t_{k}} }.

X is the same but without the restriction that k must be ≤ n.

Lemma (5.2) The X_{k} defined above satisfy the conditions of lemma (5.1). Thus X = ⋂_{k=0}^{∞} X_{k} is minimal.

Proof. Since w_{k} is a word of X_{k−1}, that is, it has “enough” w_{k′} substrings for k′ < k, and it begins and ends with a w_{k−1}, the two sided infinite concatenation … w_{k} · w_{k} · w_{k} … is in X_{k}, so it is clear that X_{k} is a non-empty closed shift invariant set. Thus condition a of lemma (5.1)is satisfied.

To prove that condition b is satisfied let x be any point of X_{k}. Define a word x′ ∈ W (X_{k−1} ∩ L_{k}) as follows :

x′_{l} = ⌊2^{k}x_{l}⌋ ; 2^{k} for − t_{k−1} ; 2 ≤ l < t_{k−1} ; 2

(it is a word of X_{k−1} since all the words w_{k′}, k′ < k are composed of letters of the form integer ; 2^{k}). Now according to property 2, x′ is a substring of w_{k} and so of any y ∈ X_{k}. So we see that for any y ∈ X_{k} there is an m such that for − t_{k−1} ; 2 ≤ l < t_{k−1} ; 2 the l^{th} coordinate of x is different from that of σ^{m}y by less than 2^{−k}. But this means (since t_{k−1} ≫ k) that there exists an m such that d(x, σ^{m}y) < 2^{−k+2} — which is condition b. ⊔ ⊓

Since every w_{n} begins and ends with a w_{n−1}, there is a one sided word (with its first letter having index 1) w_{∞}, such that w_{∞}|_{1}^{l(w_{n})+1} = w_{n} for every n, and another one sided word (with its last letter having index 0) w̄_{∞} such that for every n, w̄_{∞}|_{−l(w_{n})+1}^{1} = w_{n}.

We will say a word w is legal if for every n, w_{n} · w · w_{n} ∈ W (X), or since X is closed this is equivalent to w̄_{∞} · w · w_{∞} ∈ X. This definition was chosen (in preference to the more natural definition that w be in W (X)) so that we would have an easy way to extend any legal word to an element of X. It is easy to see that a word w is legal iff for every n

(5.2.1).  w_{n} ⪯ (w_{n} · w · w_{n})|_{t}^{t+t_{n}}  for every 1 < t ≤ l(w) + 2l(w_{n}) − t_{n}

The fact, that for legal w equation (5.2.1)is satisfied for every n, is immediate from the definitions. But if (5.2.1)is satisfied then it is easy to see that w̄_{∞} · w · w_{∞} satisfies the conditions for being an element of X stated in (5.1.1).

Notice that for large enough n, t_{n} > l(w) + 2l(w_{n}) since by definition t_{n} ≥ 100l(w_{n}), and so (5.2.1) is satisfied vacuously. It is easy to see that for every n the word w_{n} is legal; and every word in W (X ∩ L_{∞}) is a substring of w_{n} for some n, so that every such word is a substring of a legal one.

The following lemma supplies the necessary degrees of freedom we need to apply the methods of §2, and is the important step in proving our system does not have any small non-trivial factors:

Lemma (5.3) If v_{0} and v_{1} are both legal word of common length l, then there exists a continuous path v(t), 0 ≤ t ≤ 1 between them consisting of legal words (a legal path).

To prove this lemma we must look more closely at legal words. Say w is a legal word. Some of its letters we can change at will without touching any substring equal to a w_{k} — indeed in general letters may even fail to be of the form integer ; 2^{k} and so by the definition of X given in terms of the t_{k} and w_{k}, (5.1.1), the changed words will all be legal. Other letters however can be crucial and changing them will destroy an essential appearance of a w_{k} substring. Even those crucial letters have different roles: some appear as part of an essential w_{0} (i.e. if they are changed, there will be a t such that w_{0} ⊀ w̄_{∞} · w · w_{∞}|_{t}^{t+t_{0}}), others as part of an essential w_{1}, etc. Notice that if the word w is shorter than t_{k} there will be no essential w_{k+1}’s. Changing an essential letter (which is a thing we might need to do to pass from the word v_{1} to v_{2}) involves constructing alternative copies of the relevant w_{k} which is harder for larger k’s. To designate such a pattern of words with some of their letters fixed, and some ranging over all values, independently of one another, we will use a word in the extended alphabet [0, 1] ∪ {∗}, were a ‘∗’ should be interpreted as a letter that can be replaced with any value. To each pattern p of length l, we will designate the set of words that fit it by ⟨p⟩. The intersection of two such sets for two patterns p_{1} and p_{2} of common length l is either an empty set or a set associated to another pattern of the same length; if this pattern is p we will say that p = p_{1} ∧ p_{2}. Thus the following definition, which will play an important role in proving lemma (5.3), is natural:

Definition (5.4) A structure of a word w is a sequence of patterns (p_{i})_{i=0}^{∞}, (i.e. a sequence of words in the extended alphabet [0, 1] ∪ {∗}) such that:

1. w is in ⟨p_{k}⟩ for every k

2. For every k, ⟨p_{k}⟩ ⊂ ⟨p_{k+1}⟩.

3. Setting

X_{≥n} =^{def}{ x ∈ [0, 1]^{Z} | ∀t, k ≥ n w_{k} ⪯ x|_{t}^{t+t_{k}} }

Every word v in ⟨p_{n}⟩ satisfies w_{k} · v · w_{k} ∈ W (X_{≥n}) for every k (or, equivalently

w̄_{∞} · v · w_{∞} ∈ X_{≥n}; we will call such words v legal words for X_{≥n}).

The structure is minimal if there is no other structure of w, say (p′_{i})_{i=0}^{∞}, such that for every i, ⟨p′_{i}⟩ ⊃ ⟨p_{i}⟩.

Remarks: Notice that if we have a structure (p_{i})_{i=0}^{∞} for a word w of length < t_{k}, then (p′_{i})_{i=0}^{∞}, where p′_{i} = p_{i} for i ≤ k and p′_{i} = ∗∗ ⋯ ∗ otherwise, is also a structure. In particular, if l(w) < t_{k} then for any minimal structure (p_{i})_{i=0}^{∞} of w, for every k′ > k p_{k′} = ∗∗ ⋯ ∗. Another thing worth noting is that every legal word w has a structure. Indeed the sequence of patterns p_{i} = w is a structure, for both the first and second conditions are trivial since all the p_{i}s are the same, and the third follows for every n from the legality of w.

We will need the following lemma, that shows that many patterns satisfying condition 3 in (5.4) for some n can be completed to form a structure.

Lemma (5.5) Let p be a pattern satisfying property 3 of the definition (5.4) for some n. Suppose in addition that for every t such that the t^{th} letter of p is not ‘∗’ there are t′ and t′′, t′ ≤ t ≤ t′′ such that (w̄_{∞} · p · w_{∞})|_{t′}^{t′′} = w_{n′} with n′ ≥ n (fix the indexing of this infinite word by taking the last letter of w̄_{∞} as its 0^{th} coordinate). Then ⟨p⟩ contains a legal word.

Proof.

By induction on n. For n = 0 the definition (5.4) says every word in ⟨p⟩ is legal so the lemma obviously holds.

If n > 0 we will show that there is a pattern p′ such that ⟨p′⟩ ⊂ ⟨p⟩ and p′ satisfies the conditions of this lemma for n − 1.

We get the pattern p′ from p by taking any l(w_{n−1}) consecutive ‘∗’s of p and replacing them by w_{n−1} until there are no l(w_{n−1}) consecutive ‘∗’s in p′. This new pattern clearly satisfies the second condition of this lemma, namely that every non ‘∗’ of p′ is part of a w_{n′} substring for n′ ≥ n − 1. We already know that for every w ∈ ⟨p′⟩, the infinite word w̄_{∞} · w · w_{∞} (again the last letter of w̄_{∞} is the 0^{th} coordinate of the infinite word) is in X_{≤n}, and wish to show that it is in X_{≤n−1}. Thus we only need to show that for every t,

(5.5.1)  w_{n−1} ⪯ (w̄_{∞} · w · w_{∞}) |_{t}^{t+t_{n−1}} .

Since w̄_{∞} ends with w_{n−1}, and any finite substring of it is a substring of some w_{n′}, (5.5.1) is true for t ≤ −l(w_{n−1}) and in the same way we can see that (5.5.1) is true for t ≥ l(w) − t_{n−1} + l(w_{n−1}). Suppose there is t′, −l(w_{n−1}) < t′ < l(w) − t_{n−1} + l(w_{n−1}) that does not satisfy (5.5.1) for some word w ∈ ⟨p′⟩. Changing the indexation we see there is a t′, 0 < t′ < l(w) − t_{n−1} + 2l(w_{n−1}) and a w ∈ ⟨p′⟩, such that w_{n−1} is not a substring of (w_{n−1} · w · w_{n−1})|_{t′}^{t′+t_{n−1}}. Such a w exists iff w_{n−1} is not a substring of the pattern (w_{n−1} · p′ · w_{n−1})|_{t′}^{t′+t_{n−1}} (patterns are simply words in an extended alphabet, and so we can use the word terminology for patterns as well). So we see that (5.5.1) is equivalent to having

(5.5.2)  w_{n−1} ⪯ (w_{n−1} · p′ · w_{n−1})|_{t}^{t+t_{n−1}}

for every 1 < t ≤ l(w) − t_{n−1} + 2l(w_{n−1}).

Now assume again that there is a t′ that does not satisfy (5.5.2). Since in p′ there are no more than l(w_{n−1}) − 1 consecutive ‘∗’s, there is a non ‘∗’ letter in the (smaller) substring (w_{n−1} · p′ · w_{n−1}) |_{t′+l(w_{n−1})}^{t′+t_{n−1}−l(w_{n−1})} which is the substring p′ |_{t′}^{t′+t_{n−1}−2l(w_{n−1})}. We know it is part of a w_{n′} substring of ŵ_{∞} · p′ · w_{∞} for n′ ≥ n − 1. If it is part of a w_{n−1} then we get a contradiction to t′ being ‘bad’. Suppose, then, that this letter is a part of a w_{n′} for n′ ≥ n. If this w_{n′} contains all the letters of w_{n−1} · p′ · w_{n−1} in the range t′ to t′ + t_{n−1} − 1 then since w_{n′} ∈ W (X) these letter must contain a w_{n−1} substring — a contradiction. So this w_{n′} substring must have either its end or its beginning in this range of letters. Assume that its end is in this range – the other case is exactly the same. We know that a letter in the range t′ + l(w_{n−1}) to t′ + t_{n−1} − 1 − l(w_{n−1}) is part of this w_{n′}, so the last l(w_{n−1}) letters of this w_{n′} are a substring of (w_{n−1} · p′ · w_{n−1})|_{t′}^{t′+t_{n−1}}. Since w_{n′} ends with a w_{n−1}, we again get a contradiction.

Thus the pattern p′ satisfies the conditions of this lemma for n − 1. By the induction hypothesis we know that ⟨p′⟩ must contain a legal word; and since ⟨p⟩ ⊃ ⟨p′⟩ so does ⟨p⟩. ⊔ ⊓

Corollary (5.6) Any pattern p satisfying the conditions of lemma (5.5) for some n, can be completed to form a structure whose n^{th} element is p.

Proof. Let w be a legal word in ⟨p⟩; then

p_{i} = \{ \begin{matrix}w, & if i < n; \\ p, & otherwise. \end{matrix}

is a structure for the word w. ⊔ ⊓

We return to lemma (5.3). Let (p_{i}^{0})_{i=0}^{∞} and (p_{i}^{1})_{i=0}^{∞} be structures for the two legal words v_{0} and v_{1} respectively. Also define k_{0} to be the smallest integer such that t_{k_{0}} > l(v_{0}).

Now suppose p_{0}^{0} and p_{0}^{1} are the same. This means that there is a pattern which both v_{0} and v_{1} satisfy, such that any word satisfying this pattern is legal. So simply vary every letter different in v_{0} and v_{1} continuously from its value in v_{0} to that in v_{1} and we get the required legal path. On the other hand from the remark following definition (5.4), if (p_{i}^{0})_{i=0}^{∞} and (p_{i}^{1})_{i=0}^{∞} are minimal structures, p_{k_{0}}^{0} and p_{k_{0}}^{1} are always the same, namely the trivial pattern ‘∗∗ ⋯ ∗’.

We will prove by induction on r the following lemma.

Lemma (5.3)′ Suppose v_{0} and v_{1} are both legal words of common length l, with structures (p_{i}^{0})_{i=0}^{∞} and (p_{i}^{1})_{i=0}^{∞} respectively. If p_{r}^{0} = p_{r}^{1} for some r, then there exists a legal path between them.

Remark: Since we can choose any structures we like for v_{0} and v_{1}, we can choose minimal structures, which we know satisfy p_{r}^{0} = p_{r}^{1} for r large enough. So (5.3)′ does imply (5.3).

Proof Of Lemma.

Note that we have proved the case r = 0. So suppose it is true for r′ = r − 1; we must show the lemma is true also for r. Let k_{0} be the smallest k such that l < t_{k_{0}}.

(5.8) We proceed to show that the structures (p_{k}^{0})_{k=0}^{∞} and (p_{k}^{1})_{k=0}^{∞} can be assumed to be minimal. As we have already seen we can assume that for k > k_{0}, p_{k}^{0} and p_{k}^{1} are ‘∗∗ ⋯ ∗’. Also we can assume that for k ≥ r we have p_{k}^{0} = p_{k}^{1} — in fact we can take both of these equal to p_{r}^{0}. Now we can start at k = k_{0} and find a pattern p′_{k} that satisfies all the properties it needs to in the definition (5.4), and in addition ⟨p_{k}^{0}⟩ ⊂ ⟨p′_{k}⟩, such that for every other pattern q satisfying these conditions, ⟨p′_{k}⟩ ⊄ ⟨q⟩. (There is such a pattern, because ⟨p′_{k}⟩ ⊂ ⟨q⟩ simply means that some of the letters ∈ [0, 1] in p′_{k} were replaced by ‘∗’s. So after a finite number of stages we must find the pattern p′_{k} we wanted). After we have defined p′_{k}, we can define in the same way but for k′ = k − 1 a pattern p′_{k−1} and continue inductively. The resulting sequence of patterns together with the all ‘∗’ patterns for large k form a minimal structure for v_{0}.

Performing the same process for v_{1} we can we can construct a minimal structure p′′_{k} for v_{1} satisfying p′′_{k} = p′_{k} for k ≥ r, for in finding p′_{k} for k ≥ r we only used p_{k}^{0} for k ≥ r and these are the same as p_{k}^{1} for these ks. In particular p′′_{r} = p′_{r}, which is exactly the condition imposed on the original structures by the assumptions of lemma (5.3)′. From now on we will assume that (p_{k}^{0})_{k=0}^{∞} and (p_{k}^{1})_{k=0}^{∞} are minimal.

(5.9) Now suppose we find a pattern ξ, such that

i ξ satisfies property 3 of definition (5.4) for n = r − 1 ii Any non ‘∗’ letter of ξ is part of a w_{k} substring of w̄_{∞} · ξ · w_{∞} (k ≥ r − 1). iii ⟨ξ⟩ ∩ ⟨p_{r−1}^{0}⟩ ≠ ∅ and ⟨ξ⟩ ∩ ⟨p_{r−1}^{1}⟩ ≠ ∅ iv ⟨ξ⟩ ⊂ p_{r}^{0} (= p_{r}^{1})

Assume further that we can find a legal words v′_{0} in ⟨ξ ∧ p_{r−1}^{0}⟩; and v′_{1} in ⟨ξ ∧ p_{r−1}^{1}⟩. We can easily find structures (ξ_{k}^{0})_{k=0}^{∞} and (ξ_{k}^{1})_{k=0}^{∞} for v′_{0} and v′_{1} respectively, such that ξ = ξ_{r−1}^{0} = ξ_{r−1}^{1} (simply take ξ_{k}^{0} = v′_{0} for k < r − 1 and ξ_{k}^{0} = ξ for k ≥ r − 1 and the same also applies to ξ_{k}^{1}). So according to the induction hypothesis v′_{0} and v′_{1} can be connected with a legal path. From exactly the same reasons, completing p_{r−1}^{0} to structures for v_{0} and v′_{0} we see they can be connected, and the same is true for v_{1} and v′_{1}. Combining these three legal paths we get a legal path from v_{0} to v_{1}.

We can see that there is a legal word in ⟨ξ ∧ p_{r−1}^{0}⟩ using lemma (5.5). For assumption i on ξ says any word in ⟨ξ⟩ is legal for X_{≥r−1} and so the same is true for the smaller set ⟨ξ ∧ p_{r−1}^{0}⟩, which is the first part of the conditions of lemma (5.5). In addition p_{r−1}^{0} as the r − 1^{th} element of a minimal structure also satisfies assumption ii, and so every non ‘∗’ letter of ξ ∧ p_{r−1}^{0} being a non ‘∗’ letter in either ξ or p_{r−1}^{0} is part of a w_{k} substring in the concatenation of this pattern with w̄_{∞} and w_{∞} and so is part of a w_{k} in the concatenation of ξ∧p_{r−1}^{0} with these two one sided infinite words, which is the second part of the conditions of that lemma. The same is of course true for the pattern ξ ∧ p_{r−1}^{1}, so we see that the existence of v′_{0} and v′_{1} is a consequence of the conditions set on ξ.

(5.10) We will now construct ξ by starting with ξ′_{0} =^{def}p_{r}^{0}, and modifying it until at last we get a pattern that will satisfy the conditions on ξ, in a way very similar to the way we got the pattern p′ from the pattern p in the proof of lemma (5.5). Suppose we have defined already ξ′_{k}. There are two cases:

A: There is a 1 ≤ t ≤ l − l(w_{r−1}) + 1 such that

ξ′_{k} |_{t}^{t+l(w_{r−1})}= p_{r−1}^{0} |_{t}^{t+l(w_{r−1})}= ∗∗ ⋯ ∗ = p_{r−1}^{1} |_{t}^{t+l(w_{r−1})},

in which case we get ξ′_{k+1} by replacing the l(w_{r−1}) ‘∗’s starting at place t with w_{r−1}; B: There is no such t in which case we take ξ to be ξ′_{k}.

Case A can be applied only a finite number of times. So eventually we do get to case B and thus ξ is defined. ξ clearly satisfies the assumptions iii and iv. Also, since we started with p_{r}^{0} which is an r^{th} element of a minimal structure and replaced ‘∗∗ ⋯ ∗’ substrings with w_{r−1} assumption ii also holds.

To show that assumption i holds, we need to show that for every word w in ⟨ξ⟩,

w̄_{∞} · w · w_{∞} ∈ X_{≥r−1}.

Since we know that ⟨ξ⟩ ⊂ ⟨p_{r}^{0}⟩, a set of legal words for X_{≥r}, then exactly as in the proof of condition (5.5.2) in lemma (5.5) it remains to be shown that for every 1 ≤ t ≤ l + 2l(w_{r−1}) − t_{r−1},

(5.10.1)  w_{r−1} ⪯ (w_{r−1} · ξ · w_{r−1}) |_{t}^{t+t_{r−1}} .

If there is a t in the range 0 < t ≤ l + 2l(w_{r−1}) − t_{r−1} that does not satisfy (5.10.1) then (w_{r−1} · ξ · w_{r−1}) |_{t}^{t+t_{r−1}} is a pattern with t_{r−1} letters but no w_{r−1} substrings. We want to show that this contradicts the fact that ξ satisfies case B, and we start by showing (using arguments very similar to those in the proof of (5.5)) that

(5.10.2)  (w_{r−1} · p_{r}^{0} · w_{r−1}) |_{t+l(w_{r−1})}^{t+t_{r−1}−l(w_{r−1})}= ∗∗ ⋯ ∗.

Since (p_{k}^{0})_{k=0}^{∞} is minimal every non ‘∗’ letter in p_{r}^{0} must be part of a w_{k} substring for k ≥ r. If all the letters between t and t + t_{r−1} are part of this w_{k} then since w_{k} ∈ W (X) there is in this range of letters a w_{r−1} substring, which is a contradiction. We know w_{k} begins and ends with w_{r−1}, so if the letters t,…,t + t_{r−1} − 1 of (w_{r−1} · p_{r}^{0} · w_{r−1}) are not part of the w_{k}, the first or last letters of this w_{k} are in this range of letters; in order not to get a w_{k−1} substring (and a contradiction to the existence of a ‘bad’ t) we see that the letters of p_{r}^{0}, in the range that interests us, that can appear as part of any w_{k} for k ≥ r are at most the first and last l(w_{k−1}) − 1 letters. Thus the central letters between t + l(w_{r−1}) and t + t_{r−1} − 1 − l(w_{r−1}) are ‘∗’s.

Both p_{r−1}^{0} and p_{r−1}^{1} can be derived from p_{r}^{0} by replacing ‘∗’s with letters in [0, 1]. If we look at the letters from t + l(w_{r−1}) to t + t_{r−1} − l(w_{r−1}) in p_{r−1}^{0} we claim that they cannot contain more than two complete appearances of w_{r−1}. For if they contain three appearances of w_{r−1} the middle one is redundant — any t_{r−1} consecutive letters that contain the middle w_{r−1} contains either the first of last w_{r−1} — and so we can replace by ‘∗’ all the letters that compose the middle w_{r−1} and are not part of the other two w_{r−1}s contradicting minimality. So the letters t + 2l(w_{r−1}) to t + t_{r−1} − 2l(w_{r−1}) cannot contain more than two blocks of non ‘∗’ letters, each of length ≤ l(w_{r−1}) (the two blocks can merge together to form a larger block but no bigger than twice this size). The same is true for p_{r−1}^{1}, and so since t_{r−1} ≥ 100l(w_{r−1}) there must be in the range t + 2l(w_{r−1}) to t + t_{r−1} − 2l(w_{r−1}) a subrange from t′ to t′ + 3l(w_{r−1}) such that in both p_{r−1}^{0} and p_{r−1}^{1} the letters in this range are only ‘∗’. We know however that the letters of ξ in this range do not have a w_{r−1} substring. recall that ξ was derived from p_{r}^{0} by changing substrings of l(w_{r−1}) ‘∗’s by w_{r−1}. Since the letters t′ to t′ + 3l(w_{r−1}) do not contain any full copy of w_{r−1}, and since they were in p_{r}^{0} all ‘∗’, we see that the middle third of these letters must remain all ‘∗’s. Thus at last we see that ξ has an all ∗ substring satisfying the conditions of case A — a contradiction.

Since we have seen in the previous subsection that the existence of a ξ satisfying assumptions i– iv implies the existence of a legal path between v_{0} and v_{1} we are done. ⊔ ⊓

We need one more simple lemma:

Lemma (5.11) If v_{1},…,v_{n} are all legal words of common length l ≤ t_{k} − 2l(w_{k}), and n(l + l(w_{k})) ≤ t_{k+1} − 2l(w_{k+1}) then v_{1} · w_{k} · v_{2} ⋯ w_{k} · v_{n} is a legal word.

The proof of this lemma is very similar to the proof of lemma (5.5) and the last subsection of the proof of lemma (5.3)′, and involves no new ideas, so it will be omitted.

At last we are ready to prove:

Theorem (5.12) Every factor of the system (X, σ|_{X}) has infinite topological entropy.

Proof.

Recall the notations and the definition of entropy given at the beginning of subsection (4.3).

Again, as in theorem (2.1), we may consider only factors of the form (Y, σ|_{Y}), Y a subset of [0, 1]^{Z}, with a factor map ϕ̂ generated from a continuous function ϕ: X → [0, 1] (with both 0 and 1 in its image) as in (1.2.1).

There is an N_{0} such that for any two points x, y ∈ X, if x|_{−N_{0}}^{N_{0}} = y|_{−N_{0}}^{N_{0}} then |ϕ(x) − ϕ(y)| < 1 20. Since the points of X ∩ L_{∞} are dense in X, we can find x^{0} and x^{1} ∈ X such that ϕ(x^{0}) < 1 20, ϕ(x^{1}) > 19 20.

Let v′_{0} = x^{0}|_{−N_{0}}^{N_{0}}, and in the same way define the word v′_{1} using x^{1}. Both v′_{0} and v′_{1} are of course in W (X ∩ L_{∞}), and so both of them are substrings of w_{n′} for an n′ large enough. Suppose that v′_{i} begins at letter s_{i} of w_{n′} for i = 0, 1. Set for i = 0, 1

p_{i} = \underbrace{∗∗ ⋯ ∗}_{s_{1−i} times} ·w_{n′} · \underbrace{∗∗ ⋯ ∗}_{s_{i} times} .

Both p_{0} and p_{1} satisfy the conditions of lemma (5.5) for n = n′, and so there are legal words v_{0} ∈ ⟨p_{0}⟩ and v_{1} ∈ ⟨p_{1}⟩ of common length l=^{def}l(w_{n′}) + s_{0} + s_{1}. These words satisfy v_{i}|_{s_{0}+s_{1}}^{s_{0}+s_{1}+2N_{0}} = v′_{i} for i = 0, 1 and so we see there is (an easily computable) s such that for any x ∈ X,

x|_{−s}^{−s+l−1} = v_{i} ⟹ \{ \begin{matrix}ϕ(x) < 1/10, & if i = 0 \\ ϕ(x) > 9/10, & if i = 1 \end{matrix}

According to lemma (5.3), there exists a legal path, which we will denote by v(t), 0 ≤ t ≤ 1, such that v(0) = v_{0} and v(1) = v_{1}.

Using lemma (5.11), and since we know l < t_{n′} − 2l(w_{n′}), for every s_{1}, …, s_{r_{1}} ∈ [0, 1]

with r_{1} = ⌊ t_{n′+1}−2l(w_{n′+1}) ; t_{n′}⌋ the word

η_{1}(s_{1}, …, s_{r_{1}})=^{def}v(s_{1}) · w_{n_{0}} · v(s_{2}) ⋯ w_{n_{0}} · v(s_{r_{1}})

is legal. Let τ_{1}, …, τ_{r_{1}} be the indices of the first letter of v(s_{1}), …, v(s_{r_{1}}) respectively. By definition of legality, w̄_{∞} · η_{1}(s_{1}, …, s_{r_{1}}) · w_{∞} (the 0^{th} coordinate is the last letter of w̄_{∞}) is in X. If we take the projection of

ϕ̂ (w̄_{∞} · η_{1}(s_{1}, …, s_{r_{1}}) · w_{∞})

to the coordinates τ_{1} + s,…,τ_{r_{1}} + s we get a function from [0, 1]^{r_{1}} to [0, 1]^{r_{1}} satisfying the conditions of lemma (2.2). Thus the whole cube [ 1 4, 3 4]^{r_{1}} is in its image. Since there are m^{r_{1}} points that differ from each other in at least one coordinate by more that 1 ; 2m in this cube, we see that s(Y; t_{n′+1}, 1 ; 2m) ≥ s(Y; τ_{r_{1}} + s, 1 ; 2m) ≥ m^{r_{1}}. We can continue in this way

one more step. Set r_{2} = ⌊ t_{n′+2}−2l(w_{n′+2}) ; t_{n′+1}⌋, and let

\begin{matrix}η_{2}(s_{1}, s_{2}, …, s_{r_{2}r_{1}}) & =^{def}η_{1}(s_{1}, …, s_{r_{1}}) · w_{n′+1} · η_{1}(s_{r_{1}+1}, …, s_{2r_{1}}) ⋯ \\ · w_{n′+1} · η_{1}(s_{(r_{2}−1)r_{1}+1}, …, s_{r_{2}r_{1}}). \end{matrix}

And in the same way as before show that s(Y; t_{n′+2}, 1 ; 2m) ≥ m^{r_{2}r_{1}}. We can proceed further

in this way to show that if r_{i} = ⌊ t_{n′+i}−2l(w_{n′+i}) ; t_{n′+i−1}⌋ then

s(Y; t_{n′+i}, 1 ; 2m) ≥ m^{r_{i}r_{i−1}···r_{1}}.

Now since t_{n′+i−1} ≤ l(w_{n′+i}) = t_{n′+i} ; 50·2^{n′+i},

\begin{aligned}r_{i} & ≥ (1 − 2 ; 50 · 2^{n′+i} ) t_{n′+i} ; t_{n′+i−1} − 1 \\ & ≥ (1 − 1 ; 2^{n′+i} ) t_{n′+i} ; t_{n′+i−1} , \end{aligned}

and so we have

(5.12.1)

1 ; t_{n′+i} log s(Y; t_{n′+i}, 1 ; 2m) ≥ r_{i}r_{i−1} ⋯ r_{1} ; t_{n′+i} log m

≥ 1 ; t_{n′} log m ∏_{i=1}^{∞} (1 − 1 ; 2^{n′+i} )

≥ c log m

for a positive constant c > 0 that does not depend on i and m, and c > 0 since the product above converges. But (5.12.1) exactly says that h_{1 ; 2m} (Y) ≥ c log m and thus h_{top}(Y) = ∞. ⊔ ⊓ §6. Two Minimal Systems with a Common Almost 1-1 Extension but No Common Factors

In this section we present an application, due to S. Glasner, of the results of the previous section.

We will say a map ϕ: X → Y is almost 1-1 if the set Y_{0} ⊂ Y of points y such that the set ϕ^{−1}(y) consists of exactly one point is dense. It is a well known result of elementary topology that the set Y_{0} so defined is a G_{δ} set for any map ϕ, and so if ϕ is almost 1-1 then |ϕ^{−1}(y)| = 1 on a dense G_{δ} subset of Y. We will say (Z, R) is an almost 1-1 extension of (X, T) if (X, T) is a factor of (Z, R) and the factor transformation Z → X is almost 1-1.

Let (X̄, T̄) denote the minimal system we have constructed in the previous section (or any other minimal system with no finite entropy factors). We wish to find an almost 1-1, zero dimensional extension of X̄. The existence of such an extension is part of the folklore on almost 1-1 extensions (see [5]). However, there does not appear to be any published proof of this fact.

Lemma (6.1) Let (X, T) be a minimal system, and assume X is infinite. Then X has a minimal, almost 1-1, zero dimensional extension (Z, R).

Proof.

X is, by assumption, a compact metric space. Thus, since (X, T) is minimal and infinite, X has no isolated points. It is well known that under these circumstances there exist an almost 1-1 map ϕ from the cantor set C = {0, 1}^{N} onto X. Let X_{0} be the set of points where |ϕ^{−1}(x)| = 1. Recall that X_{0} is a dense G_{δ} subset of X, by definition of ϕ.

The system Z we will construct will be a subsystem of C^{Z} with the shift operation σ. We first define

Z′ = {(…, c_{−1}, c_{0}, c_{1}, …) ∈ C^{Z} : ∀i ∈ Z ϕ(c_{i}) = T^{i}ϕ(c_{0})} .

The map ϕ̄: Z′ → X, defined by

ϕ̄: (…, c_{0}, …) ↦ ϕ(c_{0}),

is a factor transformation Z′ → X.

Now, any dynamical system (and in particular Z′) has a minimal subsystem. Indeed, any subsystem of Z′ with no subsystems is minimal, and such subsystems exist by Zorn’s Lemma. Let (Z, σ|_{Z}) be a minimal subsystem of Z′.

ϕ̄(Z) is a nonempty, closed and T invariant subset of X. Thus, since X is minimal ϕ̄(Z) = X and Z is an extension of X. Clearly Z is zero dimensional, so we only need to prove that ϕ̄ is almost 1-1. Since X is minimal, we only need to prove |ϕ̄^{−1}(x_{0})| = 1 for some point x_{0} ∈ X.

Let

x_{0} ∈ ⋂_{i=−∞}^{∞} T^{i}(X_{0}).

This set is nonempty since it is the intersection of a countable number of dense G_{δ} sets. Suppose c and d are two distinct points of Z′ such that

ϕ̄(c) = ϕ̄(d) = x_{0}.

Then for some i,

c_{i} ≠ d_{i}  but  ϕ(c_{0}) = ϕ(d_{0}) = x_{0}

so that

ϕ(c_{i}) = T^{i}ϕ(c_{0}) = T^{i}(x_{0}) = T^{i}ϕ(d_{0}) = ϕ(d_{i}),

and so T^{i}(x_{0}) ∉ X_{0} — a contradiction. ⊔ ⊓

Let (Z, R) be a minimal, almost 1-1 zero dimensional extension of X̄. It is not hard to modify slightly the proof of theorem (4.10) and show that any finite dimensional dynamical system has a factor with any positive entropy we wish (smaller than the entropy of the original system) such that the factor transformation is almost 1-1. For the very special case of a zero dimensional minimal systems we can give a direct proof of this fact, due to B. Weiss:

Theorem (6.2) Any zero dimensional dynamical system (Z, R) has, for any η > 0, a factor (Y, S) with entropy h_{top}(Y) < η such that the factor transformation is almost 1-1.

Proof.

Take any z ∈ Z. Since Z is minimal it suffice to show that the factor transformation Z → Y is 1-1 at z. Let d be the metric on Z.

Pick some large integer N. Let

(6.2.1)  δ′ = 1 2 min_{i=1,...,N} d(z, T^{i}z).

Choose any δ > 0 small enough so that for any x_{1} and x_{2}, and any −N ≤ n ≤ N, if d(x_{1}, x_{2}) < δ then d(T^{n}x_{1}, T^{n}x_{2}) < δ′.

Since Z is zero dimensional, there is some clopen set U such that

z ∈ U ⊂ B(z, δ).

Notice that for any 0 < k ≤ N,

U ∩ T^{k}(U) = ∅.

For assume u ∈ U ∩ T^{k}(U). Then d(u, z) < δ and d(T^{−k}u, z) < δ. Thus we see that d(u, T^{k}z) < δ′ and so

d(z, T^{k}z) < 2δ′,

in contradiction to (6.2.1).

Define the map ϕ : Z → {0, 1}^{Z} by

ϕ: z ↦ (… 1_{U}(T^{−1}z), 1_{U}(z), 1_{U}(T z) …).

Since U is clopen, ϕ is continuous. Any infinite word of 0’s and 1’s in ϕ(X) must have at least N zero’s between every two occurrences of 1’s, and so it is immediate that by taking N large enough the entropy of the factor ϕ(Z) can be made as small as we wish. Notice also that if d(x, z) > δ then ϕ(x) ≠ ϕ(y).

To summarize, we have so far seen that for every ϵ, δ > 0, there exists a factor Y_{ϵ,δ} of Z such that h_{top}(Y_{ϵ,δ}) < ϵ, and such that if d(x, z) > δ then x and z are mapped into distinct points of Y_{ϵ,δ}.

For every n > 0, let Y_{n} = Y_{2^{−n}η,n^{−1}}, and let ϕ_{n} be the associated factor transformations. Define ϕ_{∞}: Z → ∏_{n=1}^{∞} Y_{n} by

ϕ_{∞}: x ↦ (ϕ_{1}(x), ϕ_{2}(x), …),

and set Y = ϕ_{∞}(Z).

Clearly,

h_{top}(Y) ≤ ∑_{n=1}^{∞} h_{top}(Y_{n}) < η

and if x ≠ z then for every n large enough ϕ_{n}(x) ≠ ϕ_{n}(z) and so ϕ_{∞}(x) ≠ ϕ_{∞}(z). ⊔ ⊓

Thus Z has a finite entropy factor (Y, S) with an almost 1-1 factor transformation. Every factor of (X̄, T̄) has infinite entropy; every factor of (Y, S) finite entropy. So these two minimal systems have a common almost 1-1 extension but no common factors, answering the question posed by H. Furstenberg.

REFERENCES

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

[2] J. Kulesza, Zero–Dimensional Covers of Finite Dimensional Dynamical Systems, to appear.

[3] S. Glasner and D. Maon, An Inverted Tower of Almost 1-1 Extensions, J. D’Analyse Math., 44 (1984/85)

[4] S. Glasner and B. Weiss, Minimal transformations with no common factor need not be disjoint, Isr. J. Math. 45 (1983), 1–8

[5] H. Furstenberg and B. Weiss, On Almost 1-1 Extensions, Isr. J. Math., Vol. 65, No. 3 (1989), 311–322.

[6] R. Engelking, ”Dimension Theory”, PWN, Warsaw, 1977

[7] J. van Mill, ”Infinite Dimensional Topology”, North-Holland, Amsterdam, 1989