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POSITIVE ENTROPY HOMEOMORPHISMS OF CHAINABLE
CONTINUA AND INDECOMPOSABLE SUBCONTINUA
CHRISTOPHER MOURON
Abstract. It is shown that if X is a chainable continuum and h : X −→ X is a homeomorphism such that the topological entropy of h is greater than 0, then X must contain an indecomposable subcontinuum. This answers a question of Barge.
- Introduction In dynamics, topological entropy is a number in [0, ∞] that gives a measure of the chaotic behavior of a function on a space. The connection between entropy and the dynamics of a continuous function is well documented. Many simple functions such as the tent map have positive entropy. However, for a homeomorphism to have positive entropy, it appears that the local structure must be complex. For example, in [5] it is shown that no homeomorphism of a regular continuum can have positive entropy. In [7] and [8], Ye showed that homeomorphisms of hereditarily decomposable chainable continua that are induced by square commuting diagrams on inverse systems of intervals must have zero entropy. In this paper, it is shown that every chainable continuum that admits a positive entropy homeomorphism must contain a nondegenerate indecomposable subcontinuum. Indecomposable continua are known to have a very complicated local structure. This answers a question due to Barge [2]. A continuum X is a compact connected metric space. A map is a continuous function. A continuum X is decomposable provided there exist proper subcon- tinua H and K such that X = H ∪ K. A continuum is indecomposable if it is not decomposable. If U is a collection of open sets, the mesh of U is defined as mesh(U) = sup{diam(U ) : U ∈ U}. A chain C is an indexed collection of open sets {C 1 , C 2 , ..., Cn} such that Ci ∩ Cj 6 = ∅ if and only if |i − j| ≤ 1. Here, C 1 and Cn are the endlinks of the chain. A continuum X is chainable if for every > 0 there exists a chain covering X such that mesh(C ) < . Chainable continua are also called arc-like and snake-like continua. Let X be a compact metric space, f : X −→ X be a map, and U be a finite open cover of X. Define N (U) be the number of sets in a finite subcover of U with smallest cardinality. If U and V are two open covers of X, let U ∨V = {U ∩V |U ∈ U, V ∈ V} and f −^1 (U) = {f −^1 (U )|U ∈ U}. Also, define ∨n− 1 i=0 f^
−i(U) = U ∨ f − (^1) (U) ∨ ... ∨ f −n+1(U), where f 0 = id
and
Ent(f, U) = limn→∞(1/n) log N (
∨n− 1 i=0 f^
−i(U)).
Then the topological entropy of f is defined as
1991 Mathematics Subject Classification. Primary: 54H20, 54F50, Secondary: 54E40. Key words and phrases. entropy, chainable continuum, indecomposable continuum. 1
2 C. MOURON
Ent(f ) = sup{Ent(f, U, X) : U is an open cover of X}. If U and V are finite open covers of X such that for every V ∈ V there exists a U ∈ U such that V ⊂ U , then V refines U. If for every V ∈ V there exists a U ∈ U such that V ⊂ U , then V closure refines U. The following propositions are well known and can be found in several texts such as [3] and [6].
Proposition 1. If V, U are open covers of X such that V refines U, then Ent(f, V) ≥ Ent(f, U).
Proposition 2. For each positive integer k, Ent(f k) = kEnt(f ).
To prove the main result, the following steps are taken (terms in italics will be be defined later):
- Entropy is defined on collections of infinite words on finite alphabets.
- Relative entropy between 2 symbols in the alphabet is defined.
- If a collection of infinite words has positive entropy, then it is shown through combinatorical techniques that “embedded” in the collection of words is a subcol- lection of words on 2 symbols that is complete.
- Collections of words are then generated by finite open covers and inverse images of continuous function f.
- Combinatorical techniques are used to show that if Ent(f ) > log(2), then there exist disjoint open sets A, B whose relative entropy is positive.
- It is shown that for every > 0 there exist a chain that is crooked between A and B whose endlinks have positive relative entropy.
- Using the well-used techniques of Barrett [1], it is shown that there must be an indecomposable subcontinuum.
- The Entropy of Words For a more complete treatment on the entropy of words see [4]. An alphabet A is a finite set of symbols. A word is a sequence composed from the elements of A. Words can be finite, Wn = 〈Ai〉ni=1, or infinite, W = 〈Ai〉∞ i=1. The length of a word is the number of elements in the sequence. Define πk(〈Ai〉∞ i=1) = Ak and Πk(〈Ai〉∞ i=1) = 〈Ai〉ki=1. Likewise, define πk(〈Ai〉ni=1) = Ak and Πk(〈Ai〉ni=1) = 〈Ai〉ki=1 if the word is finite and k ≤ n. If W is a word, then Πk(W ) is called the prefix of length k of W. If Wn is a word of length n then 〈(Wn), A〉 is a new word of length n + 1 formed by adding the symbol A to the end of Wn. W(A) (W when there is no confusion) will represent some collection of infinite words on A. If B ⊂ A, then
W(B) = {〈Bi〉∞ i=1|Bi ∈ B}.
Likewise, Wn will represent some collection of words of length n. Next, define Πk(W) = {Πk(W )|W ∈ W}. Πk(Wn) can be defined in a similar manner provided n ≤ k. It will be useful to define distance between words. Let A, B ∈ A and let
dA(A, B) =
0 A = B
1 A 6 = B
4 C. MOURON
Notice that if W is a complete collection of words on B ⊂ A, then |Πk(W)| ≥ |B|k and it follows that log |B| ≤ ENT(W) ≤ log |A|. Suppose that W is a collection of infinite words on A and that B ⊂ A. Again, let G = {g(i)}∞ i=1 be an increasing sequence of positive integers. Define the entropy of W relative to (B, G) by
ENT(W, B, G) = lim sup i→∞
log |Πi((W(B))G)| g(i) and the entropy of W relative to B by
ENT(W, B) = sup G
{ENT(W, B, G)}.
Notice that (W)G itself is a collection of words and it will be useful to con- sider EN T ((W)G). Also, since |Πg(i)((W)G)| ≥ |Πi((W)G)|, then EN T ((W)G) ≥ EN T (W, G). The following 2 theorems are Theorem 14 and Corollary 20 respectively in [4].
Theorem 7. Suppose that W is a closed shift-invariant collection of infinite words on A and B ⊂ A. (1) If (W(B))G (or (W)G) is a complete collection of words on B, where G =
{g(i)}∞ i=1, r ≥ 1 , and g(i) ≤ ri for all i, then ENT(W, B, G) ≥
log |B| r
(2) Conversely, if there exist A, B ∈ A such that EN T (W, {A, B}) > 0 , then there exists an increasing sequence G = {g(i)}∞ i=1 and r ≥ 1 such that g(i) ≤ ri for all i and (W({A, B}))G is a complete collection on {A, B}.
Theorem 8. Suppose that W is a closed shift-invariant collection of words on A, G = {g(i)}∞ i=1 is an increasing sequence of positive integers and r > 1 such that g(i) < ri. If (W)G is complete on A, then ENT(W, {A, B}) > 0 for every distinct A, B ∈ A.
Suppose W is a collection of words on A, C 6 ∈ A and A, B ∈ A. For W ∈ W define
φi(W ) =
πi(W ) if πi(W ) 6 ∈ {A, B} C if πi(W ) ∈ {A, B}
and let φ(W ) = 〈φi(W )〉∞ i=1. Then define W/{A, B} = {φ(W )|W ∈ W}. If Wn is a finite word of length n then take φ(Wn) = 〈φi(W )〉ni=1. The following 3 theorems are Corollary 17, Corollary 19 and Theorem 22 respec- tively in [4].
Theorem 9. Suppose that W is a closed shift-invariant collection of words on A, G = {g(i)}∞ i=1 is a increasing sequence of positive integers and r > 1 such that g(i) < ri. If ENT(W, {A, B}) = 0, then ENT(W, G) = ENT(W/{A, B}, G), ENT((W)G) = ENT((W/{A, B})G) and hence ENT(W) = ENT(W/{A, B}).
Theorem 10. Suppose that W is a closed shift-invariant collection of words on A, G = {g(i)}∞ i=1 is an increasing sequence of positive integers and r > 1 such that g(i) < ri. If ENT((W)G) > log(|A| − 1)
then ENT(W, {A, B}) > 0 for every distinct A, B ∈ A.
ENTROPY HOMEOMORPHISMS 5
Theorem 11. Suppose that W is a closed shift-invariant collection of words on A and A, B, and D are distinct elements of A. If ENT(W, {A, B, D}) > 0 and ENT(W/{A, B}, {C, D}) > 0 ,where C (C 6 ∈ A) is the identification of A and B given by the previously defined φ, then ENT(W, {A, D}) > 0 or ENT(W, {B, D}) >
Suppose that G = {g(i)}∞ i=1 is an increasing sequence of positive integers. Let Kn = (k 1 , ..., kn) be a collection of not necessarily distinct positive integers. We say that G has pattern Kn if
g(2i) − g(2i − 1) = k 1 , g(2(2i − 1) + 1) − g(2(2i − 1)) = k 2 , .. . g(2n−^1 (2i − 1) + 1) − g(2n−^1 (2i − 1)) = kn
for each positive integer i. However, it is possible that g(2ni + 1) − g(2ni) is not constant as i varies. The next theorem is Theorem 29 in [4].
Theorem 12. Suppose that W is a closed shift-invariant collection of words on A such that ENT(W) > 0. Then there exists A, B ∈ A such that
ENT(W, {A, B}) > 0 Furthermore, for each n there exists and increasing sequence of positive integers G(n) = {gn(i)}∞ i=1, rn > 1 and Kn = (k 1 , ..., kn) such that
(1) (W({A, B})G(n) is complete on {A, B}. (2) G(n) has pattern Kn. (3) gn(i) ≤ rni for each i.
- The Relationship between the Entropy of Words and the Entropy of Functions In this section, we will use the structure of words to aid in finding results on the entropy of self maps. Let f : X → X be an onto continuous function and let A be a finite open cover of X. Cover A can be thought of as an alphabet where the elements of A are the symbols. Inverse images of f will then be used to create the words. Let f : X −→ X and A be an open cover on X. A finite sequence 〈A 1 , A 2 , ..., Ai〉 of elements Ak ∈ A is a (f, A)-word of length i provided
A 1 ∩ f −^1 (A 2 ) ∩ ... ∩ f −i+1(Ai) 6 = ∅ Let Wi(f, A) be the set of all (f, A)-words of length i and W(f, A) = {W |Πi(W ) ∈ Wi(f, A)}. The following proposition, although obvious, is very important.
Proposition 13. W(f, A) is a closed shift-invariant collection of words on A.
Proof. The fact that W(f, A) is closed is obvious from the definition. To show that W(f, A) is shift invariant, let 〈Ai〉∞ i=1 ∈ W(f, A). Then for each k,
A 1 ∩ f −^1 (A 2 ) ∩ ... ∩ f −k+1(Ak) 6 = ∅
which implies that
f (A 1 ) ∩ A 2 ∩ ... ∩ f −k+2(Ak) = f (A 1 ∩ f −^1 (A 2 ) ∩ ... ∩ f −k+1(Ak)) 6 = ∅.
ENTROPY HOMEOMORPHISMS 7
Proposition 17. Suppose that B = {B 1 , ..., Bn} and B′^ = {B 1 ′, ..., B n′} are collec- tions of open sets such that Bi ⊂ B′ i for each i, then ENT(f, B) ≤ ENT(f, B′)
Theorem 18. Suppose that B is a finite collection of open sets and there exist A, B ∈ B such that ENT(f, {A, B}) = 0. Then ENT(f, (B − {A, B}) ∪ {A ∪ B}) = ENT(f, B).
Proof. For every increasing sequence G, it follows from the hypothesis that
ENT(W(f, {A, B}), G) = ENT(f, {A, B}, G) = 0.
Let C = A ∪ B. Then by Theorem 8, for each sequence G
ENT(f, (B − {A, B}) ∪ {A ∪ B}) = ENT(W(f, B)/{A, B}, G) = ENT(W(f, B), G) = ENT(f, B).
Let Kn = (k 1 , ..., kn) be a collection of positive integers (not necessarily distinct). Define RKn (B) recursively in the following way: Let D 0 = B and given Dj− 1 define Dj = Dj− 1 ∨ f −kj^ (Dj− 1 ). Let RKn (B) = Dn. Notice that if B is disjoint, then RKn (B) must be disjoint. Also |RKn (B)| = |B|^2 n . The next result is the main theorem of this section:
Theorem 19. Suppose that A, B are open sets such that ENT(f, {A, B}) > 0. Then for each positive integer n, there exists a finite set of positive integers Kn = {ki}ni=1, r ≥ 1 , and G = {g(i)}∞ i=1 such that g(i) ≤ ri and W(f, RKn {A, B}, G) is complete on RKn ({A, B}). Furthermore, for every distinct A′, B′^ ∈ RKn ({A, B}), ENT(f, {A′, B′}) > 0.
Proof. By Theorem 12, there exists an increasing sequence of positive integers G(n) = {gn(i)}∞ i=1 , rn > 0 and Kn = (k 1 , ..., kn) such that G(n) has pattern Kn and gn(i) ≤ rni. Let g(i) = gn(2n(i − 1) + 1). Then
U 1 ∩ f −g(1)(U 2 ) ∩ ... ∩ f −g(p−1)(Up) 6 = ∅
if and only if
X 1 ∩ f −gn(1)(X 2 ) ∩ ... ∩ f −g(
np−1) (X 2 np) 6 = ∅
where Xi ∈ {A, B} and
Ui =
⋂^2 n
k=
f −gn(
n(i−1)+k−1) (X 2 n(i−1)+k) ∈ RKn ({A, B}).
Thus, it follows that W(f, RKn ({A, B}), G) is complete on RKn ({A, B}) since W(f, {A, B}, G(n)) is complete on {A, B}. Furthermore, if we let r = 2n+1rn, then
g(i) = gn(2n(i − 1) + 1) ≤ (2n(i − 1) + 1)rn ≤ 2 n+1rni = ri.
Hence it follows from Theorem 8 that ENT(f, {A′, B′}) > 0 for every distinct A′, B′^ ∈ RKn ({A, B}).
8 C. MOURON
- Entropy and Open Covers Recall that C = {C 1 , C 2 , ..., Cn} is a chain cover of X provided Ci ∩ Cj 6 = ∅ if and only if |i − j| ≤ 1. C is a proper chain cover provided Ci ∩ Cj 6 = ∅ if and only if |i − j| ≤ 1. The next proposition is well known and easy to show.
Proposition 20. If X is chainable, then for every > 0 there exists a proper chain cover of X with mesh less that .
The goal of this section is to show that if C is a chain cover such that ENT(f, C) > log(2), then there exist disjoint open sets A, B such that ENT(f, {A, B}) > 0. If U is a collection of sets, define the star of U as U∗^ =
U ∈U U^.
Lemma 21. Suppose that G = {g(i)}∞ i=1 is a sequence of positive integers, r ≥ 1 such that g(i) ≤ ri and ENT(W(f, B, G)) > log(|B| − 1). Let H ∈ B and U(H) be a finite collection of open sets such that H ⊂ U(H)∗, U(H) ∩ (B − {H}) = ∅ and |B| > |U(H)|. Then there exists a U ∈ U(H) and a B′^ ⊂ B − {H} such that |B′| ≥ |B| − |U(H)| and ENT(f, {U, B}) > 0 for every B ∈ B′.
Proof. Let U(H) = {U 1 , U 2 , ..., Un} and suppose that the lemma is false for {U 1 , U 2 , ..., Un− 1 }. Then there exist distinct C 1 , ..., Cn− 1 ∈ B − {H} such that ENT(f, {Ui, Ci}) = 0 for each 1 ≤ i ≤ n − 1. Let
Bi = B − {H, C 1 , ..., Ci},
and Di = Bi ∪ {U 1 ∪ C 1 , ..., Ui ∪ Ci, Ui+1, ..., Un}.
Claim ENT(W(f, (B − {H}) ∪ U(H), G)) ≥ ENT(W(f, B, G)). Let Wn be a (f, B, G)-word of length n and let {qn(j)}p j=1n ⊂ { 1 , ..., n} such that πqn(j)(Wn) = H. Then there exists {Uj }p jn=1 such that Uj ∈ U(H) and a (f, (B − {H}) ∪ U(H), G)-word Qn of length n such that
πi(Qn) = πi(Wn) if i 6 ∈ {qn(j)}p j=1n πi(Qn) = Uj if i = qn(j).
Then let
Ψn : Πn(W(f, (B − {H}) ∪ U(H))G) −→ Πn(W(f, B, G))
be defined by
πi(Ψ(〈Ai〉∞ i=1)) = Ai if Ai ∈ B − {H} πi(Ψ(〈Ai〉∞ i=1)) = H if Ai ∈ U(H).
Then Ψn(Qn) = Wn in the previous definitions of Qn and Wn. Therefore, Ψn is onto so |Πn(W(f, (B − {H}) ∪ U(H), G))| ≥ |Πn(W(f, B, G))|
and the claim now follows. Next, because U 1 , C 1 ∈ (B − {H}) ∪ U(H) and ENT(W(f, {U 1 , C 1 })) = 0, it follows from Theorem 9 that
ENT(W(f, D 1 , G)) = ENT(W(f, (B − {H}) ∪ U(H), G)),
where it is assumed that A = U 1 , B = C 1 and C = U 1 ∪ C 1.
10 C. MOURON
{B 1 , ..., Bk}. Then clearly B ⊂ Bk ∪ {Bk+1} so it follows from Theorem 23 that ENT(f, {A, B k∗}) > 0. Then by the induction hypothesis, it follows that there exist Bi ∈ Bk such that ENT(f, {A, Bi}) > 0.
Corollary 25. Suppose ENT(f, {A, B}) > 0 , A, B are collections of open sets such that A ⊂ A∗^ and B ⊂ B∗. Then there exists A′^ ∈ A and B′^ ∈ B such that ENT(f, {A′, B′}) > 0.
Proof. Since A ⊂ A∗^ and B ⊂ B∗^ it follows that ENT(f, {A∗, B∗}) > 0. Hence by Corollary 24, there exists A′^ ∈ A∗^ such that ENT(f, {A′, B∗}) > 0. But then by an- other application of Corollary 24, there exists B′^ ∈ B∗^ such that ENT(f, {A′, B′}) >
Lemma 26. Suppose that f : X −→ X and that A, B are open sets such that A ∩ B = ∅ and ENT(f, {A, B}) > 0. Then for every positive integer m, there exists a δ > 0 such that if U is a finite open cover of X with mesh(U) < δ, then U has the following properties:
- There exist an integer p(m) and a sequence of positive integers Kp(m) = {k 1 , ..., kp(m)} such that |RKp(m) ({A, B})| ≥ m + 1.
- There exists disjoint {V, U 1 , ..., Um} ⊂ U such that ENT(f, {V, Ui}) > 0 for each i.
- Each element of {V, U 1 , ..., Um} intersects a unique element of RKp(m) ({A, B}).
Proof. Let p(m) be such that m+1 ≤ 22
p(m)
. By Theorem 19, there exist an increas- ing sequence of positive integers G = {g(i)}∞ i=1, r ≥ 1 and a set of (not necessarily distinct) positive integers Kp(m) = {k 1 , ..., kp(m)} such that W(f, RKp(m) ({A, B}), G) is complete on RKp(m) ({A, B}), where g(i) ≤ ri. Since A ∩ B = ∅, it follows that if A′, B′^ are distinct elements of RKp(m) ({A, B}),
then A′^ ∩ B′^ = ∅. So choose δ > 0 and such that
δ <
min {d(A′, B′)|A′^6 = B′^ and A′, B′^ ∈ RKp(m) ({A, B})}.
Let U be a chain cover of X with mesh less than δ. Notice that |RKp(m) ({A, B})| =
22
p(m) ≥ m+1, so pick any set of distinct elements {H, D 1 , ..., Dm} of RKp(m) ({A, B}). Then define U(H) = {U ∈ U|U ∩ H 6 = ∅}
and U(Di) = {U ∈ U|U ∩ Di 6 = ∅}.
Let n = |U(H)| and find α > p(m) such that n + 1 ≤ 22 α / 22 p(m)
. By Theorem 19, there exist positive integers Kα = {k 1 , k 2 , ..., kα} (where the first p(m) terms are the same as before), a subsequence G′^ = {g′(i)}∞ i=1 of G, and r′^ ≥ 1 such that g′(i) ≤ r′i and W(f, RKα ({A, B}), G′) is complete on RKα ({A, B}). Next let Q(H) = {Q ∈ RKα ({A, B})|Q ⊂ H}
and Q(Di) = {Q ∈ RKα ({A, B})|Q ⊂ Di}.
Then |Q(H)| = |Q(Di)| = 2^2
α / 22
p(m) ≥ n + 1.
ENTROPY HOMEOMORPHISMS 11
Let QH be any element of Q(H) and U(QH ) = {U ∩ QH |U ∈ U(H)}. Then |U(QH )| ≤ |U(H)| = n. Also, since W(f, RKα ({A, B}), G′) is complete on RKα ({A, B}), it follows that
ENT(W(f, RKα ({A, B}), G′)) = lim sup i→∞
log(|RKα ({A, B})|i) i = log(|RKα ({A, B})|.
Furthermore, the following are all true
QH ⊂ U(QH )∗
Q(Di) ⊂ RKα ({A, B}) − {QH }
|Q(Di)| ≥ n + 1 > |U(QH )|
For each UQH ∈ U(QH ) and QDi ∈ Q(Vi), UQH ∩ QDi = ∅.
Therefore, by Lemma 22, there exists V ′^ ∈ U(QH ) such that for each i there exists QDi ∈ Q(Di) such that ENT(f, {V ′, QDi }) > 0. Consequently, QDi ⊂ Di and there exists a V ∈ U(H) such that V ′^ ⊂ V. So by Proposition 17, ENT(f, {V, Di}) > 0 for each i. Additionally, it follows from Corollary 24 that there exists Ui ∈ U(Di) such that ENT(f, {V, Ui}) > 0. It follows from the fact that mesh(U) < δ that {V, U 1 , ..., Um} are all disjoint.
The next theorem gives a result about chain covers.
Theorem 27. If C is a chain cover continuum such that ENT(f, C) > log(2), then there exist disjoint open sets Ci, Cj of C such that ENT(f, {Ci, Cj }) > 0.
Proof. The proof is by induction on the number of elements n in C. Since ENT(f, C) > log(2), we can assume that n ≥ 3. Base Case. Suppose n = 3, then by Theorem 10, ENT(f, {C 1 , C 3 }) > 0. Induction Step. Suppose the theorem is true for n = k and let Ck+1 = {C 1 , ..., Ck+1} be a chain cover. If ENT(f, {C 1 , C 3 }) > 0, then we are done; so suppose that ENT(f, {C 1 , C 3 }) = 0. Let C k′ = {C 1 ′, ..., C k′} where C 1 ′ = C 2 , C 2 ′ = C 1 ∪ C 3 and C i′ = Ci+1 for i ≥ 3. Then C′ k is a proper chain cover with k elements and by The- orem 9, ENT(f, C k′) > log(2). Hence, from the induction hypothesis, there exist disjoint C i′, C′ j ∈ C′ k such that ENT(f, {C′ i, C j′ }) > 0. Suppose that i < j; we have 3 cases: Case 1. Suppose that i = 1. Then since C i′ and C j′ are disjoint, j ≥ 3. Hence ENT(f, {C 2 , Cj+1}) = ENT(f, {C 1 ′, C j′ }) > 0.
Since j + 1 ≥ 4, C 2 and Cj+1 must be disjoint and the theorem follows. Case 2. Suppose that i = 2. Then since C i′ and C j′ are disjoint, j ≥ 4. Hence ENT(f, {C 1 ∪ C 3 , Cj+1}) = ENT(f, {C′ 2 , C′ j }) > 0.
Then by Theorem 23, one of the following must be true:
ENT(f, {C 1 , Cj+1}) > 0 or ENT(f, {C 3 , Cj+1}) > 0.
ENTROPY HOMEOMORPHISMS 13
- W 5 = W 6 which is a contradiction in each case.
Lemma 30. Let h : X −→ X be a homoeomorphism, A, B be open sets such that A ∩ B = ∅, and U = {U 1 , ..., Up 1 , ..., Up 2 , ..., Up 6 , ..., Um} be a chain cover of X such that {Upi }^6 i=1 all intersect unique elements of RK 2 ({A, B}) where K 2 = {k 1 , k 2 }. Then there exist 1 < i < k ≤ 6 and n ∈ { 0 , k 1 , k 2 , k 1 + k 2 } such that either
- hn(Up 1 ) ∩ B 6 = ∅, hn(Upk ) ∩ B 6 = ∅, and hn(Upj ) ∩ A 6 = ∅ or
- hn(Up 1 ) ∩ A 6 = ∅, hn(Upk ) ∩ A 6 = ∅, and hn(Upj ) ∩ B 6 = ∅.
Proof. Let Wj = 〈Y 1 j , Y 2 j , Y 3 j , Y 4 j 〉 be the word associated with the unique element
Y 1 j ∩ hk^1 (Y 2 j ) ∩ hk^2 (Y 3 j ) ∩ hk^1 +k^2 (Y 4 j ) = (Y 1 j ∩ hk^1 (Y 2 j )) ∩ hk^2 (Y 3 j ∩ hk^1 (Y 4 j ))
of RK 2 ({A, B}) that intersects Upj , where Y (^) ij ∈ {A, B}. Then by Lemma 29, there
exists 1 < j < k and i′^ ∈ { 1 , ..., 4 } such that Y (^) i^1 ′ = Y (^) ik′ and Y (^) i^1 ′ 6 = Y (^) ij′. First, suppose that Y (^) i^1 ′ = Y (^) ik′ = B and Y (^) ij′ = A. Then using the value of ni′ defined by n 1 = 0, n 2 = k 1 , n 3 = k 2 and n 4 = k 1 + k 2 , it follows that
Up 1 ∩ hni′^ (B) 6 = ∅, Upk ∩ hni′^ (B) 6 = ∅ and Upj ∩ hni′^ (A) 6 = ∅,
so 1) is true. On the other hand, if Y (^) i^1 ′ = Y (^) ik′ = A and Y (^) ij′ = B, then 2) follows in a similar manner. The next lemma shows the relationship between entropy and crookedness.
Lemma 31. Suppose that h : X −→ X is a homeomorphism of a chainable contin- uum and A, B are disjoint elements of proper chain cover C such that ENT(h, {A, B}) >
- Then for every > 0 , there exists a chain cover U and elements U, V ∈ U with the following properties:
- mesh(U) < .
- Ent(h, {U, V }) > 0.
- The subchain from U to V is crooked between A and B.
Proof. Let > 0, m = 9 and p(m) = 2. By Theorem 19, there exist an increasing sequence of positive integers G = {g(i)}∞ i=1, r ≥ 1 and positive integers k 1 , k 2 such that W(h, RK 2 ({A, B}, G) is complete on RK 2 ({A, B}), where K 2 = {k 1 , k 2 }, and g(i) ≤ ri. By uniform continuity, there exists a δ > 0 such that if d(x, y) < δ then d(hn(x), hn(y)) < for all −k 1 − k 2 ≤ n ≤ k 1 + k 2. Let U be a chain cover of X with mesh less than δ. Then by Lemma 26, we can assume that there exist distinct elements {V, U 1 , ..., U 9 } of U such that each element intersects a unique element of RK 2 ({A, B}) and ENT(h, {U, Vi}) > 0 for each i. In the ordering of chain U, at least 5 elements of {U 1 , ..., U 9 } either follow V or precede V. Since each element of RK 2 ({A, B}) is of the form Y 1 ∩ h−k^1 (Y 2 ) ∩ h−k^2 (Y 3 ) ∩ h−k^1 −k^2 (Y 4 ) where Yi ∈ {A, B}, it follows from Lemma 41 that there exists j ∈ { 0 , k 0 , k 1 , k 0 + k 1 } and Ua, Ub such that
- Ua is between V and Ub in the ordering of U and either
- hj^ (V ) ∩ B 6 = ∅, hj^ (Ub) ∩ B 6 = ∅, and hj^ (Ua) ∩ A 6 = ∅ or
14 C. MOURON
2’) hj^ (V ) ∩ A 6 = ∅, hj^ (Ub) ∩ A 6 = ∅, and hj^ (Ua) ∩ B 6 = ∅. Hence, the lemma follows.
The next Theorem is the main result of this paper.
Theorem 32. Suppose that X is a chainable continuum and h : X −→ X is a homeomorphism such that Ent(h) > 0. Then X must contain an indecomposable subcontinuum.
Proof. By Proposition 2, we know that there exists a homeomorphism h such that Ent(h) > log(3). Then by Propositions 1 and 3 and the fact that Ent(h) can be estimated by the entropy on finite cover, there exists a proper chain cover C 0 of X such that ENT(h, C 0 ) ≥ Ent(h, C 0 ) > log(3). Then by Theorem 27, there exist A, B ∈ C 0 such that A ∩ B = ∅ and ENT(h, {A, B}) > 0. Let γ < 13 d(A, B). Continuing inductively suppose that a chain cover Ci has been found with the following properties:
- mesh(Ci) < γ/ 3 i
- There exist Ai, Bi ∈ Ci such that Ai ∩ Bi = ∅ and ENT(h, {Ai, Bi}) > 0. Then by Lemma 31, there exists a chain-cover Ci+1 such that
- Ci+1 closure refines Ci.
- mesh(Ci+1) < γ/ 3 i+
- There exist Ai+1, Bi+1 ∈ Ci+1 such that Ai+1 ∩ Bi+1 = ∅ and
ENT(h, {Ai+1, Bi+1}) > 0.
- The subchain of Ci+1 between Ai+1 and Bi+1 is crooked between Ai and Bi. Since X =
i→∞ Ci, it follows from Theorem 28 that^ X^ contains a nondegenerate indecomposable subcontinuum.
A function f is open if f (U ) is open for each open set U in the domain of f. A function f is monotone if f −^1 (H) is connected for every connected H in the range of f. Homeomorphisms are monotone and open. So the following are natural questions: Question 1: If f : X −→ X is a monotone open continuous function such that Ent(f ) > 0, then must X contain a non-degenerate indecomposable subcontinuum? If so, then consider the next question. Question 2: If f : X −→ X is a monotone continuous function such that Ent(f ) > 0, then must X contain a non-degenerate indecomposable subcontinuum? However, it is well known that the tent map f : [0, 1] −→ [0, 1] defined by
f (x) =
2 x if x ∈ [0, 1 /2] 2 − 2 x if x ∈ (1/ 2 , 1]
in an open continuous function such that Ent(f ) = log(2). Also, the cone over the Cantor set (also known as the Cantor Fan) admits a positive entropy homeomor- phism. The cone over the Cantor set is tree-like but not chainable and does not contain a non-degenerate indecomposable subcontinuum.
