Download Hereditarily Indecomposable Compacta Cannot Admit Expansive Homeomorphisms and more Papers Mathematics in PDF only on Docsity!
HEREDITARILY INDECOMPOSABLE COMPACTA DO NOT ADMIT
EXPANSIVE HOMEOMORPHISMS
HISAO KATO AND CHRISTOPHER MOURON
Abstract. A homeomorphism h : X −→ X of a compactum X is expansive provided that for some fixed c > 0 and every x, y ∈ X(x 6 = y) there exists an integer n, dependent only on x and y, such that d(hn(x), hn(y)) > c. It is shown that if X is a hereditarily indecomposable compactum, then h cannot be expansive.
- Introduction Let X be a compactum with a metric d. A homeomorphism h : X −→ X is expansive provided that for some fixed c > 0 and every distinct x, y ∈ X there exists an integer n, dependent only on x and y, such that d(hn(x), hn(y)) > c. A continuum is a compact, connected metric space. A homeomorphism h is continuum-wise expansive provided that for some fixed c > 0 and every non- degenerate subcontinuum A, there exists an integer n such that diam(hn(A)) > c. A compactum X is 0-dimensional provided that for every > 0 there exists a finite open cover V with mesh less than of X such that every point of X is in exactly 1 element of V. Similarly, a compactum X has dimension ≤ n, dim X ≤ n, provided that for every > 0 there exists a finite open cover V with mesh less than of X such that every point of X is in at most n + 1 elements of V. A compactum X is n-dimensional, dim X = n, provided that dim X ≤ n but not dim X ≤ n − 1. Every n-dimensional compactum must contain an n-dimensional continuum. For n ≥ 1, every expansive homeomorphism of an n-dimensional compactum is continuum-wise expansive, but the converse is not so. It should be noted that h is expansive (continuum-wise expansive) if and only if h−^1 is expansive (continuum-wise expansive). A continuum is decomposable if it is the union of two of its proper subcontinua. A continuum is indecomposable if it is not decomposable. In order for a homeomorphism to be continuum-wise expansive, subcontinua must be continually stretched; this creates indecomposable subcontinua. There is strong evidence that continua that admit expansive homeomorphisms must contain inde- composable subcontinua (see [4] and [7]). However, there is also strong evidence that in order for a homeomorphism to be expansive, subcontinua must be stretched and wrapped around the contin- uum as opposed to stretched and folded. For example, tree-like continua do not admit expansive homeomorphisms [5]. A continuum is hereditarily indecomposable if every subcontinuum is indecomposable. Likewise, a compactum X is hereditarily indecomposable if dim X ≥ 1 and every nondegenerate subcontin- uum is indecomposable. Hereditarily indecomposable continua are created by “infinite folding” of subcontinua. There are examples of 1-dimensional hereditarily indecomposable continua that admit continuum-wise expansive homeomorphisms. However, the infinite folding also raises the entropy of the homeomorphism, and is known that expansive homeomorphisms must have finite entropy. In this paper it will be shown that continuum-wise expansive homeomorphisms on hereditarily
2000 Mathematics Subject Classification. Primary: 54H20, 54F50, Secondary: 54E40. Key words and phrases. expansive homeomorphism, hereditarily indecomposable continuum. 1
2 H. KATO AND C. MOURON
indecomposable compacta must have infinite entropy. It then will be concluded that hereditarily indecomposable compacta cannot admit expansive homeomorphisms. Ma˜n´e has shown that infinite dimensional compacta do not admit expansive homemorphisms [3].
- Chain Covers of Indecomposable Continua Let U be a finite open cover of a compactum X. The mesh of U is defined by mesh(U) = sup{diam(U )|U ∈ U}.
A cover V refines a cover U if for each V ∈ V there exists U ∈ U such that V ⊂ U. V closure refines U if for each V ∈ V there exists U ∈ U such that V ⊂ U. A cover U is taut if U ∩ V = ∅ for all disjoint U, V ∈ U. From here on out, we will assume that all covers are taut. For U ∈ U, the core of U is defined as core(U ) =
{U − V |V ∈ U − {U }}.
Notice that if U is taut, then core(U ) 6 = ∅ for every U ∈ U. The star of U is defined by
U∗^ =
U ∈U
U.
A chain [C 1 , C 2 , ..., Cn] is a collection of open sets such that Ci ∩ Cj 6 = ∅ if and only if |i − j| ≤ 1. The elements of a chain are called links and C 1 and Cn are called end-links for n ≥ 2. Suppose that chain C has 7q links for some positive integer q. Then the subchains of the form Cp = [C 7 p+1, ..., C 7 p+7] where p ∈ { 0 , ..., q − 1 } are called Lucky 7 subchains. A subcontinuum H runs through [C 7 p+1, ..., C 7 p+7] if H ⊂
⋃n i=1 Ci, core(C^7 p+1)^ ∩^ H^6 =^ ∅^ and core(C^7 p+7)^ ∩^ H^6 =^ ∅. Notice that if H intersects at least 15 element of C, then H must run through some Lucky 7 subchain. If m < n are positive integers, then we denote the set {m, m + 1, ..., n} by [m, n]. A function f : [1, m] −→ [1, n] is called a pattern provided |f (i + 1) − f (i)| ≤ 1 for i ∈ { 1 , 2 , ..., m − 1 }. A pattern f : [1, km + 2] −→ [1, m + 2] is a proper simple k-fold provided that
f (i) =
1 if i = 1 (i − 1 mod 2m − 2) + 1 if i ∈ [p(2m − 2) + 2, p(2m − 2) + m + 1] m − 1 − (i − 1 mod 2m − 2) if i ∈ [p(2m − 2) + m + 2, (p + 1)(2m − 2) + 1] m + 2 if i = km + 2, where p ≥ 0. Let V = [V 1 , .., Vm] and U = [U 1 , ..., Un] be chain covers of a compactum X and let f : [1, m] −→ [1, n] be a pattern. We say that V follows pattern f in U provided that Vi ⊂ Uf (i) for each i = 1, ..., m. If f is a proper simple k-fold then we say that V is a k-fold refinement of U (see Figure 1). In a k-fold refinement, the links of the form Vf (i) where f (i − 1) = f (i + 1) are called the bend links of the k-fold. Suppose that U is a taut finite open cover. Then define
d(U) = min{d(U (^) i, U (^) j )|Ui ∩ Uj = ∅ for Ui, Uj ∈ U}.
Lemma 1. Let U = [U 1 , ..., Un] be a taut open chain cover of a continuum Y where n ≥ 7 and suppose that V = [V 1 , ..., Vm] is a k-fold refinement of U. Then there exist k subcontinua {Yi}ki=1 of Y such that diam(Yi) ≥ d(U) and d(Yi, Yj ) ≥ d(V) whenever i 6 = j.
Proof. Let {n(i)}k i=1−^1 be an increasing sequence of integers such that Vf (n(i)) is a bend link of V. Define C 1 = [V 3 , ..., Vf (n(1))− 1 ], Ci = [Vf (n(i−1))+1, ..., Vf (n(i))− 1 ] for 2 ≤ i ≤ k − 1 and Ck = [Vf (n(k−1))+1, ..., Vf (m−2)]. Then for each i there exist a subcontinuum Yi contained in the chain Ci intersects both end-links of Ci. Since the endlinks of Ci are contained in U 3 and Un− 2 , it follows that
4 H. KATO AND C. MOURON
Next we must find ways to cover higher dimensional compacta with chains so that “large” subcontinua must run through some Lucky 7 subchain. The following results give the prescription for this: We define a map g : X −→ Y to be light if g−^1 (y) is 0-dimensional for y ∈ g(X).
Theorem 4 (1). Let X be a compactum. Then dim X ≤ m if and only if there exists a light map g : X −→ Im.
Lemma 5. Let g : X −→ Y be a light map and let X, Y be compacta. For each δ > 0 , there exists a finite open cover Uδ of Y such that if U ∈ Uδ , then every component of g−^1 (U ) has diameter less than δ.
Proof. Suppose on the contrary that there exists a sequence of finite open covers, {Ui}∞ i=1, with the following properties:
(1) mesh(Ui) → 0 as i → ∞. (2) There exists Ui ∈ Ui such that some component, Ci, of g−^1 (Ui) has diameter greater than or equal to δ.
Since X and Y are compacta, we may assume that there is a point y ∈ Y and a subcontinuum C of X such that limi→∞ dH (Ui, y) = 0 and limi→∞ dH (Ci, C) = 0, where dH denotes the Hausdorff metric. Then C ⊂ g−^1 (y) and diam C ≥ δ. This contradicts the fact that g is light.
If g : X −→ Im, then define gi = πi ◦ g where πi : Im^ −→ I is the ith coordinate map.
Theorem 6. Let X be a compactum and let g : X −→ Im^ be a light map and δ > 0. Then there exists a chain cover C of I such that if H is a subcontinuum of X with diam(H) ≥ δ, then there exists i ∈ { 1 , ..., m} such that H runs through some Lucky 7 chain of g i− 1 (C).
Proof. By Lemma 5, there exists a finite cover U of Im^ such that every component of g−^1 (U ) has diameter less than δ for each U ∈ U. Choose C 0 to be a chain cover of I with mesh sufficiently small so that ∩mi=1π i− 1 (C 0 ) refines U. Then choose C to be a chain cover of I with 7q links and with mesh sufficiently small so that every subchain of 14 or fewer links is contained in some element of C 0. If for each i there exists a subchain Vi of C with 14 or fewer links such that gi(H) ⊂ V i∗ then g(H) would be contained in some U ∈ U which would violate the fact that every component of g−^1 (U ) has diameter less than δ. Thus there exists some i such that no subchain of C with 14 or fewer links can completely contain gi(H). Thus, H must run through some Lucky 7 subchain of g− i 1 (C). Note that in general, g i− 1 (C) may not be a chain, but g i− 1 (C) is a disjoint union of finite chains for each i = 1, 2 , .., m. If necessary, we consider such chains.
- Entropy and Expansive Homeomorphisms Entropy is a measure of how fast points move apart. Continuum-wise expansive homeomorphisms have positive entropy. The following definition of entropy is due to Bowen [8]. If h : X −→ X is a map and n a non-negative integer, define
d+ n (x, y) = max 0 ≤i<nd(hi(x), hi(y)). Let K be a compact subset of X and n a positive integer. A finite subset En of K is said to be (n, )-separated with respect to map h if x and y are distinct elements of En implies that d+ n (x, y) > . Let sn(, K, h) denote the largest cardinality of any (n, )-separated subset of K with respect to h. Then
EXPANSIVE HOMEOMORPHISMS 5
s(, K, h) = lim sup n→∞
log sn(, K, h) n
The entropy of h on X is then defined as
Ent(h, X) = sup{lim → 0 s(, K, h)|K is a compact subset of X}.
It can be shown that if h is a homeomorphism then Ent(h−^1 , X) = Ent(h, X). The following are Corollary 2.4 and Proposition 2.5 in [2]. They are stated for a continuum X, but the proof is exactly the same for a compactum X.
Theorem 7. Let h : X −→ X be a continuum-wise expansive homeomorphism on a continuum X. Then there exists a δ > 0 such that for every γ > 0 there exists Nγ > 0 such that if A is a subcontinuum of X with diam(A) > γ then diam(hn(A)) > δ for all n ≥ Nγ or diam(h−n(A)) > δ for all n ≥ Nγ.
Theorem 8. Let h : X −→ X be a continuum-wise expansive homeomorphism on a continuum X. Then there exists a non-degenerate subcontinuum A such that either lim n−→−∞ diam(hn(A)) = 0 or lim n−→∞ diam(hn(A)) = 0.
Corollary 9. Suppose that lim n−→−∞ diam(hn(A)) = 0. Then there exists δ > 0 such that for every
integer m, subcontinuum B ⊂ hm(A) and γ > 0 there exists Nγ > 0 such that if diam(B) > γ, then diam(hn(B)) > δ for every n ≥ Nγ.
Then next theorem is the main theorem of this section:
Theorem 10. Continuum-wise expansive homeomorphisms on hereditarily indecomposable com- pacta have infinite entropy.
Proof. Let h be a continuum-wise expansive homeomorphisms on a hereditarily indecomposable compactum X. Without loss of generality we may assume that there exists a subcontinuum A such that lim n−→−∞ diam(hn(A)) = 0 (otherwise, consider ̂h = h−^1 which is also a continuum-wise
expansive homeomorphisms on X with the same entropy.) Let δ be defined from Corollary 9. Since X admits a continuum-wise expansive homeomorphism, we see that dim X < ∞ (see [2]). By Theorem 4, there exists a light map g : X −→ Im^ where m = dimX. Let C be a chain cover of I with 7q links that satisfies the conclusion of Theorem 6. Let γ = min{d(g− α 1 (C)) | α ∈ { 1 , ..., m}} and let k be any positive integer. By Corollary 3, for each α ∈ { 1 , ..., m} there exists a taut refinement Vαk of g− α 1 (C) such that every Lucky 7 subchain of g− α 1 (C) is refined by a proper k-fold. Let k = min{d(Vαk ) | α ∈ { 1 , ..., m}}. Also, there exists an integer n such that diam(hn(A)) > δ. Let H = hn(A). Then by Theorem 6, for some α ∈ { 1 , ..., m} there exists a Lucky 7 subchain Cαp = [C 7 p+1, C 7 p+2, ..., C 7 p+7] of g α− 1 (C) such that H runs through Cαp. Then by Lemma 1, there exist k subcontinua {H(i)}ki=1 of H such that
(1) diam(H(i)) > γ (2) d(H(i), H(j)) > k for i 6 = j.
It follows from Corollary 9 that
diam(hNγ^ (H(i))) > δ for each i ∈ { 1 , ..., k}.
EXPANSIVE HOMEOMORPHISMS 7
- References [1] M. L. Curtis and G. S. Young, A theorem on dimension, Proc. Amer. Math. Soc. 3 (1952), 159-161.
[2] H. Kato, Continuum-wise expansive homeomorphisms, Can. J. Math. 45 (1993), no. 3, 576-
[3] R. Ma˜n´e, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319.
[4] C. Mouron, Expansive homeomorphisms and indecomposable subcontinua, Topology Appl. 1 26 (2002), no. 1-2, 13–28.
[5] C. Mouron, Tree-like continua do not admit expansive homeomorphisms, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3409-3413.
[6] L. Oversteegen and E. Tymchatyn, On hereditarily indecomposable compacta, Geometric and Algebraic Topology, 407–417, Banach Center Publ., 18 , PWN, Warsaw, 1986.
[7] R.V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys 39 (1974), 85-131.
[8] P. Walters, A Introduction to Ergodic Theory, Graduate Texts in Math. 79 , Springer, New York, 1982.
Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571 Japan E-mail address: hisakato@sakura.cc.tsukuba.ac.jp Department of Mathematics, The University of Alabama at Birmingham, Birmingham, AL 39254 E-mail address: mouron@math.uab.edu Department of Mathematics and Computer Science, Rhodes College, Memphis, TN 38112 E-mail address: mouronc@rhodes.edu
