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Mixing homeomorphisms and indecomposability
Topology and its Applications 254 (2019) 50–58
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Topology and its Applications www.elsevier.com/locate/topol
Mixing homeomorphisms and indecomposability Verónica Martínez-de-la-Vega b , Jorge M. Martínez-Montejano a,∗ , Christopher Mouron c a
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Cd. Universitaria, Ciudad de México, 04510, México b Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Cd. Universitaria, Ciudad de México, 04510, México c Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, United States of America
a r t i c l e
i n f o
Article history: Received 26 August 2018 Received in revised form 18 December 2018 Accepted 24 December 2018 Available online 28 December 2018
a b s t r a c t In this paper we show that if h : X → X is a mixing homeomorphism on a G-like continuum, then X must be indecomposable and if X is finitely cyclic, then X must be n1 -indecomposable for some natural number n. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum. © 2018 Elsevier B.V. All rights reserved.
MSC: primary 54H20, 54F15 secondary 54E40 Keywords: Mixing homeomorphism 1 n -indecomposable G-like continuum
1. Introduction There are many examples and results showing that in order for a homeomorphism on a continuum to have “interesting” or “chaotic” dynamics, the continuum must have complicated topological structure (see [1], [8], [10], [13] and [15]). In this paper we contribute to this, in particular, we show the following: Theorem 2.2. Let X be an m-cyclic continuum. If there exists a mixing homeomorphism h : X → X, then 1 X must be m+2 -indecomposable. * Corresponding author. E-mail addresses: [email protected] (V. Martínez-de-la-Vega), [email protected] (J.M. Martínez-Montejano), [email protected] (C. Mouron). https://doi.org/10.1016/j.topol.2018.12.012 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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Theorem 2.9. Let X be a G-like continuum where G is a graph with m ≥ 0 cycles. If h : X → X is a mixing homeomorphism, then X must be indecomposable. Example (Section 3). There exists a mixing homeomorphism on a hereditarily decomposable tree-like continuum. To give the results in this paper some context, let us examine a few theorems on some related definitions: In [13], the third named author shows that positive entropy homeomorphisms do not exist on hereditarily decomposable chainable continua. This result has been extended to G-like continua, where G is a graph, by Darji and Kato [6]. Theorem 2.2 and Theorem 2.9 generalize these results. Boroński and Oprocha [4] and Boroński and Kupka [3] have exhibited examples of Li–Yorke chaotic homeomorphisms on hereditarily decomposable continua; our example extends these results in the sense that the chaosity is much stronger than just Li–Yorke chaos. It is known that every positively continuum-wise fully expansive map is mixing, however the converse is not true [9]. It was shown by Kato [7] that only indecomposable continua admit positively continuum-wise fully expansive homeomorphisms. Furthermore, it has been shown [11] that in order for a finitely cyclic continuum to admit an expansive homeomorphism, then the continuum must contain a nondegenerate indecomposable subcontinuum. However, tree-like continua cannot admit expansive homeomorphisms [12]. The continuum given in the example is obviously not G-like but it is finitely cyclic. It was the first example known to the authors of a hereditary decomposable continuum that admits some form of a “globally chaotic” homeomorphism. Recently, in [5], Boroński and Oprocha have shown that the Sierpiński carpet admits a mixing homeomorphism; their result shows that our theorems are optimal in the sense that finite cyclicity is necessary for n1 -indecomposability. 2. Mixing homeomorphisms, indecomposability and
1 -indecomposability n
A continuum is a compact connected metric space with more than one point. A map is a continuous function. Given a continuum X, a map f : X → X is mixing if for every pair of nonempty open sets U, V of X, there exists a positive integer M such that f m (U ) ∩ V = ∅ for all m ≥ M . That is, if U is open, then dH (f n (U ), X) → 0 as n → ∞, where dH is the Hausdorff metric. A graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both of their end points. A tree is a graph that contains no simple closed curve. Given a graph G, a G-cover is an open cover whose nerve is G. A continuum is G-like if for every ε > 0 there exists a G-cover with mesh less than ε; equivalently, if it is the inverse limit of the same graph G. A continuum is finitely cyclic if it is the inverse limit of graphs {Gi }∞ i=1 such that the number of simple closed curves of Gi is bounded as i varies. This is equivalent to the following: there exists k such that for every ε > 0 there exist an open cover U with mesh less than ε and whose nerve has at most k simple closed curves. When we have a particular k, we say that the continuum is k-cyclic. A continuum is tree-like if it is the inverse limit of trees. A continuum is indecomposable if every proper subcontinuum has empty interior. The Knaster bucket handle continuum is an example of an indecomposable continuum. Generalizing this notion, given a positive integer n, a continuum is n1 -indecomposable if whenever {Ai }ni=1 is a collection of pairwise disjoint, nonempty subcontinua, at least one of A1 , . . . , An has empty interior. Note: 21 -indecomposable is also called semi-indecomposable. The cone over the Cantor set (also known as the Cantor fan) is an example of a semi-indecomposable continuum. First, we must establish some results on n1 -indecomposability. To do this we must look at open covers of continua in a precise way.
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A chain C = [C1 , ..., Cn ] is a finite collection of open sets of X satisfying that Ci ∩ Cj = ∅ if and only if |i − j| ≤ 1. C is a weak chain if Ci ∩ Ci+1 = ∅ for each i. A circle-chain C = [C1 , ..., Cn ] is a finite collection of open sets of X satisfying that Ci ∩ Cj = ∅ if and only if |i − j| ≤ 1 or i, j ∈ {0, n}. Let U be a collection of open sets. We say that U is connected if for every A, B ∈ U, there exists a weak chain in U from A to B. We say that V refines U if for every V ∈ V there exists U ∈ U such that V ⊂ U . Also, let U ∗ = U ∈U U . We say that two chains C and D are pairwise disjoint if C ∗ ∩ D∗ = ∅. If U is an open cover of X and Y ⊂ X, then define U(Y ) = {U ∈ U | Y ∩ U = ∅}. A cover U of X is taut if Ui ∩ Uj = ∅ if and only if Ui ∩ Uj = ∅ for every Ui , Uj ∈ U. U is a minimal cover for X if for every U ∈ U there exists x ∈ X such that x ∈ / U∗ \ U. Given a cover U of X, we say that U ∈ U is a ramification link if there exist at least three different elements U1 , U2 , U3 of U such that Ui ∩ U = ∅ and U = Ui for each i ∈ {1, 2, 3}. A free chain in U is a chain with no ramification links. The first theorem shows how semi-indecomposable continua are realized by disjoint approximating subcontinua. Theorem 2.1. Let X be an m-cyclic continuum. If for every ε > 0 there exists a collection {Aε1 , . . . , Aεm+2 } of pairwise disjoint subcontinua of X such that, for each i ∈ {1, . . . , m + 2}, dH (Aεi , X) < ε, then X is semi-indecomposable. Proof. Suppose on the contrary that X has two disjoint subcontinua, H and K, that have nonempty interior. Let xH ∈ int(H) and xK ∈ int(K). Then there exist εH > 0 and εK > 0 such that BεH (xH ) ⊂ int(H) and BεK (xK ) ⊂ int(K). Let 0 < ε < (1/3) min{εH , εK , d(H, K)} and U be a minimal taut open cover of X with mesh less than ε and whose nerve contains at most m simple closed curves. Note the following facts: (1) (2) (3) (4) (5)
U ∩ V = ∅ for every U ∈ U(H) and V ∈ U(K). U(H) is a connected cover of H. U(K) is a connected cover of K. Let UH be an element of U(H) that contains xH . Then UH ⊂ int(H) ⊂ H. Let UK be an element of U(K) that contains xK . Then UK ⊂ int(K) ⊂ K.
Let λ be a Lebesgue number for U. By hypothesis, there exist disjoint subcontinua, Aλ1 , . . . , Aλm+2 , such that, for each i ∈ {1, . . . , m + 2}, dH (X, Aλi ) < λ. Since U is minimal Aλi ∩ U = ∅ for every U ∈ U. Define γ = min{dH (Aλi , Aλj ) : i = j} and choose ∗
∗
0 < δ < (1/3) min{λ, γ, dH (H, X \ U(H) ), dH (K, X \ U(K) )}. Let V be a minimal taut open cover of X with mesh less that δ and whose nerve contains at most m simple closed curves. Notice the following facts: (1) (2) (3) (4) (5) (6) (7)
V refines U, V(H) refines U(H), and V(K) refines U(K). V ∩ V = ∅ for every V ∈ V(H) and V ∈ V(K). V ∩ V = ∅ for every V ∈ V(Aλi ) and V ∈ V(Aλj ) if i = j. For each i, there exists VHi ∈ V(Aλi ) such that VHi ⊂ UH . For each i, there exists VKi ∈ V(Aλi ) such that VKi ⊂ UK . Since for each i, VHi ⊂ UH ⊂ H; it follows that V(H) ∩ V(Aλi ) = ∅. Since for each i, VKi ⊂ UK ⊂ K; it follows that V(K) ∩ V(Aλi ) = ∅.
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i H H Since V(H) is a connected cover, for each i ∈ {1, . . . m + 1} there exists a chain CH = [C(i,1) , ..., C(i,n ]⊂ i) H λ H λ H λ λ V(H) such that C(i,1) ∈ V(Ai ), C(i,ni ) ∈ V(Ai+1 ), and C(i,j) ∈ / V(Ai )∪V(Ai+1 ) for all j such that j ∈ / {1, ni }. i K K K K Likewise, there exists a chain CK = [C(i,1) , ..., C(i,m ] ⊂ V(K) such that C(i,1) ∈ V(Aλi ), C(i,m ∈ V(Aλi+1 ), i) i) K and C(i,j) ∈ / V(Aλi ) ∪ V(Aλi+1 ) for all j such that j ∈ / {1, mi }. Since for each i ∈ {1, . . . , m + 2} the covers
V(Aλi ) are connected and pairwise disjoint, there exist pairwise disjoint chains CAi = [C1Ai , ..., CpAii ] in V(Aλi ), A Ai+1 H K H K such that for i ∈ {1, . . . , m + 1}, C1Ai = C(i,1) , CpAii = C(i,1) , C1 i+1 = C(i,n , and Cpi+1 = C(i,m . Hence, it i) i) i i can be easily verified that for each i ∈ {1, . . . , m + 1}, Di = CH ∪ CAi ∪ CK ∪ CAi+1 is a circle-chain contained in V and that each D1 , . . . , Dm+1 are different circle-chains in V. This is a contradiction to the fact that the nerve of V has at most m simple closed curves. Thus, X is semi-indecomposable. 2 Now we can immediately apply the first theorem to mixing homeomorphisms to get one part of our main result: Theorem 2.2. Let X be an m-cyclic continuum. If there exists a mixing homeomorphism h : X → X, then 1 X must be m+2 -indecomposable. 1 Proof. Suppose to the contrary that X is not m+2 -indecomposable. Then there exists a collection {A1 , . . . , Am+2 } of pairwise disjoint subcontinua of X that have non empty interior. Let ε > 0. Since h is mixing, there exists a positive integer N such that for each i ∈ {1, . . . , m + 2}, dH (hN (Ai ), X) ≤ dH (hN (int(Ai )), X) < ε. We have that the collection {hN (A1 ), . . . , hN (Am+2 )} satisfies the conditions of 1 Theorem 2.1. Hence, X is semi-indecomposable and therefore m+2 -indecomposable, which is a contradiction. 2
Now, we will examine “how much” decomposition a continuum might have. Given a continuum X and a n positive integer n, a collection of n subcontinua of X, {Ai }ni=1 , is an n-decomposition for X if X = i=1 Ai , but i=j Ai is a proper subset of X for each j ∈ {1, . . . , n}. We say that X is n-indecomposable if there exists an n-decomposition but no (n + 1)-decomposition for X. Proposition 2.3. Let n be a positive integer. If a continuum X is n-indecomposable, then there exists an n-decomposition into indecomposable subcontinua with nonempty interior. Proof. Take {Ai }ni=1 an n-decomposition for X. By Zorn’s lemma, we may assume that {Ai }ni=1 is a i is a proper subcontinuum of Ai , then minimal decomposition; that is, for each i ∈ {1, . . . , n}, if A i ∪ Aj = X. Note that, for each i ∈ {1, . . . , n}, int(Ai ) = ∅ because X \ Aj is a nonempty A j=i
j=i
open subset of X contained in Ai . Suppose to the contrary that there is i ∈ {1, . . . , n} such that Ai is decom posable. Then there exists proper subcontinua, A1i and A2i , such that Ai = A1i ∪A2i . Then A1i ∪ j=i Aj = X, A2i ∪ j=i Aj = X but A1i ∪ A2i ∪ j=i Aj = X. Hence, {A1i , A2i } ∪ {Aj }j=i is an (n + 1)-decomposition. However, this contradicts the fact that X is n-indecomposable. 2 Lemma 2.4. Let X be a G-like continuum where G is a graph, G a G-cover of X, and C = [C1 , . . . , Cs ] a free chain of G. If A is a proper non-degenerate subcontinuum of X such that A ∩C ∗ = ∅ and A ∩C1 = ∅ = A ∩Cs , then G(A) ⊂ C. Proof. Suppose to the contrary that G(A) is not contained in C. Then there exists Ci ∈ G(A) ∩(C \{C1 , Cs }) and U ∈ G(A) \ C. Furthermore, since G(A) is connected there exists a weak chain E from Ci to U such that C1 , Cs ∈ / E. Therefore, there exists a Cj ∈ E ∩ [C2 , ..., Cs−1 ] such that Cj intersects at least 3 distinct elements of G − {Cj }. However, this contradicts the fact that C is a free chain. 2
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Lemma 2.5. Let C = [C1 , . . . , Cs ] be a chain in a continuum X and {A1 , . . . , Ak } be a collection of subcontinua of X such that for every i ∈ {1, . . . , k}, Ai ⊂ C ∗ . If for each i ∈ {1, . . . , k} there exists an element Cmi ∈ C such that Cmi ⊂ Ai \ ∪j=i Aj and mi < mj whenever i < j, then Ai ∩ Aj = ∅ if |i − j| > 1. Proof. Suppose to the contrary that Ai ∩Aj = ∅ with |i − j| > 1. We may assume without loss of generality that i < j. Take t ∈ {1, . . . , k} such that i < t < j. We have that C(Ai ∪Aj ) is a chain that contains Cmi , Cmj and each Ck for mi < k < mj . Thus, Cmt ∈ C(Ai ∪ Aj ) and it follows that Cmt ∩ (Ai ∪ Aj ) = ∅ which is a contradiction. 2 The next lemma relates G-like
1 n -indecomposable
continua to k-decomposability.
Lemma 2.6. Let G be a graph with N maximal free arcs and r ramification points. Suppose that X is G-like and n1 -indecomposable. Then X is k-indecomposable for some 1 ≤ k ≤ M with M = 2N (n + 1) + r. +1 Proof. Suppose on the contrary that there exists an (M + 1)-decomposition, {Ai }M Since i=1 , for X. int(Ai ) = ∅, there exist ε > 0 and a collection of points {a1 , . . . , aM +1 } such that ai ∈ B(ε, ai ) ⊂ Ai \ j=i Aj . Define γ = min{d(ai , Aj ) : i = j} and let 0 < δ < 13 min{ε, γ}. Let G be a minimal G-cover of X with mesh less than δ. Since G has N maximal free arcs, we have that G has a collection {C1 , . . . , CN } of pairwise disjoint maximal free chains and a collection {U1 , . . . , Ur } of ramification links such that G = C1 ∪ . . . ∪ CN ∪ {U1 , . . . , Ur }. By our choice of δ, we have that there are M + 1 − r points of {a1 , . . . , aM +1 } that belong in C1 ∪ . . . ∪ CN . Then, by the pigeon-hole principle, there exists a k ∈ {1, . . . , N } such that Ck = [C1 , . . . , Cs ] contains i the elements of the decomposition, 2(n+1) different points { a1 , . . . , a2(n+1) } ⊂ {a1 , . . . , aM +1 }. Denote by A M +1 {Ai }i=1 , such that ai ∈ Ai . We may assume, without loss of generality, that if ai ∈ Cji , ar ∈ Cjr and i < r, then ji < jr . Furthermore, by the choice of δ we have
i \ ai ∈ Cji ⊂ B(ε, ai ) ⊂ A
j . A
j=i
Consider the chain D = [Cj1 , . . . , Cj2(m+2) ] in Ck , where a1 ∈ Cj1 and a2(n+1) ∈ Cj2(n+1) . By Lemma 2.4, ∗ 2 , A 4 , . . . , A 2n } consists of for each i ∈ {2, . . . 2(n + 1) − 1}, Ai ⊂ D . By Lemma 2.5, the collection {A n pairwise disjoint subcontinua of X, each with non-empty interior. This contradicts the fact that X is 1 n -indecomposable. Hence, X is k-indecomposable for some k ≤ M . 2 A continuum X is k-coherent if whenever A, B are subcontinua of X such that A ∪ B = X it follows that A ∩ B has at most k components. A continuum X is hereditarily k-coherent if every subcontinuum is k-coherent. A continuum is called unicoherent if it is 1-coherent. Note that every tree-like continuum is hereditarily unicoherent. Proposition 2.7. If X is a k-cyclic continuum, then X is hereditarily k + 1-coherent. Proof. Suppose that X is k-cyclic and, for a contradiction, that there exist subcontinua A, B such that A ∩B 1 has k + 2 distinct components {Hi }k+2 i=1 . Let 0 < ε < 3 mini=j {d(Hi , Hj )} and U be a taut minimal open cover of X with mesh less than ε. Since U(A) is connected there exist k + 1 distinct chains {A(in , jn )}k+1 n=1 in U(A) such that (1) in = jn where in , jn ∈ {1, ..., k + 1}. (2) A(in , jn ) = [U1n , ..., Upnn ] where U1n ∩ Hin = ∅ and Upnn ∩ Hjn = ∅. (3) Utn ∩ (A ∩ B) = ∅ for all t ∈ {2, ..., pn − 1}.
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(Note: the above statement follows from the fact that every connected graph with M vertices must have at least M − 1 edges.) For each n ∈ {1, ..., k + 1}, let Bn be a chain in U(B) whose endlinks are U1n and Upnn and notice that A(in , jn ) ∩ Bn = {U1n , Upnn }. Hence, Cn = A(in , jn ) ∩ Bn = {U1n , Upnn } is a circle-chain. Since, A(it , jt ) ⊂ Cn if and only if t = n, it follows that {Cn }k+1 n=1 is a collection of k + 1 distinct circle-chains in U. Since ε was chosen arbitrarily, this contradicts the fact that X is k-cyclic. 2 Proposition 2.8. Suppose that {Ai }ni=1 is an n-decomposition of a finitely cyclic continuum X where each Ai is indecomposable. If h : X → X is a homeomorphism, then for each i there exists an Ni such that hNi (Ai ) = Ai . Proof. Since, Ai and hp (Ai ) have nonempty interior for each i, there exists a ji ∈ {1, . . . , n} such that int(hp (Ai ) ∩Aji ) = ∅. Since X is finitely cyclic and hence hereditarily k-coherent for some positive integer k, hp (Ai ) ∩ Aji must have a component C with nonempty interior. Since, Aji and hp (Ai ) are indecomposable, it follows that C cannot be a proper subcontinuum of either Aji or hp (Ai ). Hence, Aji = C = hp (Ai ). It now follows, from the pigeon-hole principle, that at least two of Ai , h(Ai ), ..., hn (Ai ), say hp (Ai ) and hq (Ai ), are the same. Let Ni = q − p then hNi (Ai ) = Ai . 2 Now we may conclude the second part of our main result: Theorem 2.9. Let X be a G-like continuum where G is a graph with m ≥ 0 cycles. If h : X → X is a mixing homeomorphism, then X must be indecomposable. Proof. It follows from Theorem 2.2 and Lemma 2.6 that X is 1j -indecomposable for some j and hence X is k-indecomposable for some k. Thus, it follows from Proposition 2.3 that there exists a k-decomposition, {Ai }ki=1 , for X where each Ai is an indecomposable subcontinuum of X with nonempty interior. If k ≥ 2, then A1 is a proper subcontinuum of X with nonempty interior. Hence, since h is mixing, dH (hn (A1 ), X) → 0 as n → ∞. However, it follows from Proposition 2.8 that there is an N such that hN (A1 ) = A1 for some N = 0. So dH (hpN (A1 ), X) = dH (A1 , X) > 0 for all p. This contradicts the fact that dH (hn (A1 ), X) → 0 as n → ∞. Hence, X is indecomposable. 2 3. Example of a mixing homeomorphism on a hereditary decomposable tree-like continuum We are going to construct a hereditary decomposable tree-like continuum and a mixing homeomorphism simultaneously by taking the inverse limit of the universal dendrite with the same bonding map and using the shift homeomorphism. Let Dω be the universal dendrite. Let D(a, b) be the dyadic rationals on (a, b), that is, D(a, b) =
p
∈ (a, b) : p is an odd integer and n is a non-negative integer . 2n
If s ∈ D(a, b), then ρ(s) = n where s = where
p 2n .
C is a dyadic Dω comb on [a, b] if C = [a, b] ∪
s∈D(a,b)
Dωs
1 (1) Dωs is a universal dendrite with diameter 2ρ(s) . s (2) [a, b] ∩ Dω = {s} is a ramification point of C.
∞ Here [a, b] is called the spine of the comb. Let 0, n1 n n=1 be a collection of distinct arcs, let Cn be a
ω (1) = ∞ Cn such that 0 is identified from each dyadic Dω ω = D dyadic Dω comb on 0, n1 n and let D n=1 ω (j) ⊂ D ω (j) = ∞ Cn . Note that each ω such that D comb–that is, Ci ∩ Cj = {0} whenever i = j. Let D n=j ω (j) is homeomorphic to the universal dendrite Dω . Define g : D ω → D ω in the following way (see Fig. 1): D
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Fig. 1. g : Dω → Dω is mixing. Note that hi = g|Ci : Ci → Ci−1 is a homeomorphism for i ≥ 2 and M = g|C1 : C1 → that the corresponding colors are mapped homeomorphically with M ([0, 1]1 ) = {0}.
∞ i=2
Ci such
(1) g(0) = 0.
1 (2) Ci is mapped homeomorphically onto Ci−1 for i ≥ 2 such that g 0, 1i i = 0, i−1 . i−1 s (3) Let C1 = [0, 1]1 ∪ s∈D(0,1) Dω (1). Then (a) g([0, 1]1 ) = 0, ω (ρ(s) + 1). (b) Dωs (1) is mapped homeomorphically onto D Note the following facts about g: ω (j)) → 0 as j → ∞ and diam(Ds (1)) → 0 and ρ(s) → ∞, hence g is continuous. (1) diam(D ω ω such that 0 ∈ (2) Since g(0) = 0, if A ⊂ D / A, then 0 ∈ / g −n (A) for each n ∈ N. (3) If [0, x] is an arc from 0 to x, then g([0, x]) = [0, g(x)] is an arc from 0 to g(x). (Let [0, x] ∩ [0, 1]1 = [0, t]. Then g([0, t]) = 0 and g|[t,x] is a homeomorphism.) ω (j)) = D ω (j − 1) for j ≥ 2. (4) g(D ω such that 0 ∈ Y , then g −1 (Y ) is connected. (5) If Y is a connected subset of D (6) If Y is a subcontinuum such that Y ∩ [0, 1]1 = ∅, then g|Y is a homeomorphism. ω. Proposition 3.1. g ρ(s)+1 (Dωs (1)) = D ω (ρ(s) + 1) and g ρ(s) (D ω (ρ(s) + 1)) = D ω (1). 2 Proof. This follows from the fact that g(Dωs (1)) = D ω − {a, b}} and let S([a, b]) ∈ C(a, b) such that [a, b] ⊂ S([a, b]). Let C(a, b) = {C | C is a component of D S([a, b]) is called the subdendrite of Dω strung by [a, b]. ω. Proposition 3.2. If [a, b] ⊂ [0, 1]1 (a = b), then there exists n such that g n (S([a, b])) = D Proof. Since D([0, 1]) is dense in [0, 1], there exists s ∈ D([0, 1]) such that s ∈ (a, b). Hence, Dωs (1) ⊂ S([a, b]) and the proposition follows from Proposition 3.1. 2 ∞ Let L1 = n=1 [0, 1/n]n . Then 0 ∈ L1 = g(L1 ). So it follows that L1 ⊂ g −1 (L1 ) and g −1 (L1 ) is connected. Hence, g −n (L1 ) ⊂ g −n−1 (L1 ) for each n. Let L2 = g −1 (L1 ) and continuing inductively define Lk = g −1 (Lk−1 ). Then 0 ∈ Lk and Lk is connected. Proposition 3.3. If [a, b] ⊂ Lk − Lk−1 for some k, then there exists m such that g m ([a, b]) is a nondegenerate subarc of [0, 1]1 . Proof. Suppose that [a, b] ⊂ L1 . There are two cases:
Case 1: There exists an n such that [a, b] ⊂ 0, n1 n . Then g n−1 ([a, b]) is a nondegenerate subarc of [0, 1]1 .
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1
Case 2: There exist m, n such that a ∈ 0, m and b ∈ 0, n1 n . m n−m ([a, b]) is a nondegenerate subarc of Without loss of generality, we may assume m < n. Then g 1 and the result follows from Case 1. 0, n−m n−m
Next suppose that [a, b] ⊂ Lp where p > 1. Then there exists q such that [a, b] ⊂ Cq . Then g q−1 |[a,b] is a homeomorphism, so g q−1 ([a, b]) a subarc of C1 that does not intersect [0, 1]1 . Hence, g q ([a, b]) is a non-degenerate subarc of Lp−1 . Now the result follows by induction. 2 ω = ∞ Ln . Notice that D n=1 The following is a well known fact about dendrites. A good starting point to learn more basic facts about dendrites is [14]. Theorem 3.4. If X is a dendrite and ε > 0, then there exists at most a finite number of pairwise disjoint subcontinua with diameter greater than ε. Proof. This follows from the fact that dendrites are hereditarily locally connected and hence do not contain a subcontinuum of convergence. 2 Corollary 3.5. Let [x, y] be a subarc of dendrite X and {xn }∞ n=1 be a sequence in (x, y) such that xn → x and x < xn+1 < xn in the ordering of [x, y]. Then limn→∞ S([x, xn ]) = 0. Proof. Let ε > 0. There is N1 such that diam([x, xn )] < 3ε for all n ≥ N1 . Furthermore, there exists N2 such that diam(S([xn , xn+1 ])) < 3ε for every n ≥ N2 , otherwise, Theorem 3.4 above is violated. Let N = ∞ max{N1 , N2 }. Then S([x, xN ]) = {x} ∪ n=N S([xn , xn+1 ]) and it follows that diam(S([x, xN ])) < 3ε . 2 ω , then there exists an arc [y, x] and a k such that [y, x] ⊂ Proposition 3.6. If U is an open subset of D Lk − Lk−1 and S([y, x]) ⊂ U . ∞ ω , there exists m = 1 such that U ∩ Lm = ∅. Let x ∈ U ∩ Lm and Proof. Since n=1 Ln dense in D k be the largest integer such that x ∈ Lk . Notice that [0, x] ∩ Lk−1 is an arc, say [0, z]. So it follows that (z, x] ⊂ Lk − Lk−1 . Let {xn } be a sequence in (z, x) such that xn → x. Then diam(S([xn , x])) → 0 by Corollary 3.5. Hence, there exists an N such that S([xN , x]) ⊂ U . Let y = xN and the proposition follows. 2 ω → D ω is mixing. Theorem 3.7. g : D Proof. Let U be open. Then it follows from Proposition 3.6 that there is an k and an arc [a, b] such that S([a, b]) ⊂ U ∩Lk −{0}. Then there exists a p such that g p |S([a,b]) is a homeomorphism and g p ([a, b]) ⊂ [0, 1]1 . There exists s ∈ g p ((a, b)) ∩D(0, 1). Then Dωs (1) ⊂ g p (S([a, b])). Hence g p+ρ(s)+1 (U ) = g p+ρ(s)+1 (S([a, b])) = ω . Hence, g is mixing. 2 g p+ρ(s)+1 (Dωs (1)) = D ω , g}∞ be an inverse limit of universal dendrites. Since g is mixing, the shift homeomor{D Let D = lim n=1 ←−− ∞ phism g : D → D defined by g( xi ∞ i=1 ) = g(xi ) i=2 is mixing. Lemma 3.8. D is hereditary decomposable. Proof. Let A be a subcontinuum of D. Then there exists subcontinua {Ai }∞ i=1 of Dω such that A = ∞ lim{Ai , gi }i=1 where gi = g|Ai+1 . ←−−
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Case 1: Suppose that 0 ∈ Ai for every i. ˆ ω is locally connected, we may assume that d is a convex metric for D ˆ ω [2]. Define B1 = Since D B(ε, 0) ∩ A1 , then by hereditary unicoherence, B1 is a proper subcontinuum of A1 and 0 ∈ intA1 (B1 ). Let B2 = g −1 (B1 ) ∩ A2 . By Fact (5) of g (preceding Proposition 3.1), 0 ∈ g −1 (B1 ) and g −1 (B1 ) is a continuum. So, again by hereditary unicoherence, B2 is a subcontinuum of A2 and 0 ∈ B2 . Define Bi+1 = g −1 (Bi ) ∩ Ai+1 , then gi−1 (Bi ) = Bi+1 . Since A ∩ π1−1 (B(ε, 0)) ⊂ B, it follows that B = lim ←−{Bi , gi | Bi+1 } is a proper subcontinuum of A with non-empty interior. Since A contains a proper subcontinuum with non-empty interior, it follows that A is decomposable. / Ak . Case 2: Suppose that there exists k such that 0 ∈ Since g([0, 1]1 ) = {0}, it follows that Aj ∩ [0, 1]1 = ∅ for every j > k. Thus, gj is a homeomorphism for all j > k. Since, Aj is locally connected for each j, it follows that A is locally connected and hence decomposable. 2 Note: D is semi-indecomposable. Hence the following question is natural: Question. If X is a one-dimensional continuum that admits a mixing homeomorphism, must X contain a semi-indecomposable subcontinuum? Acknowledgements This research was partially supported by the project “Teoría de Continuos, Hiperespacios y Sistemas Dinámicos II” (IN101216) of PAPIIT, DGAPA, UNAM. References [1] Marcy Barge, Joe Martin, Chaos, periodicity, and snakelike continua, Trans. Am. Math. Soc. 289 (1) (1985) 355–365. [2] R.H. Bing, A convex metric for locally connected continuum, Bull. Am. Math. Soc. 55 (8) (1949) 812–819. [3] Jan P. Boroński, Jiří Kupka, New chaotic planar attractors from smooth zero entropy interval maps, Adv. Differ. Equ. 2015 (2015) 232. [4] Jan P. Boroński, Piotr Oprocha, On indecomposability in chaotic attractors, Proc. Am. Math. Soc. 143 (8) (2015) 3659–3670. [5] Jan P. Boroński, Piotr Oprocha, On dynamics of the Sierpiński carpet, C. R. Math. Acad. Sci. Paris 356 (3) (2018) 340–344. [6] Udayan Darji, Hisao Kato, Chaos and indecomposability, Adv. Math. 304 (2017) 793–808. [7] Hisao Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topol. Appl. 53 (3) (1993) 239–258. [8] Hisao Kato, Continuum-wise expansive homeomorphisms, Can. J. Math. 45 (3) (1993) 576–598. [9] Logan Hoehn, Christopher Mouron, Hierarchies of chaotic maps on continua, Ergod. Theory Dyn. Syst. 34 (6) (2014) 1897–1913. [10] Christopher Mouron, Mixing sets, positive entropy homeomorphisms and non-Suslinean continua, Ergod. Theory Dyn. Syst. 36 (7) (2016) 2246–2257. [11] Christopher Mouron, Expansive homeomorphisms and indecomposable subcontinua, Topol. Appl. 126 (1–2) (2002) 13–28. [12] Christopher Mouron, Tree-like continua do not admit expansive homeomorphisms, Proc. Am. Math. Soc. 130 (Nov. 2002) 3409–3413. [13] Christopher Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, Proc. Am. Math. Soc. 139 (8) (2011) 2783–2791. [14] Sam B. Nadler Jr., Continuum Theory. An Introduction, Monogr. Textb. Pure Appl. Math., vol. 158, Marcel Dekker, Inc., New York, 1992. [15] Gerald T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Am. Math. Soc. 4 (1990) 1025–1030.
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Topology and its Applications www.elsevier.com/locate/topol
Mixing homeomorphisms and indecomposability Verónica Martínez-de-la-Vega b , Jorge M. Martínez-Montejano a,∗ , Christopher Mouron c a
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Cd. Universitaria, Ciudad de México, 04510, México b Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Cd. Universitaria, Ciudad de México, 04510, México c Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, United States of America
a r t i c l e
i n f o
Article history: Received 26 August 2018 Received in revised form 18 December 2018 Accepted 24 December 2018 Available online 28 December 2018
a b s t r a c t In this paper we show that if h : X → X is a mixing homeomorphism on a G-like continuum, then X must be indecomposable and if X is finitely cyclic, then X must be n1 -indecomposable for some natural number n. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum. © 2018 Elsevier B.V. All rights reserved.
MSC: primary 54H20, 54F15 secondary 54E40 Keywords: Mixing homeomorphism 1 n -indecomposable G-like continuum
1. Introduction There are many examples and results showing that in order for a homeomorphism on a continuum to have “interesting” or “chaotic” dynamics, the continuum must have complicated topological structure (see [1], [8], [10], [13] and [15]). In this paper we contribute to this, in particular, we show the following: Theorem 2.2. Let X be an m-cyclic continuum. If there exists a mixing homeomorphism h : X → X, then 1 X must be m+2 -indecomposable. * Corresponding author. E-mail addresses: [email protected] (V. Martínez-de-la-Vega), [email protected] (J.M. Martínez-Montejano), [email protected] (C. Mouron). https://doi.org/10.1016/j.topol.2018.12.012 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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Theorem 2.9. Let X be a G-like continuum where G is a graph with m ≥ 0 cycles. If h : X → X is a mixing homeomorphism, then X must be indecomposable. Example (Section 3). There exists a mixing homeomorphism on a hereditarily decomposable tree-like continuum. To give the results in this paper some context, let us examine a few theorems on some related definitions: In [13], the third named author shows that positive entropy homeomorphisms do not exist on hereditarily decomposable chainable continua. This result has been extended to G-like continua, where G is a graph, by Darji and Kato [6]. Theorem 2.2 and Theorem 2.9 generalize these results. Boroński and Oprocha [4] and Boroński and Kupka [3] have exhibited examples of Li–Yorke chaotic homeomorphisms on hereditarily decomposable continua; our example extends these results in the sense that the chaosity is much stronger than just Li–Yorke chaos. It is known that every positively continuum-wise fully expansive map is mixing, however the converse is not true [9]. It was shown by Kato [7] that only indecomposable continua admit positively continuum-wise fully expansive homeomorphisms. Furthermore, it has been shown [11] that in order for a finitely cyclic continuum to admit an expansive homeomorphism, then the continuum must contain a nondegenerate indecomposable subcontinuum. However, tree-like continua cannot admit expansive homeomorphisms [12]. The continuum given in the example is obviously not G-like but it is finitely cyclic. It was the first example known to the authors of a hereditary decomposable continuum that admits some form of a “globally chaotic” homeomorphism. Recently, in [5], Boroński and Oprocha have shown that the Sierpiński carpet admits a mixing homeomorphism; their result shows that our theorems are optimal in the sense that finite cyclicity is necessary for n1 -indecomposability. 2. Mixing homeomorphisms, indecomposability and
1 -indecomposability n
A continuum is a compact connected metric space with more than one point. A map is a continuous function. Given a continuum X, a map f : X → X is mixing if for every pair of nonempty open sets U, V of X, there exists a positive integer M such that f m (U ) ∩ V = ∅ for all m ≥ M . That is, if U is open, then dH (f n (U ), X) → 0 as n → ∞, where dH is the Hausdorff metric. A graph is a continuum which can be written as the union of finitely many arcs any two of which are either disjoint or intersect only in one or both of their end points. A tree is a graph that contains no simple closed curve. Given a graph G, a G-cover is an open cover whose nerve is G. A continuum is G-like if for every ε > 0 there exists a G-cover with mesh less than ε; equivalently, if it is the inverse limit of the same graph G. A continuum is finitely cyclic if it is the inverse limit of graphs {Gi }∞ i=1 such that the number of simple closed curves of Gi is bounded as i varies. This is equivalent to the following: there exists k such that for every ε > 0 there exist an open cover U with mesh less than ε and whose nerve has at most k simple closed curves. When we have a particular k, we say that the continuum is k-cyclic. A continuum is tree-like if it is the inverse limit of trees. A continuum is indecomposable if every proper subcontinuum has empty interior. The Knaster bucket handle continuum is an example of an indecomposable continuum. Generalizing this notion, given a positive integer n, a continuum is n1 -indecomposable if whenever {Ai }ni=1 is a collection of pairwise disjoint, nonempty subcontinua, at least one of A1 , . . . , An has empty interior. Note: 21 -indecomposable is also called semi-indecomposable. The cone over the Cantor set (also known as the Cantor fan) is an example of a semi-indecomposable continuum. First, we must establish some results on n1 -indecomposability. To do this we must look at open covers of continua in a precise way.
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A chain C = [C1 , ..., Cn ] is a finite collection of open sets of X satisfying that Ci ∩ Cj = ∅ if and only if |i − j| ≤ 1. C is a weak chain if Ci ∩ Ci+1 = ∅ for each i. A circle-chain C = [C1 , ..., Cn ] is a finite collection of open sets of X satisfying that Ci ∩ Cj = ∅ if and only if |i − j| ≤ 1 or i, j ∈ {0, n}. Let U be a collection of open sets. We say that U is connected if for every A, B ∈ U, there exists a weak chain in U from A to B. We say that V refines U if for every V ∈ V there exists U ∈ U such that V ⊂ U . Also, let U ∗ = U ∈U U . We say that two chains C and D are pairwise disjoint if C ∗ ∩ D∗ = ∅. If U is an open cover of X and Y ⊂ X, then define U(Y ) = {U ∈ U | Y ∩ U = ∅}. A cover U of X is taut if Ui ∩ Uj = ∅ if and only if Ui ∩ Uj = ∅ for every Ui , Uj ∈ U. U is a minimal cover for X if for every U ∈ U there exists x ∈ X such that x ∈ / U∗ \ U. Given a cover U of X, we say that U ∈ U is a ramification link if there exist at least three different elements U1 , U2 , U3 of U such that Ui ∩ U = ∅ and U = Ui for each i ∈ {1, 2, 3}. A free chain in U is a chain with no ramification links. The first theorem shows how semi-indecomposable continua are realized by disjoint approximating subcontinua. Theorem 2.1. Let X be an m-cyclic continuum. If for every ε > 0 there exists a collection {Aε1 , . . . , Aεm+2 } of pairwise disjoint subcontinua of X such that, for each i ∈ {1, . . . , m + 2}, dH (Aεi , X) < ε, then X is semi-indecomposable. Proof. Suppose on the contrary that X has two disjoint subcontinua, H and K, that have nonempty interior. Let xH ∈ int(H) and xK ∈ int(K). Then there exist εH > 0 and εK > 0 such that BεH (xH ) ⊂ int(H) and BεK (xK ) ⊂ int(K). Let 0 < ε < (1/3) min{εH , εK , d(H, K)} and U be a minimal taut open cover of X with mesh less than ε and whose nerve contains at most m simple closed curves. Note the following facts: (1) (2) (3) (4) (5)
U ∩ V = ∅ for every U ∈ U(H) and V ∈ U(K). U(H) is a connected cover of H. U(K) is a connected cover of K. Let UH be an element of U(H) that contains xH . Then UH ⊂ int(H) ⊂ H. Let UK be an element of U(K) that contains xK . Then UK ⊂ int(K) ⊂ K.
Let λ be a Lebesgue number for U. By hypothesis, there exist disjoint subcontinua, Aλ1 , . . . , Aλm+2 , such that, for each i ∈ {1, . . . , m + 2}, dH (X, Aλi ) < λ. Since U is minimal Aλi ∩ U = ∅ for every U ∈ U. Define γ = min{dH (Aλi , Aλj ) : i = j} and choose ∗
∗
0 < δ < (1/3) min{λ, γ, dH (H, X \ U(H) ), dH (K, X \ U(K) )}. Let V be a minimal taut open cover of X with mesh less that δ and whose nerve contains at most m simple closed curves. Notice the following facts: (1) (2) (3) (4) (5) (6) (7)
V refines U, V(H) refines U(H), and V(K) refines U(K). V ∩ V = ∅ for every V ∈ V(H) and V ∈ V(K). V ∩ V = ∅ for every V ∈ V(Aλi ) and V ∈ V(Aλj ) if i = j. For each i, there exists VHi ∈ V(Aλi ) such that VHi ⊂ UH . For each i, there exists VKi ∈ V(Aλi ) such that VKi ⊂ UK . Since for each i, VHi ⊂ UH ⊂ H; it follows that V(H) ∩ V(Aλi ) = ∅. Since for each i, VKi ⊂ UK ⊂ K; it follows that V(K) ∩ V(Aλi ) = ∅.
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i H H Since V(H) is a connected cover, for each i ∈ {1, . . . m + 1} there exists a chain CH = [C(i,1) , ..., C(i,n ]⊂ i) H λ H λ H λ λ V(H) such that C(i,1) ∈ V(Ai ), C(i,ni ) ∈ V(Ai+1 ), and C(i,j) ∈ / V(Ai )∪V(Ai+1 ) for all j such that j ∈ / {1, ni }. i K K K K Likewise, there exists a chain CK = [C(i,1) , ..., C(i,m ] ⊂ V(K) such that C(i,1) ∈ V(Aλi ), C(i,m ∈ V(Aλi+1 ), i) i) K and C(i,j) ∈ / V(Aλi ) ∪ V(Aλi+1 ) for all j such that j ∈ / {1, mi }. Since for each i ∈ {1, . . . , m + 2} the covers
V(Aλi ) are connected and pairwise disjoint, there exist pairwise disjoint chains CAi = [C1Ai , ..., CpAii ] in V(Aλi ), A Ai+1 H K H K such that for i ∈ {1, . . . , m + 1}, C1Ai = C(i,1) , CpAii = C(i,1) , C1 i+1 = C(i,n , and Cpi+1 = C(i,m . Hence, it i) i) i i can be easily verified that for each i ∈ {1, . . . , m + 1}, Di = CH ∪ CAi ∪ CK ∪ CAi+1 is a circle-chain contained in V and that each D1 , . . . , Dm+1 are different circle-chains in V. This is a contradiction to the fact that the nerve of V has at most m simple closed curves. Thus, X is semi-indecomposable. 2 Now we can immediately apply the first theorem to mixing homeomorphisms to get one part of our main result: Theorem 2.2. Let X be an m-cyclic continuum. If there exists a mixing homeomorphism h : X → X, then 1 X must be m+2 -indecomposable. 1 Proof. Suppose to the contrary that X is not m+2 -indecomposable. Then there exists a collection {A1 , . . . , Am+2 } of pairwise disjoint subcontinua of X that have non empty interior. Let ε > 0. Since h is mixing, there exists a positive integer N such that for each i ∈ {1, . . . , m + 2}, dH (hN (Ai ), X) ≤ dH (hN (int(Ai )), X) < ε. We have that the collection {hN (A1 ), . . . , hN (Am+2 )} satisfies the conditions of 1 Theorem 2.1. Hence, X is semi-indecomposable and therefore m+2 -indecomposable, which is a contradiction. 2
Now, we will examine “how much” decomposition a continuum might have. Given a continuum X and a n positive integer n, a collection of n subcontinua of X, {Ai }ni=1 , is an n-decomposition for X if X = i=1 Ai , but i=j Ai is a proper subset of X for each j ∈ {1, . . . , n}. We say that X is n-indecomposable if there exists an n-decomposition but no (n + 1)-decomposition for X. Proposition 2.3. Let n be a positive integer. If a continuum X is n-indecomposable, then there exists an n-decomposition into indecomposable subcontinua with nonempty interior. Proof. Take {Ai }ni=1 an n-decomposition for X. By Zorn’s lemma, we may assume that {Ai }ni=1 is a i is a proper subcontinuum of Ai , then minimal decomposition; that is, for each i ∈ {1, . . . , n}, if A i ∪ Aj = X. Note that, for each i ∈ {1, . . . , n}, int(Ai ) = ∅ because X \ Aj is a nonempty A j=i
j=i
open subset of X contained in Ai . Suppose to the contrary that there is i ∈ {1, . . . , n} such that Ai is decom posable. Then there exists proper subcontinua, A1i and A2i , such that Ai = A1i ∪A2i . Then A1i ∪ j=i Aj = X, A2i ∪ j=i Aj = X but A1i ∪ A2i ∪ j=i Aj = X. Hence, {A1i , A2i } ∪ {Aj }j=i is an (n + 1)-decomposition. However, this contradicts the fact that X is n-indecomposable. 2 Lemma 2.4. Let X be a G-like continuum where G is a graph, G a G-cover of X, and C = [C1 , . . . , Cs ] a free chain of G. If A is a proper non-degenerate subcontinuum of X such that A ∩C ∗ = ∅ and A ∩C1 = ∅ = A ∩Cs , then G(A) ⊂ C. Proof. Suppose to the contrary that G(A) is not contained in C. Then there exists Ci ∈ G(A) ∩(C \{C1 , Cs }) and U ∈ G(A) \ C. Furthermore, since G(A) is connected there exists a weak chain E from Ci to U such that C1 , Cs ∈ / E. Therefore, there exists a Cj ∈ E ∩ [C2 , ..., Cs−1 ] such that Cj intersects at least 3 distinct elements of G − {Cj }. However, this contradicts the fact that C is a free chain. 2
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Lemma 2.5. Let C = [C1 , . . . , Cs ] be a chain in a continuum X and {A1 , . . . , Ak } be a collection of subcontinua of X such that for every i ∈ {1, . . . , k}, Ai ⊂ C ∗ . If for each i ∈ {1, . . . , k} there exists an element Cmi ∈ C such that Cmi ⊂ Ai \ ∪j=i Aj and mi < mj whenever i < j, then Ai ∩ Aj = ∅ if |i − j| > 1. Proof. Suppose to the contrary that Ai ∩Aj = ∅ with |i − j| > 1. We may assume without loss of generality that i < j. Take t ∈ {1, . . . , k} such that i < t < j. We have that C(Ai ∪Aj ) is a chain that contains Cmi , Cmj and each Ck for mi < k < mj . Thus, Cmt ∈ C(Ai ∪ Aj ) and it follows that Cmt ∩ (Ai ∪ Aj ) = ∅ which is a contradiction. 2 The next lemma relates G-like
1 n -indecomposable
continua to k-decomposability.
Lemma 2.6. Let G be a graph with N maximal free arcs and r ramification points. Suppose that X is G-like and n1 -indecomposable. Then X is k-indecomposable for some 1 ≤ k ≤ M with M = 2N (n + 1) + r. +1 Proof. Suppose on the contrary that there exists an (M + 1)-decomposition, {Ai }M Since i=1 , for X. int(Ai ) = ∅, there exist ε > 0 and a collection of points {a1 , . . . , aM +1 } such that ai ∈ B(ε, ai ) ⊂ Ai \ j=i Aj . Define γ = min{d(ai , Aj ) : i = j} and let 0 < δ < 13 min{ε, γ}. Let G be a minimal G-cover of X with mesh less than δ. Since G has N maximal free arcs, we have that G has a collection {C1 , . . . , CN } of pairwise disjoint maximal free chains and a collection {U1 , . . . , Ur } of ramification links such that G = C1 ∪ . . . ∪ CN ∪ {U1 , . . . , Ur }. By our choice of δ, we have that there are M + 1 − r points of {a1 , . . . , aM +1 } that belong in C1 ∪ . . . ∪ CN . Then, by the pigeon-hole principle, there exists a k ∈ {1, . . . , N } such that Ck = [C1 , . . . , Cs ] contains i the elements of the decomposition, 2(n+1) different points { a1 , . . . , a2(n+1) } ⊂ {a1 , . . . , aM +1 }. Denote by A M +1 {Ai }i=1 , such that ai ∈ Ai . We may assume, without loss of generality, that if ai ∈ Cji , ar ∈ Cjr and i < r, then ji < jr . Furthermore, by the choice of δ we have
i \ ai ∈ Cji ⊂ B(ε, ai ) ⊂ A
j . A
j=i
Consider the chain D = [Cj1 , . . . , Cj2(m+2) ] in Ck , where a1 ∈ Cj1 and a2(n+1) ∈ Cj2(n+1) . By Lemma 2.4, ∗ 2 , A 4 , . . . , A 2n } consists of for each i ∈ {2, . . . 2(n + 1) − 1}, Ai ⊂ D . By Lemma 2.5, the collection {A n pairwise disjoint subcontinua of X, each with non-empty interior. This contradicts the fact that X is 1 n -indecomposable. Hence, X is k-indecomposable for some k ≤ M . 2 A continuum X is k-coherent if whenever A, B are subcontinua of X such that A ∪ B = X it follows that A ∩ B has at most k components. A continuum X is hereditarily k-coherent if every subcontinuum is k-coherent. A continuum is called unicoherent if it is 1-coherent. Note that every tree-like continuum is hereditarily unicoherent. Proposition 2.7. If X is a k-cyclic continuum, then X is hereditarily k + 1-coherent. Proof. Suppose that X is k-cyclic and, for a contradiction, that there exist subcontinua A, B such that A ∩B 1 has k + 2 distinct components {Hi }k+2 i=1 . Let 0 < ε < 3 mini=j {d(Hi , Hj )} and U be a taut minimal open cover of X with mesh less than ε. Since U(A) is connected there exist k + 1 distinct chains {A(in , jn )}k+1 n=1 in U(A) such that (1) in = jn where in , jn ∈ {1, ..., k + 1}. (2) A(in , jn ) = [U1n , ..., Upnn ] where U1n ∩ Hin = ∅ and Upnn ∩ Hjn = ∅. (3) Utn ∩ (A ∩ B) = ∅ for all t ∈ {2, ..., pn − 1}.
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(Note: the above statement follows from the fact that every connected graph with M vertices must have at least M − 1 edges.) For each n ∈ {1, ..., k + 1}, let Bn be a chain in U(B) whose endlinks are U1n and Upnn and notice that A(in , jn ) ∩ Bn = {U1n , Upnn }. Hence, Cn = A(in , jn ) ∩ Bn = {U1n , Upnn } is a circle-chain. Since, A(it , jt ) ⊂ Cn if and only if t = n, it follows that {Cn }k+1 n=1 is a collection of k + 1 distinct circle-chains in U. Since ε was chosen arbitrarily, this contradicts the fact that X is k-cyclic. 2 Proposition 2.8. Suppose that {Ai }ni=1 is an n-decomposition of a finitely cyclic continuum X where each Ai is indecomposable. If h : X → X is a homeomorphism, then for each i there exists an Ni such that hNi (Ai ) = Ai . Proof. Since, Ai and hp (Ai ) have nonempty interior for each i, there exists a ji ∈ {1, . . . , n} such that int(hp (Ai ) ∩Aji ) = ∅. Since X is finitely cyclic and hence hereditarily k-coherent for some positive integer k, hp (Ai ) ∩ Aji must have a component C with nonempty interior. Since, Aji and hp (Ai ) are indecomposable, it follows that C cannot be a proper subcontinuum of either Aji or hp (Ai ). Hence, Aji = C = hp (Ai ). It now follows, from the pigeon-hole principle, that at least two of Ai , h(Ai ), ..., hn (Ai ), say hp (Ai ) and hq (Ai ), are the same. Let Ni = q − p then hNi (Ai ) = Ai . 2 Now we may conclude the second part of our main result: Theorem 2.9. Let X be a G-like continuum where G is a graph with m ≥ 0 cycles. If h : X → X is a mixing homeomorphism, then X must be indecomposable. Proof. It follows from Theorem 2.2 and Lemma 2.6 that X is 1j -indecomposable for some j and hence X is k-indecomposable for some k. Thus, it follows from Proposition 2.3 that there exists a k-decomposition, {Ai }ki=1 , for X where each Ai is an indecomposable subcontinuum of X with nonempty interior. If k ≥ 2, then A1 is a proper subcontinuum of X with nonempty interior. Hence, since h is mixing, dH (hn (A1 ), X) → 0 as n → ∞. However, it follows from Proposition 2.8 that there is an N such that hN (A1 ) = A1 for some N = 0. So dH (hpN (A1 ), X) = dH (A1 , X) > 0 for all p. This contradicts the fact that dH (hn (A1 ), X) → 0 as n → ∞. Hence, X is indecomposable. 2 3. Example of a mixing homeomorphism on a hereditary decomposable tree-like continuum We are going to construct a hereditary decomposable tree-like continuum and a mixing homeomorphism simultaneously by taking the inverse limit of the universal dendrite with the same bonding map and using the shift homeomorphism. Let Dω be the universal dendrite. Let D(a, b) be the dyadic rationals on (a, b), that is, D(a, b) =
p
∈ (a, b) : p is an odd integer and n is a non-negative integer . 2n
If s ∈ D(a, b), then ρ(s) = n where s = where
p 2n .
C is a dyadic Dω comb on [a, b] if C = [a, b] ∪
s∈D(a,b)
Dωs
1 (1) Dωs is a universal dendrite with diameter 2ρ(s) . s (2) [a, b] ∩ Dω = {s} is a ramification point of C.
∞ Here [a, b] is called the spine of the comb. Let 0, n1 n n=1 be a collection of distinct arcs, let Cn be a
ω (1) = ∞ Cn such that 0 is identified from each dyadic Dω ω = D dyadic Dω comb on 0, n1 n and let D n=1 ω (j) ⊂ D ω (j) = ∞ Cn . Note that each ω such that D comb–that is, Ci ∩ Cj = {0} whenever i = j. Let D n=j ω (j) is homeomorphic to the universal dendrite Dω . Define g : D ω → D ω in the following way (see Fig. 1): D
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Fig. 1. g : Dω → Dω is mixing. Note that hi = g|Ci : Ci → Ci−1 is a homeomorphism for i ≥ 2 and M = g|C1 : C1 → that the corresponding colors are mapped homeomorphically with M ([0, 1]1 ) = {0}.
∞ i=2
Ci such
(1) g(0) = 0.
1 (2) Ci is mapped homeomorphically onto Ci−1 for i ≥ 2 such that g 0, 1i i = 0, i−1 . i−1 s (3) Let C1 = [0, 1]1 ∪ s∈D(0,1) Dω (1). Then (a) g([0, 1]1 ) = 0, ω (ρ(s) + 1). (b) Dωs (1) is mapped homeomorphically onto D Note the following facts about g: ω (j)) → 0 as j → ∞ and diam(Ds (1)) → 0 and ρ(s) → ∞, hence g is continuous. (1) diam(D ω ω such that 0 ∈ (2) Since g(0) = 0, if A ⊂ D / A, then 0 ∈ / g −n (A) for each n ∈ N. (3) If [0, x] is an arc from 0 to x, then g([0, x]) = [0, g(x)] is an arc from 0 to g(x). (Let [0, x] ∩ [0, 1]1 = [0, t]. Then g([0, t]) = 0 and g|[t,x] is a homeomorphism.) ω (j)) = D ω (j − 1) for j ≥ 2. (4) g(D ω such that 0 ∈ Y , then g −1 (Y ) is connected. (5) If Y is a connected subset of D (6) If Y is a subcontinuum such that Y ∩ [0, 1]1 = ∅, then g|Y is a homeomorphism. ω. Proposition 3.1. g ρ(s)+1 (Dωs (1)) = D ω (ρ(s) + 1) and g ρ(s) (D ω (ρ(s) + 1)) = D ω (1). 2 Proof. This follows from the fact that g(Dωs (1)) = D ω − {a, b}} and let S([a, b]) ∈ C(a, b) such that [a, b] ⊂ S([a, b]). Let C(a, b) = {C | C is a component of D S([a, b]) is called the subdendrite of Dω strung by [a, b]. ω. Proposition 3.2. If [a, b] ⊂ [0, 1]1 (a = b), then there exists n such that g n (S([a, b])) = D Proof. Since D([0, 1]) is dense in [0, 1], there exists s ∈ D([0, 1]) such that s ∈ (a, b). Hence, Dωs (1) ⊂ S([a, b]) and the proposition follows from Proposition 3.1. 2 ∞ Let L1 = n=1 [0, 1/n]n . Then 0 ∈ L1 = g(L1 ). So it follows that L1 ⊂ g −1 (L1 ) and g −1 (L1 ) is connected. Hence, g −n (L1 ) ⊂ g −n−1 (L1 ) for each n. Let L2 = g −1 (L1 ) and continuing inductively define Lk = g −1 (Lk−1 ). Then 0 ∈ Lk and Lk is connected. Proposition 3.3. If [a, b] ⊂ Lk − Lk−1 for some k, then there exists m such that g m ([a, b]) is a nondegenerate subarc of [0, 1]1 . Proof. Suppose that [a, b] ⊂ L1 . There are two cases:
Case 1: There exists an n such that [a, b] ⊂ 0, n1 n . Then g n−1 ([a, b]) is a nondegenerate subarc of [0, 1]1 .
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1
Case 2: There exist m, n such that a ∈ 0, m and b ∈ 0, n1 n . m n−m ([a, b]) is a nondegenerate subarc of Without loss of generality, we may assume m < n. Then g 1 and the result follows from Case 1. 0, n−m n−m
Next suppose that [a, b] ⊂ Lp where p > 1. Then there exists q such that [a, b] ⊂ Cq . Then g q−1 |[a,b] is a homeomorphism, so g q−1 ([a, b]) a subarc of C1 that does not intersect [0, 1]1 . Hence, g q ([a, b]) is a non-degenerate subarc of Lp−1 . Now the result follows by induction. 2 ω = ∞ Ln . Notice that D n=1 The following is a well known fact about dendrites. A good starting point to learn more basic facts about dendrites is [14]. Theorem 3.4. If X is a dendrite and ε > 0, then there exists at most a finite number of pairwise disjoint subcontinua with diameter greater than ε. Proof. This follows from the fact that dendrites are hereditarily locally connected and hence do not contain a subcontinuum of convergence. 2 Corollary 3.5. Let [x, y] be a subarc of dendrite X and {xn }∞ n=1 be a sequence in (x, y) such that xn → x and x < xn+1 < xn in the ordering of [x, y]. Then limn→∞ S([x, xn ]) = 0. Proof. Let ε > 0. There is N1 such that diam([x, xn )] < 3ε for all n ≥ N1 . Furthermore, there exists N2 such that diam(S([xn , xn+1 ])) < 3ε for every n ≥ N2 , otherwise, Theorem 3.4 above is violated. Let N = ∞ max{N1 , N2 }. Then S([x, xN ]) = {x} ∪ n=N S([xn , xn+1 ]) and it follows that diam(S([x, xN ])) < 3ε . 2 ω , then there exists an arc [y, x] and a k such that [y, x] ⊂ Proposition 3.6. If U is an open subset of D Lk − Lk−1 and S([y, x]) ⊂ U . ∞ ω , there exists m = 1 such that U ∩ Lm = ∅. Let x ∈ U ∩ Lm and Proof. Since n=1 Ln dense in D k be the largest integer such that x ∈ Lk . Notice that [0, x] ∩ Lk−1 is an arc, say [0, z]. So it follows that (z, x] ⊂ Lk − Lk−1 . Let {xn } be a sequence in (z, x) such that xn → x. Then diam(S([xn , x])) → 0 by Corollary 3.5. Hence, there exists an N such that S([xN , x]) ⊂ U . Let y = xN and the proposition follows. 2 ω → D ω is mixing. Theorem 3.7. g : D Proof. Let U be open. Then it follows from Proposition 3.6 that there is an k and an arc [a, b] such that S([a, b]) ⊂ U ∩Lk −{0}. Then there exists a p such that g p |S([a,b]) is a homeomorphism and g p ([a, b]) ⊂ [0, 1]1 . There exists s ∈ g p ((a, b)) ∩D(0, 1). Then Dωs (1) ⊂ g p (S([a, b])). Hence g p+ρ(s)+1 (U ) = g p+ρ(s)+1 (S([a, b])) = ω . Hence, g is mixing. 2 g p+ρ(s)+1 (Dωs (1)) = D ω , g}∞ be an inverse limit of universal dendrites. Since g is mixing, the shift homeomor{D Let D = lim n=1 ←−− ∞ phism g : D → D defined by g( xi ∞ i=1 ) = g(xi ) i=2 is mixing. Lemma 3.8. D is hereditary decomposable. Proof. Let A be a subcontinuum of D. Then there exists subcontinua {Ai }∞ i=1 of Dω such that A = ∞ lim{Ai , gi }i=1 where gi = g|Ai+1 . ←−−
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Case 1: Suppose that 0 ∈ Ai for every i. ˆ ω is locally connected, we may assume that d is a convex metric for D ˆ ω [2]. Define B1 = Since D B(ε, 0) ∩ A1 , then by hereditary unicoherence, B1 is a proper subcontinuum of A1 and 0 ∈ intA1 (B1 ). Let B2 = g −1 (B1 ) ∩ A2 . By Fact (5) of g (preceding Proposition 3.1), 0 ∈ g −1 (B1 ) and g −1 (B1 ) is a continuum. So, again by hereditary unicoherence, B2 is a subcontinuum of A2 and 0 ∈ B2 . Define Bi+1 = g −1 (Bi ) ∩ Ai+1 , then gi−1 (Bi ) = Bi+1 . Since A ∩ π1−1 (B(ε, 0)) ⊂ B, it follows that B = lim ←−{Bi , gi | Bi+1 } is a proper subcontinuum of A with non-empty interior. Since A contains a proper subcontinuum with non-empty interior, it follows that A is decomposable. / Ak . Case 2: Suppose that there exists k such that 0 ∈ Since g([0, 1]1 ) = {0}, it follows that Aj ∩ [0, 1]1 = ∅ for every j > k. Thus, gj is a homeomorphism for all j > k. Since, Aj is locally connected for each j, it follows that A is locally connected and hence decomposable. 2 Note: D is semi-indecomposable. Hence the following question is natural: Question. If X is a one-dimensional continuum that admits a mixing homeomorphism, must X contain a semi-indecomposable subcontinuum? Acknowledgements This research was partially supported by the project “Teoría de Continuos, Hiperespacios y Sistemas Dinámicos II” (IN101216) of PAPIIT, DGAPA, UNAM. References [1] Marcy Barge, Joe Martin, Chaos, periodicity, and snakelike continua, Trans. Am. Math. Soc. 289 (1) (1985) 355–365. [2] R.H. Bing, A convex metric for locally connected continuum, Bull. Am. Math. Soc. 55 (8) (1949) 812–819. [3] Jan P. Boroński, Jiří Kupka, New chaotic planar attractors from smooth zero entropy interval maps, Adv. Differ. Equ. 2015 (2015) 232. [4] Jan P. Boroński, Piotr Oprocha, On indecomposability in chaotic attractors, Proc. Am. Math. Soc. 143 (8) (2015) 3659–3670. [5] Jan P. Boroński, Piotr Oprocha, On dynamics of the Sierpiński carpet, C. R. Math. Acad. Sci. Paris 356 (3) (2018) 340–344. [6] Udayan Darji, Hisao Kato, Chaos and indecomposability, Adv. Math. 304 (2017) 793–808. [7] Hisao Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topol. Appl. 53 (3) (1993) 239–258. [8] Hisao Kato, Continuum-wise expansive homeomorphisms, Can. J. Math. 45 (3) (1993) 576–598. [9] Logan Hoehn, Christopher Mouron, Hierarchies of chaotic maps on continua, Ergod. Theory Dyn. Syst. 34 (6) (2014) 1897–1913. [10] Christopher Mouron, Mixing sets, positive entropy homeomorphisms and non-Suslinean continua, Ergod. Theory Dyn. Syst. 36 (7) (2016) 2246–2257. [11] Christopher Mouron, Expansive homeomorphisms and indecomposable subcontinua, Topol. Appl. 126 (1–2) (2002) 13–28. [12] Christopher Mouron, Tree-like continua do not admit expansive homeomorphisms, Proc. Am. Math. Soc. 130 (Nov. 2002) 3409–3413. [13] Christopher Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, Proc. Am. Math. Soc. 139 (8) (2011) 2783–2791. [14] Sam B. Nadler Jr., Continuum Theory. An Introduction, Monogr. Textb. Pure Appl. Math., vol. 158, Marcel Dekker, Inc., New York, 1992. [15] Gerald T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Am. Math. Soc. 4 (1990) 1025–1030.