Inverse limits of set-valued functions
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1 Iztok Banič University o Maribor Vienna, 2012
2 Continua Deinition A continuum is a nonempty, compact and connected metric space.
3 sin 1 x -continuum Let W = { } (x, sin 1 x ) R2 0 < x 1. Any continuum homeomorphic to Cl(W) is called a sin 1 x -continuum
4 A harmonic an
5 The Brouwer-Janiszewski-Knaster continuum The Brouwer-Janiszewski-Knaster continuum: B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922)
6 Pseudo-arc
7 Ważewski s universal dendrite T. Ważewski, Sour les courbes de Jordan ne renermant aucune courbe simple ermee de Jordan, Roczniki Polskiego Tow. Mat. 2 (1923)
8 Inverse limit Inverse sequence X 1 1 X2 2 X3 3 X4 4 X5 5 X The inverse limit o an inverse sequence {X k, k } k=1 is deined to be the subspace o the product space X k o all k=1 x = (x 1, x 2, x 3,...) X k, such that x k = k (x k+1 ) or each k. k=1 K = lim {X k, k } k=1
9 Inverse limit Inverse sequence X 1 1 X2 2 X3 3 X4 4 X5 5 X The inverse limit o an inverse sequence {X k, k } k=1 is deined to be the subspace o the product space X k o all k=1 x = (x 1, x 2, x 3,...) X k, such that x k = k (x k+1 ) or each k. k=1 K = lim {X k, k } k=1
10 Examples [0, 1]... For each t [0, 1], (t) = t. For each t [0, 1], (t) = 1. (t) = { 2t ; t t ; t 1 2. For each t [0, 1], (t) = t 2.
11 Examples [0, 1]... For each t [0, 1], (t) = t. For each t [0, 1], (t) = 1. (t) = { 2t ; t t ; t 1 2. For each t [0, 1], (t) = t 2.
12 Examples [0, 1]... For each t [0, 1], (t) = t. For each t [0, 1], (t) = 1. (t) = { 2t ; t t ; t 1 2. For each t [0, 1], (t) = t 2.
13 Examples [0, 1]... For each t [0, 1], (t) = t. For each t [0, 1], (t) = 1. (t) = { 2t ; t t ; t 1 2. For each t [0, 1], (t) = t 2.
14 2 X I (X, d) is a compact metric space, then 2 X denotes the set o all nonempty closed subsets o X.
15 u.s.c. unctions A unction : X 2 Y, where X and Y are compact metric spaces, is upper semi-continuous (abbreviated u.s.c.) at x X i or each open set V in Y, such that (x) V, there is an open set U in X, such that x U and or each u U, (u) V. A unction : X 2 Y is surjective, i or each y Y there is an x X, such that y (x). The graph Γ() o a unction : X 2 Y is the set o all points (x, y) X Y such that y (x).
16 u.s.c. unctions A unction : X 2 Y, where X and Y are compact metric spaces, is upper semi-continuous (abbreviated u.s.c.) at x X i or each open set V in Y, such that (x) V, there is an open set U in X, such that x U and or each u U, (u) V. A unction : X 2 Y is surjective, i or each y Y there is an x X, such that y (x). The graph Γ() o a unction : X 2 Y is the set o all points (x, y) X Y such that y (x).
17 u.s.c. unctions A unction : X 2 Y, where X and Y are compact metric spaces, is upper semi-continuous (abbreviated u.s.c.) at x X i or each open set V in Y, such that (x) V, there is an open set U in X, such that x U and or each u U, (u) V. A unction : X 2 Y is surjective, i or each y Y there is an x X, such that y (x). The graph Γ() o a unction : X 2 Y is the set o all points (x, y) X Y such that y (x).
18 u.s.c. unctions Theorem Let X and Y be compact metric spaces and : X 2 Y a unction. Then is u.s.c. i and only i its graph Γ() is closed in X Y.
19 Examples : [0, 1] 2 [0,1] (t) = { {1} ; t < 1 [0, 1] ; t = 1. { {1} ; t < 1 (t) = [ ] 0, 1 2 ; t = 1.
20 Examples : [0, 1] 2 [0,1] (t) = { {1} ; t < 1 [0, 1] ; t = 1. { {1} ; t < 1 (t) = [ ] 0, 1 2 ; t = 1.
21 Examples I : [0, 1] [0, 1] is continuous, then F : [0, 1] 2 [0,1], F(x) = {(x)}, is u.s.c. I F : [0, 1] 2 [0,1] is u.s.c., and or each x [0, 1], F(x) = {y x }, then : [0, 1] [0, 1], (x) = y x, is continuous.
22 Examples I : [0, 1] [0, 1] is continuous, then F : [0, 1] 2 [0,1], F(x) = {(x)}, is u.s.c. I F : [0, 1] 2 [0,1] is u.s.c., and or each x [0, 1], F(x) = {y x }, then : [0, 1] [0, 1], (x) = y x, is continuous.
23 Generalized inverse limits Inverse sequence X 1 1 X2 2 X3 3 X4 4 X5 5 X The (generalized) inverse limit o an inverse sequence {X k, k } k=1 is deined to be the subspace o the product space X k o all x = (x 1, x 2, x 3,...) X k, such that x k k (x k+1 ) k=1 or each k. k=1 K = lim {X k, k } k=1
24 Generalized inverse limits Inverse sequence X 1 1 X2 2 X3 3 X4 4 X5 5 X The (generalized) inverse limit o an inverse sequence {X k, k } k=1 is deined to be the subspace o the product space X k o all x = (x 1, x 2, x 3,...) X k, such that x k k (x k+1 ) k=1 or each k. k=1 K = lim {X k, k } k=1
25 Generalized inverse limits The inverse limit o {X k, k } k=1 is denoted by lim {X k, k } k=1. W.T. Ingram, W.S. Mahavier, Inverse limits o upper semi-continuous set valued unctions, Houston J. Math. 32 (2006)
26 Generalized inverse limits The inverse limit o {X k, k } k=1 is denoted by lim {X k, k } k=1. W.T. Ingram, W.S. Mahavier, Inverse limits o upper semi-continuous set valued unctions, Houston J. Math. 32 (2006)
27 Examples [0, 1]... For each t [0, 1], (t) = {0, 1}. For each t [0, 1], (t) = [0, 1]. (t) = { {t} ; t < 1 [0, 1] ; t = 1.
28 Examples [0, 1]... For each t [0, 1], (t) = {0, 1}. For each t [0, 1], (t) = [0, 1]. (t) = { {t} ; t < 1 [0, 1] ; t = 1.
29 Examples [0, 1]... For each t [0, 1], (t) = {0, 1}. For each t [0, 1], (t) = [0, 1]. (t) = { {t} ; t < 1 [0, 1] ; t = 1.
30 Examples (t) = { { 1 2 } ; t < 1 [0, 1] ; t = 1 For each t [0, 1], (t) = {0, t}.
31 Examples (t) = { { 1 2 } ; t < 1 [0, 1] ; t = 1 For each t [0, 1], (t) = {0, t}.
32 Properties - compactness Theorem Let or each k, X k be a compact metric space, and let or each k k : X k+1 X k be a continuous unction. Then lim {X k, k } k=1 is compact. Question Let or each k, X k be a compact metric space, and let or each k k : X k+1 2 X k be a u.s.c. unction. Is it true that lim {X k, k } k=1 is also compact?
33 Properties - compactness Theorem Let or each k, X k be a compact metric space, and let or each k k : X k+1 X k be a continuous unction. Then lim {X k, k } k=1 is compact. Question Let or each k, X k be a compact metric space, and let or each k k : X k+1 2 X k be a u.s.c. unction. Is it true that lim {X k, k } k=1 is also compact?
34 Properties - compactness YES!
35 Properties - connectedness Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a continuous unction. Then lim {X k, k } k=1 is a continuum. Question Let or each k, X k be a continuum, and let or each k k : X k+1 2 X k be a u.s.c. unction. Is it true that lim {X k, k } k=1 is also connected?
36 Properties - connectedness Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a continuous unction. Then lim {X k, k } k=1 is a continuum. Question Let or each k, X k be a continuum, and let or each k k : X k+1 2 X k be a u.s.c. unction. Is it true that lim {X k, k } k=1 is also connected?
37 Properties - connectedness NO! [0, 1] For each t [0, 1], (t) = {0, 1}....
38 Properties - connectedness NO! [0, 1] For each t [0, 1], (t) = {0, 1}....
39 Properties - connectedness Question Let or each k, X k be a continuum, and let or each k k : X k+1 2 X k be a u.s.c. unction with connected graph. Is it true that lim{x k, k } k=1 is connected?
40 Properties - connectedness NO! [0, 1] (t) = { { t, t} ; t 1 2 {0, 1} ; t
41 Properties - connectedness NO! [0, 1] (t) = { { t, t} ; t 1 2 {0, 1} ; t
42 Properties - connectedness Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a u.s.c. unction with connected graph. I or each x X k+1, (x) is connected, then lim {X k, k } k=1 is connected. Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a u.s.c. unction with connected graph. I or each y X k, {x X k+1 y (x)} is connected, then lim {X k, k } k=1 is connected.
43 Properties - connectedness Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a u.s.c. unction with connected graph. I or each x X k+1, (x) is connected, then lim {X k, k } k=1 is connected. Theorem Let or each k, X k be a continuum, and let or each k k : X k+1 X k be a u.s.c. unction with connected graph. I or each y X k, {x X k+1 y (x)} is connected, then lim {X k, k } k=1 is connected.
44 Properties - connectedness I : [0, 1] 2 [0,1] is a surjective u.s.c. unction, such that Γ() is a pseudo-arc, then the inverse limit lim {[0, 1], } is totally disconnected. There is a u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim {[0, 1], } is totally disconnected. Question Is there a surjective u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim{[0, 1], } is totally disconnected?
45 Properties - connectedness I : [0, 1] 2 [0,1] is a surjective u.s.c. unction, such that Γ() is a pseudo-arc, then the inverse limit lim {[0, 1], } is totally disconnected. There is a u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim {[0, 1], } is totally disconnected. Question Is there a surjective u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim{[0, 1], } is totally disconnected?
46 Properties - connectedness I : [0, 1] 2 [0,1] is a surjective u.s.c. unction, such that Γ() is a pseudo-arc, then the inverse limit lim {[0, 1], } is totally disconnected. There is a u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim {[0, 1], } is totally disconnected. Question Is there a surjective u.s.c. unction : [0, 1] 2 [0,1], such that 1 Γ() is an arc, 2 lim{[0, 1], } is totally disconnected?
47 Properties - the subsequence lemma Theorem Let or each k, X k be a compact metric space, and let or each k k : X k+1 X k be a continuous unction. I {i k } k=1 is a strictly increasing sequence o integers and or each k, g ik : X ik+1 X ik is the composition g ik = ik+1 1 ik+1 2 ik ik, then lim {X k, k } k=1 and lim{x i k, g ik } k=1 are homeomorphic. Question Let or each k, X k be a compact metric space, and let or each k k : X k+1 2 X k be a u.s.c. unction. I {i k } k=1 is a strictly increasing sequence o integers and or each k, g ik : X ik+1 2 X i k is the composition g ik = ik+1 1 ik+1 2 ik ik, is it true that then lim{x k, k } k=1 and lim{x i k, g ik } k=1 are homeomorphic?
48 Properties - the subsequence lemma Theorem Let or each k, X k be a compact metric space, and let or each k k : X k+1 X k be a continuous unction. I {i k } k=1 is a strictly increasing sequence o integers and or each k, g ik : X ik+1 X ik is the composition g ik = ik+1 1 ik+1 2 ik ik, then lim {X k, k } k=1 and lim{x i k, g ik } k=1 are homeomorphic. Question Let or each k, X k be a compact metric space, and let or each k k : X k+1 2 X k be a u.s.c. unction. I {i k } k=1 is a strictly increasing sequence o integers and or each k, g ik : X ik+1 2 X i k is the composition g ik = ik+1 1 ik+1 2 ik ik, is it true that then lim{x k, k } k=1 and lim{x i k, g ik } k=1 are homeomorphic?
49 Properties - the subsequence lemma NO!
50 What can one get? [0, 1]... I lim {[0, 1], } k=1 is a graph, then it is an arc.
51 lim {[0, 1], } k=1 is never a disc. There is a u.s.c. unction : [0, 1] 2 [01] such that lim {[0, 1], } k=1 is homeomorphic to the Ważewski s universal dendrite.
52 lim {[0, 1], } k=1 is never a disc. There is a u.s.c. unction : [0, 1] 2 [01] such that lim {[0, 1], } k=1 is homeomorphic to the Ważewski s universal dendrite.
53 [0, 1] Can any continuum be represented as an inverse limit lim {[0, 1], k} k=1?
54 Tent-unctions For any a, b [0, 1], the tent unction (a,b) : [0, 1] [0, 1] is deined as the set-valued unction with the graph Γ( (a,b) ) being the union o the segment (possibly degenerate) rom (0, 0) to (a, b) and the segment (possibly degenerate) rom (a, b) to (1, 0). The inverse limit obtained rom the inverse sequence o closed unit intervals [0, 1] and bonding unction (a,b) is denoted by K (a,b) = lim {[0, 1], (a,b) } n=1.
55 ( 1 2,1)
56 The Ingram conjecture The whole amily o continua K ( 1,b), < b 1, has been called Knaster continua and the amous Ingram conjecture, stated in 1992, claimed that all o them are pairwise non-homeomorphic. The conjecture was proved in 2009 by M. Barge, H. Bruin and S. Štimac
57 Classiying tent maps inverse limits Question What is K (a,b) or a given tent unction (a,b)?
58 Theorem I a, b [0, 1] and b < a or (a, b) = (0, 0), then K (a,b) = {(0, 0, 0,...)}
59 Theorem I a (0, 1), then K (a,a) is an arc rom (0, 0, 0,...) to (a, a, a,...)
60 (0,b), b (0, 1) Theorem I b (0, 1), then K (0,b) is an arc rom (0, 0, 0,...) to ( b b 1+b,...). 1+b, b 1+b,
61 (1,1) Theorem K (1,1) is homeomorphic to the harmonic an.
62 (a,1), a (0, 1) Theorem I a (0, 1), then K (a,1) is homeomorphic to the Brouwer-Janiszewki-Knaster continuum
63 (0,1) Theorem (a) K (0,1) contains sin 1 x continua. (b) K (0,1) contains harmonic ans. (c) For each integer n 3, there are simple n ods contained in K (0,1) that are not contained in any simple m-od or any m > n. (d) K (0,1) is one-dimensional. (e) K (0,1) is tree-like.
64 a,b, a, b (0, 1), a < b and b < 1 a Theorem Let a, b (0, 1), a < b and b < 1 a. Then K (a,b) is an arc
65 The curves C t For any t [1, ) let C t = {(x, y) [0, 1] [0, 1] x t+1 x t = y t+1 y t, 0 < x < y}
66 The curves C t For any t [1, ) let C t = {(x, y) [0, 1] [0, 1] x t+1 x t = y t+1 y t, 0 < x < y}
67 C n Theorem Let n be a positive integer and (a, b),(c, d) C n. Then K (a,b) and K (c,d) are homeomorphic.
68 Theorem I (a, b) C n and (c, d) C m or some positive integers n m, then K (a,b) and K (c,d) are not homeomorphic.
69 (a,b), (a, b) C 1 Theorem I (a, b) C 1, then K (a,b) is a sin 1 x -continuum
70 Towards the complete classiication o tent maps inverse limits
71 The Conjecture Let t [1, ) and (a, b),(c, d) C t. Then K (a,b) and K (c,d) are homeomorphic.
72 Thank you!
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