Centralizers of Cantor minimal systems

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1 Centralizers of Cantor minimal systems 1 X X X φ (X, φ) (X, φ) φ φ 2 X X X Homeo(X) Homeo(X) φ Homeo(X) x X Orb φ (x) = { φ n (x) ; n Z } x φ x Orb φ (x) X Orb φ (x) x n N 1

2 φ n (x) = x 1. (X, φ) (i) (X, φ) topologically transitive x X Orb φ (x) X (ii) (X, φ) minimal x X Orb φ (x) X (X, φ) X X Z φ n Z φ(n) = n + 1 n Z Orb φ (n) = Z (Z, φ) X R/Z α R \ Q x α x x + α X = R/Z φ (X, φ) irrational rotation (X, φ) [PY] 1.6 X φ R R/Z R 2 R 2 \ {finite points} R n (n 3) S 2n S 2n+1 2

$(Z, φ) X R/Z α R \ Q x α x x + α X = R/Z φ (X, φ) irrational rotation (X, φ)$

3 X R [O] 18 X [0, 1] [0, 1] X R R/Z R 2 Brouwer transformation theorem [Fr1] R 2 [H] [CY] [Fr2] n R n X X [Fu] 2n S 2n S 2n+1 [FH] [FH] 3

4 3 2. (i) X perfect X (ii) X totally disconnected (iii) X extremally disconnected (ii) (iii) βn 3. {0, 1} N X X {0, 1} N [Y] dimensional 2.13 {0, 1} N {a, b, c} N 4. {0, 1} N X X X X X N X X X 4

{0, 1} N X X {0, 1} N [Y] 5.8 0-dimensional 2.

5 n n n A 1 i, j n m N A m (i, j) A m ij n n 0 1 A n {1, 2,..., n} {1, 2,..., n} Z A X X = {x = (x n ) n Z ; A(x n, x n+1 ) = 1 for all n Z} X x = (x n ) n X φ(x) = (x n+1 ) n φ(x) X φ X X (X, φ) A = A A {1, 2, 3} Z X Z φ A X (X, φ) 5

6 [PY] X φ X (X, φ) (X, φ) C(X, Z) X Z Z X C(X, Z) C(X, Z) B φ = {f f φ ; f C(X, Z)} B φ C(X, Z) C(X, Z) B φ K 0 (X, φ) f C(X, Z) K 0 (X, φ) [f] K 0 (X, φ) K 0 (X, φ) K 0 (X, φ) + K 0 (X, φ) + = {[f] ; f C(X, Z), f 0} K 0 (X, φ) + K 0 (X, φ) + K 0 (X, φ) + X [1] K 0 (X, φ) + (K 0 (X, φ), K 0 (X, φ) +, [1]) (X, φ) AF K 0 [GPS] [GPS] (K 0 (X, φ), K 0 (X, φ) +, [1]) (X, φ) [M1] [GPS] 6

7 α R \ Q A = {n + mα ; n, m Z} R A B = {b p ; p A} C = {c p ; p A} Y R \ A B C disjoint union Y R \ A R x R \ A, b p, b q B, c p, c q C x b p b p b q def def x < p, x c p def def p < q, c p c q x < p p < q b p c q def p q Y Y Y A y 1 < y 2 (y 1, y 2 ) = {y Y ; y 1 < y < y 2 } (y 1, y 2 ) Y Y [Y] 4.1 Y A R Y x R \ A p A x x + 1, b p b p+1, c p c p+1 T 1 Y T 1 T 1 Y Y α T α T 1 Y X y 1 y 2 def T n 1 (y 1 ) = y 2 for some n Z 7

$1 Y A R Y x R \ A p A x x + 1, b p b p+1, c p c p+1 T 1 Y T 1 T 1 Y Y α T α T 1 Y X y 1 y 2 def T n 1 (y 1$

8 Y X π : Y X X X T α Y X φ φ π = π T α (X, φ) φ α K 0 (X, φ) Z Z K 0 (X, φ) + = {(n, m) Z Z ; n + mα 0} [1] (1, 0) α [PSS] (X, φ) C [M1] interval exchange transformation interval exchnge n n α 1, α 2,..., α n [0, 1) β 0 = 0, β k = k α i, I i = [β i 1, β i ) i=1 I 1, I 2,..., I n {1, 2,..., n} σ σ I 1, I 2,..., I n T 8

9 0 α 1 α 2 α 3 α 4 1 T σ = ( ) 0 α 4 α 1 α 3 α 2 1 T [0, 1) [0, 1) β i A 0 = {β 0, β 1,..., β n 1 } A = k Z T k (A 0 ) A [0, 1) A A B C X = ((0, 1] \ A) B C {0} X T X X (X, φ) 7. (i) α 1, α 2,..., α n 1 Q (ii) σ({1, 2,..., j}) = {1, 2,..., j} j = n (X, φ) [K1] n = 2 (X, φ) K 0 (X, φ) Z n [P] 2.1 Z n [β i 1, β i ) 9

$.., β n 1 } A = k Z T k (A 0 ) A [0, 1) A A B C X = ((0, 1] \ A) B C {0} X T X X$

10 [1] (1, 1,, 1) T φ [0, 1) (a 1, a 2,, a n ) Z n \ {0} α 1 a 1 + α 2 a α n a n > 0 {α i } σ uniquely ergodic [KN] [K2] 5 (X, φ) C(φ) = {γ : X X ; γ is continuous, γ φ = φ γ} C(φ) φ X X C(φ) C(φ) C(φ) γ C(φ) γ K 0 (X, φ) mod (γ) g C(X, Z) g γ C(X, Z) γ C(X, Z) f C(X, Z) f f φ γ (f f φ) γ = f γ f φ γ = f γ f γ φ γ K 0 (X, φ) γ mod (γ) T (φ) = {γ C(φ) ; mod (γ) = id} C h (φ) = {γ C(φ); γ is a homeomorphism} T h (φ) = C h (φ) T (φ) 8. (i) (X, φ) T h (φ) G G 10

(γ) g C(X, Z) g γ C(X, Z) γ C(X, Z) f C(X, Z) f f φ γ (f f φ) γ = f γ f φ γ = f γ f γ φ γ K 0 (X, φ) γ mod (γ) T (φ)

11 (ii) T (φ) \ T h (φ) (X, φ) (i) C h (φ) T h (φ) T (φ) (ii) γ C(φ) mod (γ) (i)(ii) [M2] [M3] [CY] Le Calvez, P.; Yoccoz, J. C.; Un thorme d indice pour les homomorphismes du plan au voisinage d un point fixe, Ann. of Math. (2), 146 (1997), [Fu] Fuller, F. B.; The existence of periodic points, Ann. of Math. (2), 57 (1953), [FH] Fathi, A.; Herman, M; Existence de diffomorphismes minimaux, Asterisque, N0. 49, (1977), [Fr1] Franks, J.; A new proof of the Brouwer plane translation theorem, Ergodic Theory Dynam. Systems, 12 (1992), [Fr2] Franks, J.; The Conley index and non-existence of minimal homeomorphisms, Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, II, 1997), Illinois J. Math., 43 (1999), [GPS] Giordano, T.; Putnam, I. F.; Skau, C. F.; Topological orbit equivalence and C -crossed products, J. reine angew. Math., 469 (1995), [H] Handel, M.; There are no minimal homeomorphisms of the multipuctured plane, Ergodic Theory Dynam. Systems, 12 (1992), [K1] Keane, M; Interval exchange transformations, Math. Z., 141 (1975), [K2] Keane, M; Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), [KN] Keynes, H; Newton, D.; A minimal, non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), [M1] ; Cantor minimal systems and full groups, 11

; Herman, M; Existence de diffomorphismes minimaux, Asterisque, N0. 49, (1977), 37-59. [Fr1] Franks, J.; A new proof of the Brouwer plane translation theorem, Ergodic Theory Dynam.

12 1110, Hilbert C -modules and groupoid C -algebras, [M2] Matui, H.; Finite order automorphisms and dimension groups of Cantor minimal systems, to appear in J. Math. Soc. Japan. [M3] Matui, H.; Dimension groups of topological joinings and non-coalescence of Cantor minimal systems, preprint [O] Oxtoby, J. C.; Measure and category, Graduate Texts in Mathematics, Vol. 2, Springer-Verlag, New York-Berlin, [P] Putnam, I. F.; The C -algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math., 136 (1989), [PSS] Putnam, I. F.; Schmidt, K.; Skau, C. F.; C -algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory, 16 (1986), [PY] Pollicott, M; Yuri, M; Dynamical systems and ergodic theory, London Mathematical Society student Texts 40, Cambridge University Press, Cambridge, [Y] ;,, ; matui@kusm.kyoto-u.ac.jp