The UV -property of compact invariant sets

Transcription

1 The UV -property of compact invariant sets Krystyna Kuperberg, Auburn University 6th European Congress of Mathematics Kraków Jagiellonian University July 2, 2012 Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 1 / 23

2 Shape Theory- K. Borsuk Definition Let Q be the Hilbert cube or any AR. A closed set F Q is movable provided such that 1 H(x, 0) = x for all x V, 2 H(V {1}) W. U V W H H : V I U U, V, W are open neighborhoods of F ; H is a homotopy. We may assume F W V U. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 2 / 23

3 The notion of movable belongs to so called UV -properties. Armentrout, Steve UV properties of compact sets. Trans. Amer. Math. Soc. 143, 1969, Etale Homotopy A. Grothendieck infitinite cycles vs true cycles - L. Vietoris Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 3 / 23

4 Examples Movable compacta ANRs attractors planar continua (e.g. the pseudoarc, Smale horseshoes) Denjoy exceptional sets (P.A. Schweitzer) Non-movable compacta solenoids (inverse limits of circles, nontrivial) McCord solenoids topological suspensions (union of two cones) of solenoids UV -property is shape invariant Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 4 / 23

5 Let G = Z or G = R and let φ : G M M be a group action on a manifold M. Let F M be compact and invariant under φ. Definition F is Lyapunov stable provided U, V, are open neighborhoods of F. We may assume F V U. U V x V t 0 φ(t, x) U Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 5 / 23

6 Approximating by circles Remark: A solenoid Ω is not movable, but if Ω R 3, then there are embedded in R 3 cirlces K 1, K 2,... approximating Ω such that the set is movable. Ω n=1 K n Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 6 / 23

7 Stable solenoids Theorem (Buescu-Stewart) Every neighborhood of a Lyapunov stable solenoid in a C 1 flow φ on R 3 contains a periodic orbit. J. Buescu and Ian Stewart, Liapunov stability and adding machines, Ergodic Theory Dynam. Systems 15 (1995), Corollary A Lyapunov stable solenoid in a C 1 flow on R 3 is contained in an invariant movable set. Theorem (Thomas) If Ω is an invariant solenoid under a C 1 flow on a three-manifold, then Ω is not isolated. E.S. Thomas, One-dimensional minimal sets, Topology 12 (1973), Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 7 / 23

8 Approximating by Denjoy sets Theorem (Petra Šindelářová) There is a flow on R 3 with a non-movable compact invariant set Γ with approximating invariant Denjoy sets D 1, D 2,... such that the set Γ is movable. Γ is locally the Cartesian product of the Cantor set and an interval, but Γ is not a solenoid. Petra Šindelářová, An example on movable approximations of a minimal set in a continuous flow, Topology and its Applications 154 (2007), n=1 D n Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 8 / 23

9 Questions Let ψ be a flow on R 3 with an compact invariant set A. 1 Does there exist a movable invariant set B containing A? 2 If A is isolated, is A movable? Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 9 / 23

10 Adding machines For a sequence of integers (k 1, k 2, k 3,...), k i > 1, let C(k 1, k 2, k 3,...) (or shortly C) be the Cantor set Π n=1 Z/k nz. Definition An adding machine is a homeomorphism α : C C such that for α(i 1, i 2, i 3,...) = (j 1, j 2, j 3,...) 1 if there is an m 1 such that i n = k n 1 for n < m and i m < k m 1, then j n = 0 for n < m, j m = i m + 1, and j n = i n for n > m, 2 otherwise j m = 0 for all m, i.e., if i m = k m 1 for m 1, then j m = 0 for m 1. α = the Poincaré return map of the dynamical system on the solenoid Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 10 / 23

11 Adding machines cylinders Definition A cylinder of length n as the set C i1,...,i n = {(x 1, x 2,...) x 1 = i 1,..., x n = i n }. Let h : R 2 R 2 be a homeomorphism and let C = C(k 1, k 2, k 3,...) R 2 be an invariant Cantor set. Assume that α = h C is an adding machine. Theorem (Buescu-Stewart) If C is Lyapunov stable, then every neighborhood U of C contains a periodic orbit of h. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 11 / 23

12 Theorem Let h : R 2 R 2 be a homeomorphism and let C = C(k 1, k 2, k 3,...) R 2 be an invariant Cantor set such that α = h C is an adding machine. Then every neighborhood U of C contains a periodic point of h. The cylinder C i1,...,i n = {(x 1, x 2,...) x 1 = i 1,..., x n = i n } is invariant under α p iff p is a multiple of the product k 1 k n. The period of a periodic point close to C is not arbitrary. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 12 / 23

13 Theorem Let h : R 2 R 2 be a homeomorphism and let C = C(k 1, k 2, k 3,...) R 2 be an invariant Cantor set such that α = h C is an adding machine. Then every neighborhood U of C contains a periodic point of h. adding machine Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 13 / 23

14 Assume h is orientation preserving (or consider h 2 ). Let P be the set of periodic points of h in R 2 of periods k 1. Suppose that Cl(P) C =. Let U be a component of R 2 \ Cl(P) intersecting C, i.e., containing a cylinder of C. Let Ũ be the universal cover of U with π : Ũ U the covering map. If U is simply connected, then Ũ = U is a plane and by the Brouwer translation theorem applied to h U there is a fixed point of some iteration of h in U. Thus there is periodic point of h in U - a contradiction. In general, there is a cylinder C i1,...,i n contained in an open, evenly covered disk D U. Since C i1,...,i n is invariant under h k 1 k n, so is U. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 14 / 23

15 Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 15 / 23

16 Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 16 / 23

17 By composing a lift of f = h k 1 k n U with an appropriate deck transformation, we achieve a lift f with an invariant copy the cylinder C i1,...,i n, C, in Ũ. Since h has no periodic points in U, f have no fixed points in Ũ. Ũ is homeomorphic to R2. Then f is an orientation preserving homeomorphism of the plane with an invariant compact set C. By the Brouwer translation theorem, f has a fixed point a. Therefore f or equivalently h would have a periodic point π(a) / P. Remark Morton Brown used this method to give his short short proof of the Carthwright-Littlewood theorem. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 17 / 23

18 Self-insertion Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 18 / 23

19 C plug The C plug contains a huge minimal set. The remains of the annulus between the two circular orbits can be in the minimal set the topological (covering) dimension is two. Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 19 / 23

20 Differentiable plug Krystyna Kuperberg, Auburn University ( 6th European Congress of Mathematics Kraków Jagiellonian University ) The UV -property of compact invariant sets July 2, / 23

21 PL plug Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 21 / 23

22 Make only a Cantor set of trajectories of the annulus between the two circular orbits end up in the minimal set. The minimal set F is a 1-dimensional matchbox manifold. questions: Is F movable? Is F contained in a larger movable invariant set? Does there exist a similar C 0 construction with an isolated minimal set? Does there exist a similar C 0 volume preserving plug? Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 22 / 23

23 Thank you for listening Krystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23

24 Thank you for listening Thank you for participating in this mini-symposium and the STS rystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23

25 Thank you for listening Thank you for participating in this mini-symposium and the STS Thank you to the 6ECM organizers for approving the mini-symposium and the STS rystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23

26 Thank you for listening Thank you for participating in this mini-symposium and the STS Thank you to the 6ECM organizers for approving the mini-symposium and the STS Thank you for the 6ECM rystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23

27 Thank you for listening Thank you for participating in this mini-symposium and the STS Thank you to the 6ECM organizers for approving the mini-symposium and the STS Thank you for the 6ECM Thank you to UJ for hosting the Congress rystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23

28 Thank you for listening Thank you for participating in this mini-symposium and the STS Thank you to the 6ECM organizers for approving the mini-symposium and the STS Thank you for the 6ECM Thank you to UJ for hosting the Congress Thank you Kraków rystyna Kuperberg, Auburn University ( 6thThe European UV -property Congress of compact of Mathematics invariantkraków sets Jagiellonian University July 2, 2012 ) 23 / 23