A homotopical property of attractors
KeywordsAttractor, homeomorphism, fundamental group, finitely generated, Antoine necklace, Antoine sphere
AbstractWe construct a 2-dimensional torus T ⊆ R3 having the property that it cannot be an attractor for any homeomorphism of R3. To this end we show that the fundamental group of the complement of an attractor has certain finite generation property that the complement of T does not have.
L. Antoine, Sur l’homomorphie de figures et de leurs voisinages, J. Math. Pures Appl. 86 (1921), 221–325.
L. Antoine, Sur les ensembles parfaits partout discontinus, C.R. Acad. Sci. Paris 173 (1921), 284–285.
R.H. Bing, Locally tame sets are tame, Ann. Math. (2) 59 (1954), 145–158.
R.H. Bing, Approximating surfaces with polyhedral ones, Ann. Math. 65 (1957), 456–483.
N.P. Bhatia and G.P. Szego, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer–Verlag, 1970.
S. Crovisier and M. Rams, IFS attractors and Cantor sets, Topology Appl. 153 (2006), 1849–1859.
P.F. Duvall and L.S. Husch, Attractors of iterated function systems, Proc. Amer. Math. Soc. 116 (1992), 279–284.
B.M. Garay, Strong cellularity and global asymptotic stability, Fund. Math. 138 (1991), 147–154.
B. Gunther, A compactum that cannot be an attractor of a self-map on a manifold, Proc. Amer. Math. Soc. 120 (1994), 653–655.
B. Gunther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc. 119 (1993), 321–329.
V. Jimenez and J. Llibre, A topological characterization of the !-limit sets for analytic flows on the plane, the sphere and the projective plane, Adv. Math. 216 (2007), 677–710.
V. Jimenez and D. Peralta-Salas, Global attractors of analytic plane flows, Ergodic Theory Dynam. Systems 29 (2009), 967–981.
H. Kato, Attractors in Euclidean spaces and shift maps on polyhedra, Houston J. Math. 24 (1998), 671–680.
E.E. Moise, Geometric topology in dimensions 2 and 3, Springer–Verlag”, 1977.
D. Rolfsen, Knots and Links, AMS Chelsea Publishing, 2003.
J.J. Sanchez-Gabites, How strange can an attractor for a dynamical system in a 3-manifold look?, Nonlinear Anal. 74 (2011), 6162–6185.
J.J. Sanchez-Gabites, Arcs, balls, and spheres that cannot be attractors in R3, submitted, http://arxiv.org/abs/1406.5482
J.M.R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (1994), 330–344.
R.B. Sher, Concerning wild Cantor sets in E3, Proc. Amer. Math. Soc. 19 (1968), 1195–1200.
E.H. Spanier, Algebraic Topology, McGraw–Hill Book Co., 1966.
D.G. Wright, Rigid sets in En, Pacific J. Math. 121 (1986), 245–256.
How to Cite
Number of views and downloads: 136
Number of citations: 0