Bizarre topology is natural in dynamical systems

Kennedy, Judy A.; Yorke, James A.

doi:10.1090/S0273-0979-1995-00599-9

Bizarre topology is natural in dynamical systems
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by Judy A. Kennedy and James A. Yorke PDF

Bull. Amer. Math. Soc. 32 (1995), 309-316 Request permission

Abstract:

We describe an example of a ${C^{\infty }}$ diffeomorphism on a 7-manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7-manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the ${C^1}$ metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic.

References

Marcy Barge and Richard M. Gillette, Indecomposability and dynamics of invariant plane separating continua, Continuum theory and dynamical systems (Arcata, CA, 1989) Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 13–38. MR 1112800, DOI 10.1090/conm/117/1112800
Marcy Barge, Homoclinic intersections and indecomposability, Proc. Amer. Math. Soc. 101 (1987), no. 3, 541–544. MR 908665, DOI 10.1090/S0002-9939-1987-0908665-6
R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451, DOI 10.2140/pjm.1951.1.43
George D. Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1–26 (French). MR 1504983, DOI 10.24033/bsmf.1182
M. Charpentier, Sur quelques propriétés des courbes de M. Birkhoff, Bull. Soc. Math. France 62 (1934), 193–224 (French). MR 1505024, DOI 10.24033/bsmf.1221
Michael Handel, A pathological area preserving $C^{\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), no. 1, 163–168. MR 663889, DOI 10.1090/S0002-9939-1982-0663889-6
Michael-R. Herman, Construction of some curious diffeomorphisms of the Riemann sphere, J. London Math. Soc. (2) 34 (1986), no. 2, 375–384. MR 856520, DOI 10.1112/jlms/s2-34.2.375
Judy Kennedy and James A. Yorke, Basins of Wada, Phys. D 51 (1991), no. 1-3, 213–225. Nonlinear science: the next decade (Los Alamos, NM, 1990). MR 1128813, DOI 10.1016/0167-2789(91)90234-Z
Judy A. Kennedy and James A. Yorke, Pseudocircles in dynamical systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 349–366. MR 1187029, DOI 10.1090/S0002-9947-1994-1187029-5
Judy A. Kennedy and James A. Yorke, Pseudocircles, diffeomorphisms and perturbable dynamical systems, Ergodic Theory Dynam. Systems 16 (1996), no. 5, 1031–1057. MR 1417772, DOI 10.1017/S0143385700010166

The dynamics of some diffeomorphisms which admit complicated invariant sets

R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5
R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794, DOI 10.1007/BF02684369