Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Periodicity and decomposability of basin boundaries with irrational maps on prime ends

Author: Russell B. Walker
Journal: Trans. Amer. Math. Soc. 324 (1991), 303-317
MSC: Primary 54H20; Secondary 58F10, 58F11
MathSciNet review: 992609
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Planar basin boundaries of iterated homeomorphisms induce homeomorphisms on prime ends. When the basin is connected, simply connected, and has a compact connected boundary, the space of prime ends is a topological circle. If the induced homeomorphism on prime ends has rational rotation number, the basin boundary contains periodic orbits. Several questions as to basin boundary periodics, decomposability, and minimality, when the induced map on prime ends has irrational rotation number, are answered by construction of both homeomorphisms and diffeomorphisms. Examples in the literature of basin boundaries with interesting prime end dynamics have been sparse. Prime end dynamics has drawn recent interest as a natural tool for the study of strange attractors.

References [Enhancements On Off] (What's this?)

  • [AY] K. Alligood and J. Yorke, Accessible saddles on fractal basin boundaries, preprint, 1988. MR 1182653 (94b:58080)
  • [AS] K. Alligood and T. Sauer, Rotation numbers of periodic orbits in the Hénon map, preprint 1988. MR 972545 (90j:58075)
  • [BG] M. Barge and R. Gillette, Indecomposability and dynamics of invariant plane separating continua, preprint 1988. MR 1112800 (92k:54049)
  • [Be1] A. S. Besicovitch, A problem on topological transformations of the plane, Fund. Math. 28 (1937).
  • [Be2] -, A problem on topological transformations of the plane. II, Cambridge Philos. Soc. Proc. Math. 47 (1951). MR 0039247 (12:519e)
  • [Bi] G. D. Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932). MR 1504983
  • [C] C. Carathéodory, Uber die Begrenzung einfach zusammenhangender Gebiete, Math. Ann. 73 (1913).
  • [CL1] M. L. Cartwright and J. E. Littlewood, Some fixed point theorems, Ann. of Math. (2) 54 (1951), 1-37. MR 0042690 (13:148f)
  • [CL2] -, On non-linear differential equations of the second order..., J. London Math. Soc. 20 (1945), 180-189. MR 0016789 (8:68g)
  • [Den] A. Denjoy, Sur les courbes definies par les equations differentielles a la surface de tore, J. Math. Pures Appl. 11 (1932), 333-375.
  • [Dev] R. Devaney, An introduction to chaotic dynamical systems, Benjamin-Cummings, Menlo Park, Calif., 1986. MR 811850 (87e:58142)
  • [GH] W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R. I., 1955, pp. 139-142. MR 0074810 (17:650e)
  • [H] M. Handel, A pathological area preserving $ {C^ \infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982). MR 663889 (84f:58040)
  • [K] K. Kuratowski, Topology, vol. II, Academic Press, New York, 1968. MR 0259835 (41:4467)
  • [L] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc. No. 244 (1981). MR 617687 (82g:58052)
  • [M] J. Mather, Topological proofs of some purely topological consequences of Carathéodory's theory of prime ends, Selected Studies (Th. M. Rassias and G. M. Rassias, eds.), North-Holland, 1982, pp. 225-255. MR 662863 (84k:57004)
  • [Ne] M. Newman, Elements of the topology of plane sets of points, 2nd ed., Cambridge Univ. Press, 1954. MR 0044820 (13:483a)
  • [Ni] Z. Nitecki, Differentiable dynamics, an introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass., 1971. MR 0649788 (58:31210)
  • [R] M. Rees, A minmal positive entropy homeomorphism of the $ 2$-torus, J. London Math. Soc. (2) 23 (1981), 537-550. MR 616561 (82h:58045)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H20, 58F10, 58F11

Retrieve articles in all journals with MSC: 54H20, 58F10, 58F11

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society