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Weak disks of Denjoy minimal sets

Published online by Cambridge University Press:  19 September 2008

Philip Boyland
Philip Boyland
Affiliation:
Institute for Mathematical Sciences, SUNY at Stony Brook, Stony Brook, NY 11794, USA
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Abstract

We investigate the existence of Denjoy minimal sets and, more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It is shown that for the full two-shift, the collection of such invariant sets with the weak topology contains topological balls of all finite dimensions. One implication is an analogous result that holds for diffeomorphisms with transverse homoclinic points. It is also shown that the union of Denjoy minimal sets is dense in the two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 4 , December 1993 , pp. 597 - 614
Copyright
Copyright © Cambridge University Press 1993

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References

REFERENCES

[A]Auslander, J.. Minimal Flows and their Extensions, North-Holland Mathematics Studies 153. North-Holland, Amsterdam, 1988.Google Scholar
[B-F]Blanchard, P. & Franks, J.. The dynamical complexity of orientation reversing homeomorphisms of surfaces. Inv. Math. 62 (1980), 333–339.CrossRefGoogle Scholar
[Bd2]Boyland, P.. An analog of Sharkovski's theorem for twist maps. Contemp. Math. 81 (1988), 119–133.CrossRefGoogle Scholar
[C]Choquet, G.. Lectures on Analysis. Vol. 2. Benjamin, New York, 1969.Google Scholar
[F]Furstenberg, H.. Strict ergodicity and transformations of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
[F-W]Furstenberg, H. & Weiss, B.. On almost 1–1 extensions. Isr. J. Math. 65 (1989), 311–322.Google Scholar
[GH]Gottschalk, W. & Hedlund, G.. Topological Dynamics. Amer. Math. Soc. Coll. Pub., vol. 36, 1955.Google Scholar
[H]Handel, M.. The entropy of orientation reversing homeomorphisms of surfaces. Topology 21 (1982), 291–296.CrossRefGoogle Scholar
[Hm]Herman, M.. Construction d'un difféomorphisme d'entropie topologique non nulle. Ergod. Th. & Dynam. Sys. 1 (1981), 65–76.Google Scholar
[H-H1]Hockett, K. & Holmes, P.. Bifurcation to rotating Cantor sets in maps of the circle. Nonlinearity 1 (1988), 603–616.Google Scholar
[H-H2]Hockett, K. & Holmes, P.. Josephson's junction, annulus maps, Birhoff attractors, horseshoes and rotation sets. Ergod. Th. & Dynam. Sys. 6 (1986), 205–239.Google Scholar
[K]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51 (1980), 137–173.Google Scholar
[M]Mather, J.. More Denjoy minimal sets for area-preserving mappings. Comm. Math. Helv. 60 (1985), 508–557.Google Scholar
[My]Markley, N.. Homeomorphisms of the circle without periodic orbits. Proc. Land. Math. Soc. 20 (1970), 688–698.CrossRefGoogle Scholar
[M-P]Markley, N. & Paul, M.. Almost automorphic symbolic minimal sets without unique ergodicity. Isr. J. Math. 34 (1979), 259–372.Google Scholar
[O]Oxtoby, J.. Ergodic sets Bull. Amer. Math. Soc. 58 (1952), 116–136.Google Scholar
[R]Rees, M.. A minimal positive entropy homeomorphism of the two torus. J. London Math. Soc. 23 (1981), 537–550.CrossRefGoogle Scholar
[Rl]Ruelle, D.. Elements of Bifurcation Theory. Academic, New York, 1989.Google Scholar
[Sc]Schweitzer, P. A.. Counterexamples to the Seifert conjecture and opening closed leaves of foliations. Ann. Math. 100 (1974), 386–400.Google Scholar
[S]Smillie, J.. Periodic points of surface homeomorphisms with zero entropy. Ergod. Th. & Dynam. Sys. 3 (1983), 315–334.CrossRefGoogle Scholar
[W]Walters, P.. An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79. Springer, Berlin, 1982.CrossRefGoogle Scholar
[Wm]Williams, S.. Toeplitz minimal flows that are not uniquely ergodic. Z. Wahrscheinlichkeitstheori verw. Geb. 67 (1984), 95–107.CrossRefGoogle Scholar