Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T05:24:01.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity†

Published online by Cambridge University Press:  19 September 2008

J. Llibre  and
R. S. Mackay
J. Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autonoma de Barcelona, Bellaterra, 08193 Barcelona, Spain
R. S. Mackay
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV47AL, England
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 1 , March 1991 , pp. 115 - 128
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[ALMM]Alsedà, L., Llibre, J., Mañosas, F. & Misiurewicz, M.. Lower bounds of the topological entropy for continuous maps of the circle of degree one. Nonlinearity 1 (1988), 463–479.CrossRefGoogle Scholar
[AF]Asimov, D. & Franks, J.. Unremovable closed orbits. In: Geometric Dynamics, ed. Palis, J.. Springer Lect. Notes in Math. 1007 (1983), pp. 22–29.CrossRefGoogle Scholar
[BGKM]Baesens, C., Guckenheimer, J., Kim, S. & MacKay, R. S.. Three coupled oscillators: mode locking, global bifurcations and toroidal chaos. Preprint.Google Scholar
[BMPT]Bamon, R., Malta, I. P., Pacifico, M. J. & Takens, F.. Rotation intervals of endomorphisms of the circle. Ergod. Th. & Dynam. Sys. 4 (1984), 493–498.CrossRefGoogle Scholar
[BK]Birman, J. S. & Kidwell, M. E.. Fixed points of pseudo-Anosov diffeomorphisms of surfaces. Adv. Math. 46 (1982), 217–220.CrossRefGoogle Scholar
[Bow]Bowen, R.. Entropy and the Fundamental Group, Springer Lecture Notes in Math. 668 (1978), 21–29.Google Scholar
[Boy]Boyland, P.. An Analog of Sharkovski's theorem for twist maps. In: Hamiltonian dynamical systems. Contemp. Math. 81 (1988), 119–133.CrossRefGoogle Scholar
[BGMY]Block, L., Guckenheimer, J., Misiurewicz, M. & Young, L.-S.. Periodic points and topological entropy of one-dimensional maps. In: Global Theory of Dynamical Systems, eds. Nitecki, Z. and Robinson, R. C.. Springer Lecture Notes in Math. 819 (1980), pp. 18–34.CrossRefGoogle Scholar
[CGT]Chenciner, A., Gambaudo, J. M. & Tresser, C.. Une remarque sur la structure des endomorphismes de degré 1 du cercle. C.R. Acad. Sci. Paris Sér. I 299 (1984), 253–256.Google Scholar
[E1]Epstein, D. B. A.. Curves on 2-manifolds and isotopies. Acta Math. 115 (1966), 83–107.CrossRefGoogle Scholar
[E2]Epstein, D. B. A.. Pointwise periodic homeomorphisms. Proc. London Math. Soc. 42 (1981), 415–460.CrossRefGoogle Scholar
[FLP]Fathi, A., Laudenbach, F. & Poenaru, V.. Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979).Google Scholar
[Fra1]Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8 (1988), 99–107.Google Scholar
[Fra2]Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Math. Soc. 311 (1989), 107–115.CrossRefGoogle Scholar
[Fri1]Fried, D.. Growth rates of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 5 (1985), 539–563.CrossRefGoogle Scholar
[Fri2]Fried, D.. Ph.D thesis, University of California, Berkeley (1976).Google Scholar
[G]Gilman, J.. On the Nielsen type and the classification for the mapping class group. Adv. Math. 40 (1981), 68–96.CrossRefGoogle Scholar
[H1]Handel, M.. The entropy of orientation-reversing homeomorphisms of surfaces. Topology 21 (1982), 291–296.CrossRefGoogle Scholar
[H2]Handel, M.. Global shadowing of pseudo-Anosov homeomorphisms. Ergod. Th. & Dynam. Sys. 5 (1985), 373–377.CrossRefGoogle Scholar
[H3]Handel, M.. Periodic point free homeomorphisms of T2. Preprint.Google Scholar
[HT]Handel, M. & Thurston, W. P.. New proofs of some results of Nielsen. Adv. Math. 56 (1985), 173–191.CrossRefGoogle Scholar
[KMG]Kim, S., Mackay, R. S. and Guckenheimer, J.. Resonance regions for families of torus maps. Nonlinearity 2 (1989), 391–404.CrossRefGoogle Scholar
[MT]MacKay, R. S. & Tresser, C.. Badly ordered orbits of circle maps. Math. Proc. Camb. Phil. Soc. 96 (1984), 447–451.CrossRefGoogle Scholar
[Mil]Miller, R., Nielsen's viewpoint on geodesic laminations. Adv. Math. 45 (1982), 189–212.CrossRefGoogle Scholar
[Mis]Misiurewicz, M.. Twist sets for maps of the circle. Ergod. Th. & Dynam. Sys. 4 (1984), 391–404.CrossRefGoogle Scholar
[MZ1]Misiurewicz, M. & Ziemian, K.. Rotation sets for maps of tori. Preprint.CrossRefGoogle Scholar
[MZ2]Misiurewicz, M. & Ziemian, K.. Rotation sets and ergodic measures for torus homeomorphisms. Preprint.CrossRefGoogle Scholar
[N]Nielsen, J.. Untersuchung zur Topologie der geschlossen zweiseitigen Flaschen I–III. Acta Math. 50 (1927), 189–358; 53 (1929), 1–76; 58 (1932), 87–167.CrossRefGoogle Scholar
[T]Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19 (1988), 417–431.CrossRefGoogle Scholar
[W]Walters, P.. An Introduction to Ergodic Theory. Springer, New York (1982).CrossRefGoogle Scholar