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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
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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

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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