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)



Topological entropy of standard type
monotone twist maps

Author: Oliver Knill
Journal: Trans. Amer. Math. Soc. 348 (1996), 2999-3013
MSC (1991): Primary 58F11, 28D20, 28D10; Secondary 58E30, 58F05
MathSciNet review: 1373642
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study invariant measures of families of monotone twist maps $S_{\gamma }(q,p)$ $=$ $(2q-p+ \gamma \cdot V'(q),q)$ with periodic Morse potential $V$. We prove that there exist a constant $C=C(V)$ such that the topological entropy satisfies $h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3$. In particular, $h_{top}(S_{\gamma }) \to \infty $ for $|\gamma | \to \infty $. We show also that there exist arbitrary large $\gamma $ such that $S_{\gamma }$ has nonuniformly hyperbolic invariant measures $\mu _{\gamma }$ with positive metric entropy. For large $\gamma $, the measures $\mu _{\gamma }$ are hyperbolic and, for a class of potentials which includes $V(q)=\sin (q)$, the Lyapunov exponent of the map $S$ with invariant measure $\mu _{\gamma }$ grows monotonically with $\gamma $.

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

  • [Ang 90] S. Angenent. Monotone recurrence relations, their Birkhoff orbits and topological entropy. Erg. Th. Dyn. Sys., 10:15-41, 1990. MR 91b:58181
  • [Ang 92] S. Angenent. A remark on the topological entropy and invariant circles of an area preserving twist map. In ``Twist Mappings and Their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R. Mc Gehee, K. Meyer, Springer, 1992, pp. 1--5. MR 94d:58078
  • [Aub 90] S. Aubry, G. Abramovici. Chaotic trajectories in the standard map. The concept of anti-integrability. Physica D, 43:199-219, 1990. MR 91j:58100
  • [Aub 92a] S. Aubry. The concept of anti-integrability: definition, theorems and applications to the standard map. In ``Twist Mappings and Their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R. Mc Gehee, K. Meyer, Springer, 1992, pp. 7--54. MR 94c:58179
  • [Aub 92b] S. Aubry, R. S. Mackay, C. Baesens C. Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models. Physica D, 56:123-134, 1992. MR 93e:58144
  • [Aub 95] S. Aubry. Anti-integrability in dynamical and variational problems, Physica D 86:284--296, 1995. CMP 1996:2. [Added in Proof]
  • [Ban 88] V. Bangert. Mather sets for twist maps and geodesics on tori. Dynamics Reported, 1:1-55, 1988. MR 90a:58145
  • [Bow 75] R. Bowen, D. Ruelle. The ergodic theory of axiom A flows. Inventiones Math., 29:181-202, 1975. MR 52:1786
  • [Bro 75] M. Brown, W. D. Neumann. Proof of the Poincaré-Birkhoff fixed point theorem. Michigan Math. J., 24:21-31, 1975. MR 56:6646
  • [Cyc 87] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon. Schrödinger Operators. Texts and Monographs in Physics, Springer, 1987. MR 88g:35003
  • [Den 76] M. Denker, C. Grillenberger, K. Sigmund. Ergodic Theory on Compact Spaces. Lecture Notes in Math., No. 527, Springer, 1976. MR 56:15879
  • [Dua 94] P. Duarte. Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 11:359--409, 1994. MR 95k:58115. [Added in Proof]
  • [Fat 89] A. Fathi. Expansiveness, hyperbolicity and Hausdorff dimension. Commun. Math. Phys., 126: 249-262, 1989. MR 90m:58159
  • [Fon 90] E. Fontich. Transversal homoclinic points of a class of conservative diffeomorphisms. J. Diff. Equ., 5, 87:1-27, 1990. MR 91j:58095
  • [Fri 89] S. Friedland, J. Milnor. Dynamical properties of plane polynomial automorphisms. Ergod. Th. & Dynam. Sys. 9:67-99, 1989. MR 90f:58163
  • [Her 83] M. Herman. Sur les courbes invariantes par les difféomorphismes de l'anneau. Astérisque, 103-104, Société Mathématique de France, 1983. MR 85m:58062
  • [Jak 95] A. Jakobsen and O. Knill. Iteration of the coupled map lattice construction, Phys. Lett. A 205:179--183, 1995. CMP 1996:1. [Added in Proof]
  • [Kat 80] A. Katok. Lyapunov exponents, entropy, and periodic orbits for diffeomorphisms. Publ.Math. IHES, 51:137-172, 1980. MR 81i:28022
  • [Kat 82] A. Katok. Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys., 2:185-194, 1982. MR 84m:58041
  • [Kni 93] O. Knill. Isospectral deformations of random Jacobi operators. Commun. Math. Phys., 151:403-426, 1993. MR 94j:58136
  • [Kni 94] O. Knill. Isospectral deformation of discrete random Laplacians. In ``On Three Levels", Ed.: M.Fannes et al., Plenum Press, New York, 1994, pp. 312--330.
  • [Laz 84] V. F. Lazutkin, D. Ya. Terman. Percival variational principle for invariant measures and commensurate-incommensurate phase transitions in one dimensional chains. Commun. Math. Phys, 94:511-522, 1984. MR 86k:58072
  • [Man 87] R. Mané. Ergodic Theory and differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete; Ser. 3, Bd. 8, New York, Springer, 1987. MR 88c:58040
  • [Mat 82] J. Mather. Existence of quasi-periodic orbits for twist homeomorphism of the annulus. Topology, 21:457-467, 1982. MR 84g:58084
  • [Mat 84] J. Mather. Amount of rotation about a point and the Morse index. Comm. Math. Phys., 94:141-153, 1984. MR 86c:58051
  • [Mos 73] J. Moser. Stable and Random Motion in Dynamical Systems. Princeton University Press, Princeton, 1973. MR 56:1355
  • [New 89] S. Newhouse. Continuity properties of entropy. Annals of Mathematics, 129:215-235, 1989. MR 90f:58108; 90m:58117
  • [Orn 70] D. S. Ornstein. Factors of Bernoulli shifts are Bernoulli shifts. Adv. Math., 5:349-364, 1970. MR 43:478b
  • [Per 80] I. C. Percival. Variational principle for invariant tori and cantori. Amer. Instit. of Physics Conf. Proc., 57:310-320, 1980. MR 82m:58040
  • [Pet 83] K. Petersen. Ergodic Theory. Cambridge Studies in Advanced Mathematics 2. Cambridge University Press, Cambridge, 1983. MR 87i:28002
  • [You 81] L-S. Young. On the prevalence of horseshoes. Trans. Amer. Math. Soc., 263:75-88, 1981. MR 82g:58070

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F11, 28D20, 28D10, 58E30, 58F05

Retrieve articles in all journals with MSC (1991): 58F11, 28D20, 28D10, 58E30, 58F05

Additional Information

Oliver Knill
Affiliation: Division of Physics, Mathematics and Astronomy, California Institute of Technology, 91125 Pasadena, California

Keywords: Ergodic theory, monotone twist maps, topological entropy, hyperbolic sets, Lyapunov exponents, invariant measures, variational principles
Received by editor(s): July 25, 1994
Additional Notes: This material is based upon work which was supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society