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Title

Rotation sets for subshifts of finite type

Authors 1

Affiliations

  1. Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana 46202, U.S.A.

Abstract

For a dynamical system (X,f) and a function φ:X→ℝN the rotation set is defined. The case when (X,f) is a transitive subshift of finite type and φ depends on the cylinders of length 2 is studied. Then the rotation set is a convex polyhedron. The rotation vectors of periodic points are dense in the rotation set. Every interior point of the rotation set is a rotation vector of an ergodic measure.

Bibliography

  1. [ALM] Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynamics 5, World Scientific, Singapore, 1993.
  2. [A] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Grundlehren Math. Wiss. 250, Springer, New York, 1988.
  3. [Bi] G. D. Birkhoff, Sur quelques courbes fermées remarquables, Bull. Soc. Math. France 60 (1932), 1-26.
  4. [Bl1] A. Blokh, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, preprint, 1993.
  5. [Bl2] A. Blokh, Functional rotation numbers for one dimensional maps, preprint, 1993.
  6. [Ch] A. Chenciner, La dynamique au voisinage d'un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather, Séminaire Bourbaki vol. 1983/84, Astérisque 121-122 (1985), 147-170.
  7. [F] J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), 107-115.
  8. [He] M. R. Herman, Existence et non existence de tores invariants par des difféomorphismes symplectiques, Séminaire sur les Équations aux Dérivées Partielles 1987-1988, Exp. No. XIV, École Polytechnique, Palaiseau, 1988.
  9. [KMG] S. Kim, R. S. MacKay and J. Guckenheimer, Resonance regions for families of torus maps, Nonlinearity 2 (1989), 391-404.
  10. [LM] J. Llibre and R. S. MacKay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynamical Systems 11 (1991), 115-128.
  11. [MaT] B. Marcus and S. Tuncel, The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains, ibid., 129-180.
  12. [MN] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 456 (1991).
  13. [MT] M. Misiurewicz and J. Tolosa, Entropy of snakes and the restricted variational principle, Ergodic Theory Dynamical Systems 12 (1992), 791-802.
  14. [MZ1] M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), 490-506.
  15. [MZ2] M. Misiurewicz and K. Ziemian, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), 45-52.
  16. [NPT] S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5-71.
  17. [O] G. Orwell, Animal Farm, Harcourt, Brace and Co., New York, 1946.
  18. [P] H. Poincaré, Sur les courbes définies par les équations différentielles, in: Oeuvres complètes, Vol. 1, Gauthier-Villars, Paris, 1952, 137-158.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm146/fm14628.pdf

Fundamenta Mathematicae
1994-1995/146/2
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Pages:
189-201
Main language of publication
English
Received
1994-05-10
Published
1995
Exact and natural sciences