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Functional rotation numbers for one-dimensional maps

Author: A. M. Blokh
Journal: Trans. Amer. Math. Soc. 347 (1995), 499-513
MSC: Primary 58F03
MathSciNet review: 1270659
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Abstract: We introduce functional rotation numbers and sets for one-dimensional maps (we call them $ f$-rotation numbers and sets) and deduce some of their properties (density of $ {\text{f}}$-rotation numbers of periodic points in the $ {\text{f}}$-rotation set, conditions for the connectedness of the $ {\text{f}}$-rotation set) from the spectral decomposition theorem for one-dimensional maps.

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  • [ALM] L. Alseda, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, World Scientific, Singapore, 1993. MR 1255515 (95j:58042)
  • [AK] J. Auslander and Y. Katznelson, Continuous maps of the circle without periodic points, Israel J. Math. 32 (1979), 375-381. MR 571091 (81e:58048)
  • [Bl1] L. Block, Continuous maps of the Interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221-230. MR 0474240 (57:13887)
  • [Bl2] -, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391-398. MR 539925 (80m:58031)
  • [BCJM] L. Block, E. Coven, L. Jonker, and M. Misiurewicz, Primary cycles on the circle, Trans. Amer. Math. Soc. 311 (1989), 323-335. MR 974779 (90c:58082)
  • [B1] A. M. Blokh, Decomposition of dynamical systems on an interval, Russian Math. Surveys 38 (1983), no. 5, 133-134. MR 718829 (86d:54060)
  • [B2] -, On dynamical systems on one-dimensional branched manifolds. I, II, III, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen No. 46 (1986), 8-18; No. 47 (1986), 67-77; No. 48 (1987), 32-46. (Russian) MR 865783 (88j:58053)
  • [B3] -, The spectral decomposition for one-dimensional maps, SUNY at Stony Brook, Preprint #1991/14, September, Dynamics Reported (to appear).
  • [B4] -, Trees with snowflakes and zero entropy maps, Topology 33 (1994), 379-396. MR 1273790 (95b:58119)
  • [B5] -, On some properties of graph maps: pectral decomposition, Misiurewicz conjecture and abstract sets of periods, Max-Planck-Institut für Mathematik, Preprint #35, June 1991.
  • [B6] -, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, Ergodic Theory Dynamical Systems (to appear). MR 1314966 (96c:58058)
  • [DGS] M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math., vol. 527, Springer, Berlin, 1976. MR 0457675 (56:15879)
  • [I] R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), 107-111. MR 591976 (82i:58061)
  • [M1] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Ser. Sci. Math. 27 (1979), 167-169. MR 542778 (81b:58033)
  • [M2] -, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems 2 (1982), 221-227. MR 693977 (84j:58101)
  • [MS] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 6 (1980), 45-53. MR 579440 (82a:58030)
  • [MN] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 456 (1990). MR 1086562 (92h:58105)
  • [MZ1] M. Misiurewicz and K. Ziemian, Cycles for degree - $ 1$ circle maps, European Conference on Iteration Theory (ECIT 87), World Scientific, Singapore, 1989, pp. 8-25. MR 1085274 (92b:58186)
  • [MZ2] -, Rotation sets for maps of tori, J. London Math. Soc. (2) (1989), 490-506. MR 1053617 (91f:58052)
  • [N] Z. Nitecki, Periodic and limit orbits and the depth of the center for piecewise monotone interval maps, Proc. Amer. Math. Soc. 80 (1980), 511-514. MR 581016 (81j:58068)
  • [NPT] S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffemorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5-71. MR 699057 (84g:58080)
  • [P] H. Poincaré, Sur les courbes definies par les equations differentielle, Oeuvres Completes, vol. 1, Gauthier-Villars, Paris, 1952, pp. 137-158.
  • [S] A. N. Sharkovskii, Non-wandering points and the center of a continuous map of the line into itself, Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A (1964), 865-868. (Ukrainian) MR 0165178 (29:2467)
  • [Z] K. Ziemian, in preparation.

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Keywords: Periodic points, rotation numbers, interval maps
Article copyright: © Copyright 1995 American Mathematical Society

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