Functional rotation numbers for one-dimensional maps. (English) Zbl 0820.54018
The author introduces rotation numbers and sets for a continuous selfmap of a compact metric space \(X\) with respect to a bounded measurable function \(X\to \mathbb{R}^ n\). The paper is devoted mainly to studying these functional rotation numbers and sets for interval maps, for circle maps with periodic points and for degree one circle maps. From the spectral decomposition theorem for one-dimensional maps the author deduces some properties of the functional rotation numbers and sets (density of functional rotation numbers of periodic points in the functional rotation set, conditions for the connectedness of the functional rotation set).
Reviewer: Ľ.Snoha (Banská Bystrica)
MSC:
| 54H20 | Topological dynamics (MSC2010) |
| 37E99 | Low-dimensional dynamical systems |
| 37B99 | Topological dynamics |
References:
| [1] | Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. · Zbl 0963.37001 |
| [2] | J. Auslander and Y. Katznelson, Continuous maps of the circle without periodic points, Israel J. Math. 32 (1979), no. 4, 375 – 381. · Zbl 0442.54011 · doi:10.1007/BF02760466 |
| [3] | Louis Block, Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221 – 230. · Zbl 0389.54006 |
| [4] | Louis Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391 – 398. · Zbl 0386.54025 |
| [5] | Louis Block, Ethan M. Coven, Leo Jonker, and Michał Misiurewicz, Primary cycles on the circle, Trans. Amer. Math. Soc. 311 (1989), no. 1, 323 – 335. · Zbl 0662.58034 |
| [6] | A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk 38 (1983), no. 5(233), 179 – 180 (Russian). · Zbl 0557.28017 |
| [7] | A. M. Blokh, Dynamical systems on one-dimensional branched manifolds. I, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 46 (1986), 8 – 18 (Russian); English transl., J. Soviet Math. 48 (1990), no. 5, 500 – 508. · Zbl 0711.58024 · doi:10.1007/BF01095616 |
| [8] | -, The spectral decomposition for one-dimensional maps, SUNY at Stony Brook, Preprint #1991/14, September, Dynamics Reported (to appear). · Zbl 0828.58009 |
| [9] | A. M. Blokh, Trees with snowflakes and zero entropy maps, Topology 33 (1994), no. 2, 379 – 396. · Zbl 0804.54021 · doi:10.1016/0040-9383(94)90019-1 |
| [10] | -, On some properties of graph maps: pectral decomposition, Misiurewicz conjecture and abstract sets of periods, Max-Planck-Institut für Mathematik, Preprint #35, June 1991. |
| [11] | A. M. Blokh, Rotation numbers, twists and a Sharkovskiĭ-Misiurewicz-type ordering for patterns on the interval, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 1 – 14. · Zbl 0842.58018 · doi:10.1017/S014338570000821X |
| [12] | Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. · Zbl 0328.28008 |
| [13] | Ryuichi Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 107 – 111. · Zbl 0484.58027 · doi:10.1017/S0305004100057984 |
| [14] | Michał Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167 – 169 (English, with Russian summary). · Zbl 0459.54031 |
| [15] | Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems 2 (1982), no. 2, 221 – 227 (1983). · Zbl 0508.58038 |
| [16] | M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45 – 63. · Zbl 0445.54007 |
| [17] | Michał Misiurewicz and Zbigniew Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94 (1991), no. 456, vi+112. · Zbl 0745.58019 · doi:10.1090/memo/0456 |
| [18] | Michał Misiurewicz and Krystyna Ziemian, Cycles for degree -1 circle maps, European Conference on Iteration Theory (Caldes de Malavella, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 8 – 25. · Zbl 0641.58026 |
| [19] | Michał Misiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490 – 506. · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490 |
| [20] | Zbigniew Nitecki, Periodic and limit orbits and the depth of the center for piecewise monotone interval maps, Proc. Amer. Math. Soc. 80 (1980), no. 3, 511 – 514. · Zbl 0478.58019 |
| [21] | S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5 – 71. · Zbl 0518.58031 |
| [22] | H. Poincaré, Sur les courbes definies par les equations differentielle, Oeuvres Completes, vol. 1, Gauthier-Villars, Paris, 1952, pp. 137-158. |
| [23] | O. M. Šarkovs\(^{\prime}\)kiĭ, Fixed points and the center of a continuous mapping of the line into itself, Dopovidi Akad. Nauk Ukraïn. RSR 1964 (1964), 865 – 868 (Ukrainian, with Russian and English summaries). |
| [24] | K. Ziemian, in preparation. |
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