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Rotating an interval and a circle
Author(s):
Alexander
Blokh;
Michal
Misiurewicz
Journal:
Trans. Amer. Math. Soc.
351
(1999),
63-78.
MSC (1991):
Primary 54H20, 58F03, 58F08
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Abstract:
We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea.
References:
- [ALM]
- Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific, River Edge, NJ, 1993. MR 95j:58042
- [B]
- A. Blokh, Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, Ergod. Th. & Dynam. Sys. 15 (1995), 1-14. MR 96c:58058
- [BM1]
- A. Blokh and M. Misiurewicz, Entropy of twist interval maps, Isr. J. Math. 102 (1997), 61-99. CMP 98:06
- [BM2]
- -, New order for periodic orbits of interval maps, Ergod. Th. & Dynam. Sys. 17 (1997), 565-574. CMP 97:13
- [BM3]
- -, Entropy and over-rotation numbers for interval maps, Proc. Steklov Inst. Math. 216 (1997), 229-235.
- [BK]
- J. Bobok and M. Kuchta, X-minimal patterns and a generalization of Sharkovskii's theorem, Fund. Math. 156 (1998), 33-66. CMP 98:09
- [MN]
- M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94 (1991). MR 92h:58105
- [NPT]
- S. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffemorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5-71. MR 84g:58080
- [P]
- H. Poincaré, Sur les courbes définies par les équations différentielles, Oeuvres completes, vol. 1, 137-158, Gauthier-Villars, Paris, 1952.
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Additional Information:
Alexander
Blokh
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email:
ablokh@math.uab.edu
Michal
Misiurewicz
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email:
mmisiure@math.iupui.edu
DOI:
10.1090/S0002-9947-99-02367-3
PII:
S 0002-9947(99)02367-3
Keywords:
Periodic points,
rotation numbers,
interval maps
Received by editor(s):
August 26, 1996
Additional Notes:
The first author was partially supported by the NSF grant DMS-9626303. The second author was partially supported by the NSF grant DMS-9305899
Copyright of article:
Copyright
1999,
American Mathematical Society
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