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)

 
 

 

A genealogy for finite kneading sequences of bimodal maps on the interval


Authors: John Ringland and Charles Tresser
Journal: Trans. Amer. Math. Soc. 347 (1995), 4599-4624
MSC: Primary 58F03; Secondary 58F14
DOI: https://doi.org/10.1090/S0002-9947-1995-1311914-5
MathSciNet review: 1311914
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval, building each sequence uniquely from a pair of shorter ones. There is a single pair at generation 0, with members of length $ 1$. Concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits. (See Figure 0.)

Figure 0. Loci of some finite kneading sequences for a two-parameter cubic family


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

  • [B] P. L. Boyland, Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals, Comm. Math. Phys. 106 (1986), 353-381. MR 859816 (88c:58045)
  • [DGMT] S. P. Dawson, R. Galeeva, J. Milnor, and C. Tresser, Real and complex dynamical systems (B. Branner and P. Hjorth, eds.), Kluwer, Dordrecht, 1995.
  • [D] R. L. Devaney, Genealogy of periodic points of maps of the interval, Trans. Amer. Math. Soc. 265 (1981), 137-146. MR 607112 (82e:58081)
  • [FK] S. Fraser and R. Kapral, Universal vector scaling in one-dimensional maps, Phys. Rev. A 30 (1984), 1017-1025. MR 757698 (85m:58129)
  • [GT] J. M. Gambaudo and C. Tresser, A monotonicity property in one dimensional dynamics, Contemporary Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 213-222. MR 1185089 (93i:58087)
  • [GP] L. Glass and R. Perez, Fine structure of phase locking, Phys. Rev. Lett. 48 (1982), 1772-1775. MR 662183 (83m:58067)
  • [MaT1] R. S. Mackay and C. Tresser, Transition to topological chaos for circle maps, Physica 19D (1986), 206-237. MR 844701 (87k:58182)
  • [MaT2] -, Some flesh on the skeleton: the bifurcation structure of bimodal maps, Physica 27D (1987), 412-422. MR 913688 (88m:58120)
  • [MaT3] -, Boundary of topological chaos for bimodal maps of the interval, J. London Math. Soc. 37 (1988), 164-181. MR 921755 (89b:58155)
  • [Mil] J. Milnor, Remarks on iterated cubic maps, Experimental Math. 1 (1992), 5-24. MR 1181083 (94c:58096)
  • [MT] J. Milnor and W. Thurston, On iterated maps of the interval, Lecture Notes in Math., vol. 1342, Springer, 1988, pp. 465-563. MR 970571 (90a:58083)
  • [M] P. Mumbrù and I. Rodriguez, Estructura, periòdica i entropia topològica de les applications bimodals, Doctoral Thesis, Universidad Autónoma de Barcelona, 1987.
  • [R] J. Ringland, A genealogy for the periodic orbits of a class of $ 1{\text{D}}$, Physica D 79 (1994), 289-298. MR 1306464 (95k:58131)
  • [RS1] J. Ringland and M. Schell, The Farey tree embodied--in bimodal maps of the interval, Phys. Lett. A 136 (1989), 379-386. MR 993051 (91c:58057)
  • [RS2] -, Genealogy and bifurcation skeleton for cycles of the iterated two-extremum map of the interval, SIAM J. Math. Anal. 22 (1989), 1354-1371. MR 1112513 (92h:58161)
  • [STZ] R. Siegel, C. Tresser, and G. Zettler, A decoding problem in dynamics and in number theory, Chaos 2 (1992), 473-493. MR 1195880 (93i:11151)
  • [TW] C. Tresser and R. F. Williams, Splitting words and Lorenz braids, Physica D 62 (1993), 15-21. MR 1207414 (94a:58065)
  • 1. See also W.-Z. Zeng and L. Glass, Symbolic dynamics and skeletons of circle maps, Physica D 40 (1989), 218-234. MR 1029464 (91g:58070)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F03, 58F14

Retrieve articles in all journals with MSC: 58F03, 58F14


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1311914-5
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society