On the ordering of 𝑛-modal cycles

Bernhardt, Chris

doi:10.1090/S0002-9947-96-01664-9

On the ordering of $n$-modal cycles
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Trans. Amer. Math. Soc. 348 (1996), 3827-3834 Request permission

Abstract:

The forcing relation on $n$-modal cycles is studied. If $\alpha$ is an $n$-modal cycle then the $n$-modal cycles with block structure that force $\alpha$ form a $2^n$-horseshoe above $\alpha$. If $n$-modal $\beta$ forces $\alpha$, and $\beta$ does not have a block structure over $\alpha$, then $\beta$ forces a $2$-horseshoe of simple extensions of $\alpha$.

References

Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1255515, DOI 10.1142/1980
Stewart Baldwin, Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math. 67 (1987), no. 2, 111–127. MR 913178, DOI 10.1016/0012-365X(87)90021-5
C. Bernhardt, E. Coven, M. Misiurewicz, and I. Mulvey, Comparing periodic orbits of maps of the interval, Trans. Amer. Math. Soc. 333 (1992), no. 2, 701–707. MR 1079051, DOI 10.1090/S0002-9947-1992-1079051-2
Chris Bernhardt, The ordering on permutations induced by continuous maps of the real line, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 155–160. MR 896787, DOI 10.1017/S0143385700003898
C. Bernhardt, Self-similarity maps for the set of unimodal cycles, Proc. Conf. “Thirty Years After Sarkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), Internat. J. Bifurcation Chaos Appl. Sci. Ergrg. 5 (1995), 1325–1330.
K. Brucks, R. Galeeva, D. Rockmore and C. Tresser, On the *-product in kneading theory, Preprint.
Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
Jaume Llibre and Pere Mumbrú, Extending the ${}^*$-product operator, European Conference on Iteration Theory (Batschuns, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 199–214. MR 1184166
John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
MichałMisiurewicz and Zbigniew Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94 (1991), no. 456, vi+112. MR 1086562, DOI 10.1090/memo/0456