The full periodicity kernel of the trefoil
Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 219-262.

We consider the following topological spaces: O={z:|z+i|=1}, O 3 =O{z:z 4 [0,1],Imz0}, O 4 =O{z:z 4 [0,1]}, 1 =O:|z-i|=1}{z:z[0,1]}, 2 = 1 {z:z 2 [0,1]}, et T={z:z= cos (3θ)e iθ ,0θ2π}. Set E{O 3 ,O 4 , 1 , 2 ,T}. An E map f is a continuous self-map of E having the branching point fixed. We denote by Per(f) the set of periods of all periodic points of f. The set K is the full periodicity kernel of E if it satisfies the following two conditions: (1) If f is an E map and KPer(f), then Per(f)=. (2) If S is a set such that for every E map f, SPer(f) implies Per(f)=, then KS. In this paper we compute the full periodicity kernel of O 3 ,O 4 , 1 , 2 and T.

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Leseduarte, Carme; Llibre, Jaume. The full periodicity kernel of the trefoil. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 219-262. doi : 10.5802/aif.1512. https://aif.centre-mersenne.org/articles/10.5802/aif.1512/

[ALM1] L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of Y, Trans. Amer. Math. Soc., 313 (1989), 475-538. | MR | Zbl

[ALM2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial dynamics in dimension one, Advanced Series in Nonlinear Dynamics, Vol. 5, World Scientific, 1993. | MR | Zbl

[ALMT] L. Alsedà, J. Llibre, M. Misiurewicz and C. Tresser, Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier, 39-4 (1989), 929-952. | Numdam | MR | Zbl

[AM] L. Alsedà and J.M. Moreno, Linear orderings and the full periodicity kernel for the n-star, J. Math. Anal. Appl., 180 (1993), 599-616. | MR | Zbl

[Ba] S. Baldwin, An extension of Sharkovskii's Theorem to the n-od, Ergod. Th. & Dynam. Sys., 11 (1991), 249-271. | MR | Zbl

[BL] S. Baldwin and J. Llibre, Periods of maps on trees with all branching points fixed, Ergodic Th. & Dynam. Sys., 15 (1995), 239-246. | MR | Zbl

[Bc1] L. Block, Periodic orbits of continuous maps of the circle, Trans. Amer. Math. Soc., 260 (1980), 553-562. | MR | Zbl

[Bc2] L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc., 82 (1981), 481-486. | MR | Zbl

[BGMY] L. Block, J. Guckenheimer, M. Misiurewicz and L.S. Young, Periodic points and topological entropy of one dimensional maps, Lecture Notes in Math., Springer-Verlag, Heidelberg, 819 (1980), 18-34. | MR | Zbl

[Bk1] A.M. Blokh, Periods implying almost all periods for tree maps, Nonlinearity, 5 (1992), 1375-1382. | MR | Zbl

[Bk2] A.M. Blokh, On some properties of graph maps : spectral descomposition, Misiurewicz conjecture and abstract sets of periods, preprint, Max-Plank-Institut für Mathematik, Bonn.

[LL1] C. Leseduarte and J. Llibre, On the set of periods for σ maps, to appear in Trans. Amer. Math. Soc. | MR | Zbl

[LL2] C. Leseduarte and J. Llibre, On the full periodicity kernel for one-dimensional maps, preprint, 1994,. | MR | Zbl

[LM] J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664. | MR | Zbl

[LPR1] J. Llibre, J. Paraños and J. A. Rodríguez, The full periodicity kernel for σ maps, J. Math. Anal. and Appl., 182 (1994), 639-651. | MR | Zbl

[LPR2] J. Llibre, J. Paraños and J. A. Rodríguez, Sets of periods for maps on connected graphs with zero Euler characteristic having all branching points fixed, to appear in J. Math. Anal. and Appl. | MR | Zbl

[LPR3] J. Llibre, J. Paraños and J.A. Rodríguez, International Journal of Bifurcation and Chaos, 5 (1995), 1395-1405. | Zbl

[LR] J. Llibre and R. Reventós, Sur le nombre d'orbites périodiques d'une application continue du cercle en lui-même, C. R. Acad. Sci. Paris, Sér. I Math., 294 (1982), 52-54. | MR | Zbl

[LY] T. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. | MR | Zbl

[M] P. Mumbrú, Periodes 1, 2, 3, 4, 5, 7 equivalen a caos, Master Thesis, Universitat Autònoma de Barcelona, 1982.

[Sh] A.N. Sharkovskii, Co-existence of the cycles of a continuous mapping of the line into itself (Russian), Ukrain. Math. Zh., 16 (1964), 61-71. | MR

[St] P.D. Straffin, Periodic points of continuous functions, Math. Mag., 51 (1978), 99-105. | MR | Zbl

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