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October  2015, 35(10): 4683-4734. doi: 10.3934/dcds.2015.35.4683

On the set of periods of sigma maps of degree 1

Lluís Alsedà 1, and  Sylvie Ruette 2,

1. 

Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona
2. 

Laboratoire de Mathématiques, CNRS UMR 8628, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay cedex, France

Received  September 2014 Revised  January 2015 Published  April 2015

We study the set of periods of degree 1 continuous maps from $\sigma$ into itself, where $\sigma$ denotes the space shaped like the letter $\sigma$ (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration have degree 1, the rotation theory can be used. We show that, when the interior of the rotation interval contains an integer, then the set of periods (of periodic points of any rotation number) is the set of all integers except maybe $1$ or $2$. We exhibit degree 1 $\sigma$-maps $f$ whose set of periods is a combination of the set of periods of a degree 1 circle map and the set of periods of a $3$-star (that is, a space shaped like the letter $Y$). Moreover, we study the set of periods forced by periodic orbits that do not intersect the circuit of $\sigma$; in particular, when there exists such a periodic orbit whose diameter (in the covering space) is at least $1$, then there exist periodic points of all periods.
Keywords: sigma maps, Rotation set, degree one, star maps, sets of periods, large orbits..
Mathematics Subject Classification: Primary: 37E15, 37E2.
Citation: Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683
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show all references

References:
[1]

Topology, 36 (1997), 1123-1153. doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[2]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341. doi: 10.1142/S021812740300656X.  Google Scholar

[3]

Ergodic Theory Dynam. Systems, 25 (2005), 1373-1400. doi: 10.1017/S0143385704000896.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 20 (2008), 511-541.  Google Scholar

[5]

Ann. Inst. Fourier (Grenoble), 55 (2005), 2375-2398. doi: 10.5802/aif.2164.  Google Scholar

[6]

Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.2307/2001417.  Google Scholar

[7]

2nd edition, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/4205.  Google Scholar

[8]

Ann. Inst. Fourier (Grenoble), 58 (2008), 1233-1294, URL http://aif.cedram.org/item?id=AIF_2008__58_4_1233_0. doi: 10.5802/aif.2384.  Google Scholar

[9]

Proc. Amer. Math. Soc., 138 (2010), 3211-3217. doi: 10.1090/S0002-9939-10-10332-3.  Google Scholar

[10]

Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.  Google Scholar

[11]

Ergodic Theory Dynam. Systems, 15 (1995), 239-246. doi: 10.1017/S014338570000835X.  Google Scholar

[12]

Discrete Contin. Dyn. Syst., 14 (2006), 399-408. doi: 10.3934/dcds.2006.14.399.  Google Scholar

[13]

Proc. Amer. Math. Soc., 82 (1981), 481-486. doi: 10.1090/S0002-9939-1981-0612745-7.  Google Scholar

[14]

in Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., Vol. 819, Springer, Berlin, 1980, 18-34.  Google Scholar

[15]

Math. Proc. Cambridge Philos. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[16]

Trans. Amer. Math. Soc., 347 (1995), 4899-4942. doi: 10.2307/2155068.  Google Scholar

[17]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1743-1754; On the extension of Sharkovskiĭ's theorem to connected graphs with non-positive Euler characteristic, in Proceedings of the Conference "Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'' (Murcia, 1994), 5 (1995), 1395-1405. doi: 10.1142/S0218127495001071.  Google Scholar

[18]

(Spanish) Master thesis, Universidad National de Ingeniería, Peru, 2011. Available from: http://cybertesis.uni.edu.pe/bitstream/uni/277/1/malaga_sa.pdf. Google Scholar

[19]

Ergodic Theory Dynamical Systems, 2 (1982), 221-227 (1983).  Google Scholar

[20]

Ukrain. Mat. Ž., 16 (1964), 61-71.  Google Scholar

[21]

Proceedings of the Conference "Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'' (Murcia, 1994), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273; Translated from the Russian [Ukrain. Mat. Zh., 16 (1964), 61-71; MR0159905] by J. Tolosa. doi: 10.1142/S0218127495000934.  Google Scholar

[22]

Comm. Math. Phys., 54 (1977), 237-248. doi: 10.1007/BF01614086.  Google Scholar

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