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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
References:
[1]

L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153. doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[2]

L. Alsedà, D. Juher and P. Mumbrú, Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341. doi: 10.1142/S021812740300656X.  Google Scholar

[3]

L. Alsedà, D. Juher and P. Mumbrú, Periodic behavior on trees, Ergodic Theory Dynam. Systems, 25 (2005), 1373-1400. doi: 10.1017/S0143385704000896.  Google Scholar

[4]

L. Alsedà, D. Juher and P. Mumbrú, Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst., 20 (2008), 511-541.  Google Scholar

[5]

L. Alsedà, D. Juher and P. Mumbrú, On the preservation of combinatorial types for maps on trees, Ann. Inst. Fourier (Grenoble), 55 (2005), 2375-2398. doi: 10.5802/aif.2164.  Google Scholar

[6]

L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.2307/2001417.  Google Scholar

[7]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension one, vol. 5 of Advanced Series in Nonlinear Dynamics, 2nd edition, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/4205.  Google Scholar

[8]

L. Alsedà and S. Ruette, Rotation sets for graph maps of degree 1, 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]

L. Alsedà and S. Ruette, Periodic orbits of large diameter for circle maps, Proc. Amer. Math. Soc., 138 (2010), 3211-3217. doi: 10.1090/S0002-9939-10-10332-3.  Google Scholar

[10]

S. Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$, Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.  Google Scholar

[11]

S. Baldwin and J. Llibre, Periods of maps on trees with all branching points fixed, Ergodic Theory Dynam. Systems, 15 (1995), 239-246. doi: 10.1017/S014338570000835X.  Google Scholar

[12]

C. Bernhardt, Vertex maps for trees: algebra and periods of periodic orbits, Discrete Contin. Dyn. Syst., 14 (2006), 399-408. doi: 10.3934/dcds.2006.14.399.  Google Scholar

[13]

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc., 82 (1981), 481-486. doi: 10.1090/S0002-9939-1981-0612745-7.  Google Scholar

[14]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, 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]

R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[16]

M. C. Leseduarte and J. Llibre, On the set of periods for $\sigma$ maps, Trans. Amer. Math. Soc., 347 (1995), 4899-4942. doi: 10.2307/2155068.  Google Scholar

[17]

J. Llibre, J. Paraños and J. Á. Rodríguez, Periods for continuous self-maps of the figure-eight space, 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]

A. Málaga, Dinámica de Grafos de un Ciclo Para Funciones de Grado Diferente de uno, (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]

M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227 (1983).  Google Scholar

[20]

O. M. Šarkovs'kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., 16 (1964), 61-71.  Google Scholar

[21]

A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, 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]

P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248. doi: 10.1007/BF01614086.  Google Scholar

show all references

References:
[1]

L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153. doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[2]

L. Alsedà, D. Juher and P. Mumbrú, Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341. doi: 10.1142/S021812740300656X.  Google Scholar

[3]

L. Alsedà, D. Juher and P. Mumbrú, Periodic behavior on trees, Ergodic Theory Dynam. Systems, 25 (2005), 1373-1400. doi: 10.1017/S0143385704000896.  Google Scholar

[4]

L. Alsedà, D. Juher and P. Mumbrú, Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst., 20 (2008), 511-541.  Google Scholar

[5]

L. Alsedà, D. Juher and P. Mumbrú, On the preservation of combinatorial types for maps on trees, Ann. Inst. Fourier (Grenoble), 55 (2005), 2375-2398. doi: 10.5802/aif.2164.  Google Scholar

[6]

L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.2307/2001417.  Google Scholar

[7]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension one, vol. 5 of Advanced Series in Nonlinear Dynamics, 2nd edition, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/4205.  Google Scholar

[8]

L. Alsedà and S. Ruette, Rotation sets for graph maps of degree 1, 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]

L. Alsedà and S. Ruette, Periodic orbits of large diameter for circle maps, Proc. Amer. Math. Soc., 138 (2010), 3211-3217. doi: 10.1090/S0002-9939-10-10332-3.  Google Scholar

[10]

S. Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$, Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.  Google Scholar

[11]

S. Baldwin and J. Llibre, Periods of maps on trees with all branching points fixed, Ergodic Theory Dynam. Systems, 15 (1995), 239-246. doi: 10.1017/S014338570000835X.  Google Scholar

[12]

C. Bernhardt, Vertex maps for trees: algebra and periods of periodic orbits, Discrete Contin. Dyn. Syst., 14 (2006), 399-408. doi: 10.3934/dcds.2006.14.399.  Google Scholar

[13]

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc., 82 (1981), 481-486. doi: 10.1090/S0002-9939-1981-0612745-7.  Google Scholar

[14]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, 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]

R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[16]

M. C. Leseduarte and J. Llibre, On the set of periods for $\sigma$ maps, Trans. Amer. Math. Soc., 347 (1995), 4899-4942. doi: 10.2307/2155068.  Google Scholar

[17]

J. Llibre, J. Paraños and J. Á. Rodríguez, Periods for continuous self-maps of the figure-eight space, 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]

A. Málaga, Dinámica de Grafos de un Ciclo Para Funciones de Grado Diferente de uno, (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]

M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227 (1983).  Google Scholar

[20]

O. M. Šarkovs'kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., 16 (1964), 61-71.  Google Scholar

[21]

A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, 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]

P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248. doi: 10.1007/BF01614086.  Google Scholar

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