# American Institute of Mathematical Sciences

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

## On the set of periods of sigma maps of degree 1

 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.
Citation: Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete and 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. [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. [3] L. Alsedà, D. Juher and P. Mumbrú, Periodic behavior on trees, Ergodic Theory Dynam. Systems, 25 (2005), 1373-1400. doi: 10.1017/S0143385704000896. [4] L. Alsedà, D. Juher and P. Mumbrú, Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst., 20 (2008), 511-541. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984. [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. [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. [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. [19] M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227 (1983). [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. [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. [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.

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##### 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. [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. [3] L. Alsedà, D. Juher and P. Mumbrú, Periodic behavior on trees, Ergodic Theory Dynam. Systems, 25 (2005), 1373-1400. doi: 10.1017/S0143385704000896. [4] L. Alsedà, D. Juher and P. Mumbrú, Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst., 20 (2008), 511-541. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984. [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. [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. [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. [19] M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227 (1983). [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. [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. [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.
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