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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 |
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. |
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. |
[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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