A Sharkovsky theorem for non-locally connected spaces

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October 2012, 32(10): 3485-3499. doi: 10.3934/dcds.2012.32.3485

A Sharkovsky theorem for non-locally connected spaces

G. Conner ^1, , Christopher P. Grant ^1, and Mark H. Meilstrup ^2,

1.	Department of Mathematics, Brigham Young University, Provo, UT 84602, United States
2.	Mathematics Department, Southern Utah University, Cedar City, UT, 84720, United States

Received April 2011 Revised August 2011 Published May 2012

Abstract
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We extend Sharkovsky's Theorem to several new classes of spaces, which include some well-known examples of non-locally connected continua, such as the topologist's sine curve and the Warsaw circle. In some of these examples the theorem applies directly (with the same ordering), and in other examples the theorem requires an altered partial ordering on the integers. In the latter case, we describe all possible sets of periods for functions on such spaces, which are based on multiples of Sharkovsky's order.

Keywords: non-locally connected, arc-like continua., Sharkovsky ordering, combinatorial dynamics, periodicity.

Mathematics Subject Classification: Primary: 37E1.

Citation: G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485

References:

[1]	Ll. 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.
[2]	Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.1090/S0002-9947-1989-0958882-0.
[3]	Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,'' 2^nd edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
[4]	Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5-71.
[5]	S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces, Houston J. Math., 17 (1991), 39-53.
[6]	Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the n-od, Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.
[7]	Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.
[8]	Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang 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., 819, Springer, Berlin, (1980), 18-34.
[9]	Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: A natural direct proof, Amer. Math. Monthly, 118 (2011), 229-244. doi: 10.4169/amer.math.monthly.118.03.229.
[10]	A. I. Demin, Coexistence of periodic, almost periodic and recurrent points of transformations of n-od, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1996, 84-87; translation in Moscow Univ. Math. Bull., 51 (1996), 46-48.
[11]	Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' 2^nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[12]	Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space, Reprint of the paper reviewed in MR1361924 (97d:58161), World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 8, in "Thirty Years After Sharkovskiĭ's Theorem: New Perspectives" (Murcia, 1994), World Scientific Publ., River Edge, NJ, (1995), 95-106.
[13]	W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc., 107 (1989), 549-553. doi: 10.1090/S0002-9939-1989-0984796-1.
[14]	Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc., 315 (1989), 173-188. doi: 10.2307/2001378.
[15]	Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227.
[16]	Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.
[17]	H. Schirmer, A topologist's view of Sharkovsky's theorem, Houston J. Math., 11 (1985), 385-395.
[18]	A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Translated from the Russian by J. Tolosa, 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.
[19]	H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$, in "Numerical Solution of Nonlinear Equations" (Bremen, 1980), Lecture Notes in Math., 878, Springer, Berlin-New York, (1981), 351-370.
[20]	Jin Cheng Xiong, Xiang Dong Ye, Zhi Qiang Zhang and Jun Huang, Some dynamical properties of continuous maps on the Warsaw circle, (Chinese), Acta Math. Sinica (Chin. Ser.), 39 (1996), 294-299.
[21]	Li Zhen Zhou and You Cheng Zhou, Some dynamical properties of continuous self-maps on the $k$-Warsaw circle, (Chinese), J. Zhejiang Univ. Sci. Ed., 29 (2002), 12-16.

show all references

References:

[1]	Ll. 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.
[2]	Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.1090/S0002-9947-1989-0958882-0.
[3]	Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,'' 2^nd edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
[4]	Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5-71.
[5]	S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces, Houston J. Math., 17 (1991), 39-53.
[6]	Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the n-od, Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.
[7]	Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.
[8]	Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang 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., 819, Springer, Berlin, (1980), 18-34.
[9]	Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: A natural direct proof, Amer. Math. Monthly, 118 (2011), 229-244. doi: 10.4169/amer.math.monthly.118.03.229.
[10]	A. I. Demin, Coexistence of periodic, almost periodic and recurrent points of transformations of n-od, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1996, 84-87; translation in Moscow Univ. Math. Bull., 51 (1996), 46-48.
[11]	Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' 2^nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
[12]	Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space, Reprint of the paper reviewed in MR1361924 (97d:58161), World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 8, in "Thirty Years After Sharkovskiĭ's Theorem: New Perspectives" (Murcia, 1994), World Scientific Publ., River Edge, NJ, (1995), 95-106.
[13]	W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc., 107 (1989), 549-553. doi: 10.1090/S0002-9939-1989-0984796-1.
[14]	Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc., 315 (1989), 173-188. doi: 10.2307/2001378.
[15]	Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227.
[16]	Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.
[17]	H. Schirmer, A topologist's view of Sharkovsky's theorem, Houston J. Math., 11 (1985), 385-395.
[18]	A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Translated from the Russian by J. Tolosa, 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.
[19]	H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$, in "Numerical Solution of Nonlinear Equations" (Bremen, 1980), Lecture Notes in Math., 878, Springer, Berlin-New York, (1981), 351-370.
[20]	Jin Cheng Xiong, Xiang Dong Ye, Zhi Qiang Zhang and Jun Huang, Some dynamical properties of continuous maps on the Warsaw circle, (Chinese), Acta Math. Sinica (Chin. Ser.), 39 (1996), 294-299.
[21]	Li Zhen Zhou and You Cheng Zhou, Some dynamical properties of continuous self-maps on the $k$-Warsaw circle, (Chinese), J. Zhejiang Univ. Sci. Ed., 29 (2002), 12-16.

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