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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1142/S021812740300656X.  Google Scholar

[2]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$,, Trans. Amer. Math. Soc., 313 (1989), 475.  doi: 10.1090/S0002-9947-1989-0958882-0.  Google Scholar

[3]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,'', 2nd edition, 5 (2000).   Google Scholar

[4]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle,, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5.   Google Scholar

[5]

S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces,, Houston J. Math., 17 (1991), 39.   Google Scholar

[6]

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

[7]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites,, Topology Proc., 18 (1993), 19.   Google Scholar

[8]

Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18.   Google Scholar

[9]

Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: A natural direct proof,, Amer. Math. Monthly, 118 (2011), 229.  doi: 10.4169/amer.math.monthly.118.03.229.  Google Scholar

[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 (1996), 84.   Google Scholar

[11]

Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', 2nd edition, (1989).   Google Scholar

[12]

Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space,, Reprint of the paper reviewed in MR1361924 (97d:58161), 8 (1995), 95.   Google Scholar

[13]

W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua,, Proc. Amer. Math. Soc., 107 (1989), 549.  doi: 10.1090/S0002-9939-1989-0984796-1.  Google Scholar

[14]

Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua,, Trans. Amer. Math. Soc., 315 (1989), 173.  doi: 10.2307/2001378.  Google Scholar

[15]

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

[16]

Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'', Monographs and Textbooks in Pure and Applied Mathematics, 158 (1992).   Google Scholar

[17]

H. Schirmer, A topologist's view of Sharkovsky's theorem,, Houston J. Math., 11 (1985), 385.   Google Scholar

[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, 5 (1995), 1263.   Google Scholar

[19]

H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$,, in, 878 (1981), 351.   Google Scholar

[20]

Jin Cheng Xiong, Xiang Dong Ye, Zhi Qiang Zhang and Jun Huang, Some dynamical properties of continuous maps on the Warsaw circle,, (Chinese), 39 (1996), 294.   Google Scholar

[21]

Li Zhen Zhou and You Cheng Zhou, Some dynamical properties of continuous self-maps on the $k$-Warsaw circle,, (Chinese), 29 (2002), 12.   Google Scholar

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.  doi: 10.1142/S021812740300656X.  Google Scholar

[2]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$,, Trans. Amer. Math. Soc., 313 (1989), 475.  doi: 10.1090/S0002-9947-1989-0958882-0.  Google Scholar

[3]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,'', 2nd edition, 5 (2000).   Google Scholar

[4]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle,, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5.   Google Scholar

[5]

S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces,, Houston J. Math., 17 (1991), 39.   Google Scholar

[6]

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

[7]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites,, Topology Proc., 18 (1993), 19.   Google Scholar

[8]

Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18.   Google Scholar

[9]

Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: A natural direct proof,, Amer. Math. Monthly, 118 (2011), 229.  doi: 10.4169/amer.math.monthly.118.03.229.  Google Scholar

[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 (1996), 84.   Google Scholar

[11]

Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', 2nd edition, (1989).   Google Scholar

[12]

Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space,, Reprint of the paper reviewed in MR1361924 (97d:58161), 8 (1995), 95.   Google Scholar

[13]

W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua,, Proc. Amer. Math. Soc., 107 (1989), 549.  doi: 10.1090/S0002-9939-1989-0984796-1.  Google Scholar

[14]

Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua,, Trans. Amer. Math. Soc., 315 (1989), 173.  doi: 10.2307/2001378.  Google Scholar

[15]

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

[16]

Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'', Monographs and Textbooks in Pure and Applied Mathematics, 158 (1992).   Google Scholar

[17]

H. Schirmer, A topologist's view of Sharkovsky's theorem,, Houston J. Math., 11 (1985), 385.   Google Scholar

[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, 5 (1995), 1263.   Google Scholar

[19]

H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$,, in, 878 (1981), 351.   Google Scholar

[20]

Jin Cheng Xiong, Xiang Dong Ye, Zhi Qiang Zhang and Jun Huang, Some dynamical properties of continuous maps on the Warsaw circle,, (Chinese), 39 (1996), 294.   Google Scholar

[21]

Li Zhen Zhou and You Cheng Zhou, Some dynamical properties of continuous self-maps on the $k$-Warsaw circle,, (Chinese), 29 (2002), 12.   Google Scholar

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