American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm
  • DCDS Home
  • This Issue
  • Next Article
    Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system
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, 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.  Google Scholar

[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.  Google Scholar

[3]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,'' 2nd edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.  Google Scholar

[4]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5-71.  Google Scholar

[5]

S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces, Houston J. Math., 17 (1991), 39-53.  Google Scholar

[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.  Google Scholar

[7]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.  Google Scholar

[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.  Google Scholar

[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.  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, 84-87; translation in Moscow Univ. Math. Bull., 51 (1996), 46-48.  Google Scholar

[11]

Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 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), 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.  Google Scholar

[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.  Google Scholar

[14]

Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc., 315 (1989), 173-188. 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), 221-227.  Google Scholar

[16]

Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.  Google Scholar

[17]

H. Schirmer, A topologist's view of Sharkovsky's theorem, Houston J. Math., 11 (1985), 385-395.  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 "Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'' (Murcia, 1994), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  Google Scholar

[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.  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), Acta Math. Sinica (Chin. Ser.), 39 (1996), 294-299.  Google Scholar

[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.  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), 311-341. 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), 475-538. 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, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.  Google Scholar

[4]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, (1981), 5-71.  Google Scholar

[5]

S. Baldwin, Some limitations toward extending Šarkovskiĭ's theorem to connected linearly ordered spaces, Houston J. Math., 17 (1991), 39-53.  Google Scholar

[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.  Google Scholar

[7]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.  Google Scholar

[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.  Google Scholar

[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.  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, 84-87; translation in Moscow Univ. Math. Bull., 51 (1996), 46-48.  Google Scholar

[11]

Robert L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 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), 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.  Google Scholar

[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.  Google Scholar

[14]

Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc., 315 (1989), 173-188. 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), 221-227.  Google Scholar

[16]

Sam B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.  Google Scholar

[17]

H. Schirmer, A topologist's view of Sharkovsky's theorem, Houston J. Math., 11 (1985), 385-395.  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 "Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'' (Murcia, 1994), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  Google Scholar

[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.  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), Acta Math. Sinica (Chin. Ser.), 39 (1996), 294-299.  Google Scholar

[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.  Google Scholar

[1]

Alexander Blokh, Lex Oversteegen, Vladlen Timorin. Non-degenerate locally connected models for plane continua and Julia sets. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5781-5795. doi: 10.3934/dcds.2017251
[2]

Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39
[3]

Cristian Barbarosie, Anca-Maria Toader. Optimization of bodies with locally periodic microstructure by varying the periodicity pattern. Networks & Heterogeneous Media, 2014, 9 (3) : 433-451. doi: 10.3934/nhm.2014.9.433
[4]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[5]

Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205
[6]

Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264
[7]

Tomasz Kaczynski, Marian Mrozek, Thomas Wanner. Towards a formal tie between combinatorial and classical vector field dynamics. Journal of Computational Dynamics, 2016, 3 (1) : 17-50. doi: 10.3934/jcd.2016002
[8]

Manuel Fernández-Martínez. A real attractor non admitting a connected feasible open set. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 723-725. doi: 10.3934/dcdss.2019046
[9]

Yangyou Pan, Yuzhen Bai, Xiang Zhang. Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1761-1774. doi: 10.3934/dcdss.2019116
[10]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321
[11]

Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 79-93. doi: 10.3934/mbe.2017006
[12]

Yu-Chung Tsao. Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits. Journal of Industrial & Management Optimization, 2017, 13 (1) : 329-347. doi: 10.3934/jimo.2016020
[13]

Rafael Obaya, Víctor M. Villarragut. Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 185-207. doi: 10.3934/dcdsb.2013.18.185
[14]

Zhiyuan Zhen, Honglin Yang, Wenyan Zhuo. Financing and ordering decisions in a capital-constrained and risk-averse supply chain for the monopolist and non-monopolist supplier. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021104
[15]

Conrad Bertrand Tabi, Alidou Mohamadou, Timoleon Crepin Kofane. Soliton-like excitation in a nonlinear model of DNA dynamics with viscosity. Mathematical Biosciences & Engineering, 2008, 5 (1) : 205-216. doi: 10.3934/mbe.2008.5.205
[16]

Vanderlei Horita, Nivaldo Muniz. Basin problem for Hénon-like attractors in arbitrary dimensions. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 481-504. doi: 10.3934/dcds.2006.15.481
[17]

Marina Gonchenko, Sergey Gonchenko, Klim Safonov. Reversible perturbations of conservative Hénon-like maps. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1875-1895. doi: 10.3934/dcds.2020343
[18]

Uwe an der Heiden, Mann-Lin Liang. Sharkovsky orderings of higher order difference equations. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 599-614. doi: 10.3934/dcds.2004.11.599
[19]

Yeshun Sun, Chung-Chun Yang. Buried points and lakes of Wada Continua. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 379-382. doi: 10.3934/dcds.2003.9.379
[20]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (219)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy