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 - 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

[1]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408
[2]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376
[3]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[4]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[5]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[6]

Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311
[7]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[8]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[9]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[10]

Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020365
[11]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362
[12]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406
[13]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449
[14]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381
[15]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018
[16]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[17]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[18]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[19]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[20]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (128)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences