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May  2013, 33(5): 1819-1833. doi: 10.3934/dcds.2013.33.1819

Characterizations of $\omega$-limit sets in topologically hyperbolic systems

1. 

Heilbronn Institute of Mathematical Research, University of Bristol, Howard House, Queens Avenue, Bristol, BS8 1SN, United Kingdom

2. 

School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom

3. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków

4. 

Department of Mathematics, Baylor University, Waco, TX 76798–7328, United States

Received  December 2011 Revised  May 2012 Published  December 2012

It is well known that $\omega$-limit sets are internally chain transitive and have weak incompressibility; the converse is not generally true, in either case. However, it has been shown that a set is weakly incompressible if and only if it is an abstract $\omega$-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) $\omega$-limit set. In this paper we generalise these and other results, proving that the characterization for shifts of finite type holds in a variety of topologically hyperbolic systems (defined in terms of expansive and shadowing properties), and also show that the notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces.
Citation: Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819
References:
[1]

S. J. Agronsky, A. M. Bruckner, J. G. Ceder and T. L. Pearson, The structure of $\omega$-limit sets for continuous functions, Real Analysis Exchange, 15 (1989/90), 483-510.

[2]

N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems," North-Holland Publishing Co., Amsterdam, 1994.

[3]

F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps, Acta Mathematica Hungarica, 88 (2000), 291-300. doi: 10.1023/A:1026775906693.

[4]

A. D. Barwell, A characterization of $\omega$-limit sets of piecewise monotone maps of the interval, Fundamenta Mathematicae, 207 (2010), 161-174. doi: 10.4064/fm207-2-4.

[5]

A. D. Barwell, C. Good, R. Knight and B. E. Raines, A characterization of $\omega$-limit sets of shifts of finite type, Ergodic Theory and Dynamical Systems, 30 (2010), 21-31. doi: 10.1017/S0143385708001089.

[6]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Springer-Verlag, Berlin, 1992.

[7]

A. Blokh, A. M.Bruckner, P. D. Humke and J. Smítal, The space of $\omega$-limit sets of a continuous map of the interval, Transactions of the American Mathematical Society, 348 (1996), 1357-1372. doi: 10.1090/S0002-9947-96-01600-5.

[8]

R. Bowen, $\omega $-limit sets for axiom $A$ diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.

[9]

A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory and Dynamical Systems, 13 (1993), 7-19. doi: 10.1017/S0143385700007173.

[10]

L. Chen, Linking and the shadowing property for piecewise monotone maps, Proceedings of the American Mathematical Society, 113 (1991), 251-263. doi: 10.2307/2048466.

[11]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston MA, 1980.

[12]

E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Transactions of the American Mathematical Society, 308 (1988), 227-241. doi: 10.2307/2000960.

[13]

W. De Melo and S. van Strien, "One-Dimensional Dynamics," Springer-Verlag, Berlin, 1993.

[14]

C. Good, R. Knight and B. E. Raines, Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, Fundamenta Mathematicae, 192 (2006), 267-289. doi: 10.4064/fm192-3-6.

[15]

C. Good, B. E. Raines and R. Suabedissen, Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, Fundamenta Mathematicae, 205 (2009), 179-189. doi: 10.4064/fm205-2-6.

[16]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131. doi: 10.1023/A:1009044515567.

[17]

A. Kazda, The chain relation in sofic subshifts, Fundamenta Informaticae, 84 (2008), 375-390.

[18]

M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fundamenta Mathematicae, 212 (2011), 35-52. doi: 10.4064/fm212-1-3.

[19]

P. Kurka, "Topological and Symbolic Dynamics," Société Mathématique de France, Paris, 2003.

[20]

K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete and Continuous Dynamical Systems. Series A, 13 (2005), 533-540. doi: 10.3934/dcds.2005.13.533.

[21]

J. Milnor, On the concept of attractor, Communications in Mathematical Physics, 99 (1985), 177-195.

[22]

J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and Its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.

[23]

J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, Australian Mathematical Society Journal, Series A, 61 (1996), 57-72.

[24]

W. Parry, Symbolic dynamics and transformations of the unit interval, Transactions of the American Mathematical Society, 122 (1966), 368-378.

[25]

S. Y. Pilyugin, "Shadowing in Dynamical Systems," Springer-Verlag, Berlin, 1999.

[26]

F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods," Cambridge University Press, Cambridge, 2010.

[27]

D. Richeson and J. Wiseman, Positively expansive homeomorphisms of compact spaces, International Journal of Mathematics and Mathematical Sciences, 53-56 (2004), 2907-2910. doi: 10.1155/S0161171204312184.

[28]

K. Sakai, Various shadowing properties for positively expansive maps, Topology and Its Applications, 131 (2003), 15-31. doi: 10.1016/S0166-8641(02)00260-2.

[29]

A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrainskiĭ Matematicheskiĭ Zhurnal, 18 (1966), 127-130.

[30]

O. M. Šarkovskiĭ, On attracting and attracted sets, Doklady Akademii Nauk SSSR, 160 (1965), 1036-1038.

[31]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in "The Structure of Attractors in Dynamical Systems," Springer, Berlin, (1978).

[32]

R. Yang, Topological Anosov maps of non-compact metric spaces, Northeastern Mathematical Journal, 17 (2001), 120-126.

show all references

References:
[1]

S. J. Agronsky, A. M. Bruckner, J. G. Ceder and T. L. Pearson, The structure of $\omega$-limit sets for continuous functions, Real Analysis Exchange, 15 (1989/90), 483-510.

[2]

N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems," North-Holland Publishing Co., Amsterdam, 1994.

[3]

F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps, Acta Mathematica Hungarica, 88 (2000), 291-300. doi: 10.1023/A:1026775906693.

[4]

A. D. Barwell, A characterization of $\omega$-limit sets of piecewise monotone maps of the interval, Fundamenta Mathematicae, 207 (2010), 161-174. doi: 10.4064/fm207-2-4.

[5]

A. D. Barwell, C. Good, R. Knight and B. E. Raines, A characterization of $\omega$-limit sets of shifts of finite type, Ergodic Theory and Dynamical Systems, 30 (2010), 21-31. doi: 10.1017/S0143385708001089.

[6]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Springer-Verlag, Berlin, 1992.

[7]

A. Blokh, A. M.Bruckner, P. D. Humke and J. Smítal, The space of $\omega$-limit sets of a continuous map of the interval, Transactions of the American Mathematical Society, 348 (1996), 1357-1372. doi: 10.1090/S0002-9947-96-01600-5.

[8]

R. Bowen, $\omega $-limit sets for axiom $A$ diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.

[9]

A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory and Dynamical Systems, 13 (1993), 7-19. doi: 10.1017/S0143385700007173.

[10]

L. Chen, Linking and the shadowing property for piecewise monotone maps, Proceedings of the American Mathematical Society, 113 (1991), 251-263. doi: 10.2307/2048466.

[11]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston MA, 1980.

[12]

E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Transactions of the American Mathematical Society, 308 (1988), 227-241. doi: 10.2307/2000960.

[13]

W. De Melo and S. van Strien, "One-Dimensional Dynamics," Springer-Verlag, Berlin, 1993.

[14]

C. Good, R. Knight and B. E. Raines, Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, Fundamenta Mathematicae, 192 (2006), 267-289. doi: 10.4064/fm192-3-6.

[15]

C. Good, B. E. Raines and R. Suabedissen, Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, Fundamenta Mathematicae, 205 (2009), 179-189. doi: 10.4064/fm205-2-6.

[16]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131. doi: 10.1023/A:1009044515567.

[17]

A. Kazda, The chain relation in sofic subshifts, Fundamenta Informaticae, 84 (2008), 375-390.

[18]

M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fundamenta Mathematicae, 212 (2011), 35-52. doi: 10.4064/fm212-1-3.

[19]

P. Kurka, "Topological and Symbolic Dynamics," Société Mathématique de France, Paris, 2003.

[20]

K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete and Continuous Dynamical Systems. Series A, 13 (2005), 533-540. doi: 10.3934/dcds.2005.13.533.

[21]

J. Milnor, On the concept of attractor, Communications in Mathematical Physics, 99 (1985), 177-195.

[22]

J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and Its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.

[23]

J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, Australian Mathematical Society Journal, Series A, 61 (1996), 57-72.

[24]

W. Parry, Symbolic dynamics and transformations of the unit interval, Transactions of the American Mathematical Society, 122 (1966), 368-378.

[25]

S. Y. Pilyugin, "Shadowing in Dynamical Systems," Springer-Verlag, Berlin, 1999.

[26]

F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods," Cambridge University Press, Cambridge, 2010.

[27]

D. Richeson and J. Wiseman, Positively expansive homeomorphisms of compact spaces, International Journal of Mathematics and Mathematical Sciences, 53-56 (2004), 2907-2910. doi: 10.1155/S0161171204312184.

[28]

K. Sakai, Various shadowing properties for positively expansive maps, Topology and Its Applications, 131 (2003), 15-31. doi: 10.1016/S0166-8641(02)00260-2.

[29]

A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrainskiĭ Matematicheskiĭ Zhurnal, 18 (1966), 127-130.

[30]

O. M. Šarkovskiĭ, On attracting and attracted sets, Doklady Akademii Nauk SSSR, 160 (1965), 1036-1038.

[31]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in "The Structure of Attractors in Dynamical Systems," Springer, Berlin, (1978).

[32]

R. Yang, Topological Anosov maps of non-compact metric spaces, Northeastern Mathematical Journal, 17 (2001), 120-126.

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