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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 & 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 (): 483.   Google Scholar [2] N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems," North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Springer-Verlag, Berlin, 1992.  Google Scholar [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.  Google Scholar [8] R. Bowen, $\omega$-limit sets for axiom $A$ diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston MA, 1980.  Google Scholar [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.  Google Scholar [13] W. De Melo and S. van Strien, "One-Dimensional Dynamics," Springer-Verlag, Berlin, 1993.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] A. Kazda, The chain relation in sofic subshifts, Fundamenta Informaticae, 84 (2008), 375-390.  Google Scholar [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.  Google Scholar [19] P. Kurka, "Topological and Symbolic Dynamics," Société Mathématique de France, Paris, 2003.  Google Scholar [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.  Google Scholar [21] J. Milnor, On the concept of attractor, Communications in Mathematical Physics, 99 (1985), 177-195.  Google Scholar [22] J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and Its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.  Google Scholar [23] J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, Australian Mathematical Society Journal, Series A, 61 (1996), 57-72.  Google Scholar [24] W. Parry, Symbolic dynamics and transformations of the unit interval, Transactions of the American Mathematical Society, 122 (1966), 368-378.  Google Scholar [25] S. Y. Pilyugin, "Shadowing in Dynamical Systems," Springer-Verlag, Berlin, 1999.  Google Scholar [26] F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods," Cambridge University Press, Cambridge, 2010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrainskiĭ Matematicheskiĭ Zhurnal, 18 (1966), 127-130.  Google Scholar [30] O. M. Šarkovskiĭ, On attracting and attracted sets, Doklady Akademii Nauk SSSR, 160 (1965), 1036-1038.  Google Scholar [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).  Google Scholar [32] R. Yang, Topological Anosov maps of non-compact metric spaces, Northeastern Mathematical Journal, 17 (2001), 120-126.  Google Scholar

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 (): 483.   Google Scholar [2] N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems," North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Springer-Verlag, Berlin, 1992.  Google Scholar [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.  Google Scholar [8] R. Bowen, $\omega$-limit sets for axiom $A$ diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston MA, 1980.  Google Scholar [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.  Google Scholar [13] W. De Melo and S. van Strien, "One-Dimensional Dynamics," Springer-Verlag, Berlin, 1993.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] A. Kazda, The chain relation in sofic subshifts, Fundamenta Informaticae, 84 (2008), 375-390.  Google Scholar [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.  Google Scholar [19] P. Kurka, "Topological and Symbolic Dynamics," Société Mathématique de France, Paris, 2003.  Google Scholar [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.  Google Scholar [21] J. Milnor, On the concept of attractor, Communications in Mathematical Physics, 99 (1985), 177-195.  Google Scholar [22] J. Ombach, Equivalent conditions for hyperbolic coordinates, Topology and Its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.  Google Scholar [23] J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, Australian Mathematical Society Journal, Series A, 61 (1996), 57-72.  Google Scholar [24] W. Parry, Symbolic dynamics and transformations of the unit interval, Transactions of the American Mathematical Society, 122 (1966), 368-378.  Google Scholar [25] S. Y. Pilyugin, "Shadowing in Dynamical Systems," Springer-Verlag, Berlin, 1999.  Google Scholar [26] F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods," Cambridge University Press, Cambridge, 2010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrainskiĭ Matematicheskiĭ Zhurnal, 18 (1966), 127-130.  Google Scholar [30] O. M. Šarkovskiĭ, On attracting and attracted sets, Doklady Akademii Nauk SSSR, 160 (1965), 1036-1038.  Google Scholar [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).  Google Scholar [32] R. Yang, Topological Anosov maps of non-compact metric spaces, Northeastern Mathematical Journal, 17 (2001), 120-126.  Google Scholar
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