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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
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##### 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] North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar [3] Acta Mathematica Hungarica, 88 (2000), 291-300. doi: 10.1023/A:1026775906693.  Google Scholar [4] Fundamenta Mathematicae, 207 (2010), 161-174. doi: 10.4064/fm207-2-4.  Google Scholar [5] Ergodic Theory and Dynamical Systems, 30 (2010), 21-31. doi: 10.1017/S0143385708001089.  Google Scholar [6] Springer-Verlag, Berlin, 1992.  Google Scholar [7] Transactions of the American Mathematical Society, 348 (1996), 1357-1372. doi: 10.1090/S0002-9947-96-01600-5.  Google Scholar [8] Journal of Differential Equations, 18 (1975), 333-339.  Google Scholar [9] Ergodic Theory and Dynamical Systems, 13 (1993), 7-19. doi: 10.1017/S0143385700007173.  Google Scholar [10] Proceedings of the American Mathematical Society, 113 (1991), 251-263. doi: 10.2307/2048466.  Google Scholar [11] Birkhäuser, Boston MA, 1980.  Google Scholar [12] Transactions of the American Mathematical Society, 308 (1988), 227-241. doi: 10.2307/2000960.  Google Scholar [13] Springer-Verlag, Berlin, 1993.  Google Scholar [14] Fundamenta Mathematicae, 192 (2006), 267-289. doi: 10.4064/fm192-3-6.  Google Scholar [15] Fundamenta Mathematicae, 205 (2009), 179-189. doi: 10.4064/fm205-2-6.  Google Scholar [16] Journal of Dynamics and Differential Equations, 13 (2001), 107-131. doi: 10.1023/A:1009044515567.  Google Scholar [17] Fundamenta Informaticae, 84 (2008), 375-390.  Google Scholar [18] Fundamenta Mathematicae, 212 (2011), 35-52. doi: 10.4064/fm212-1-3.  Google Scholar [19] Société Mathématique de France, Paris, 2003.  Google Scholar [20] Discrete and Continuous Dynamical Systems. Series A, 13 (2005), 533-540. doi: 10.3934/dcds.2005.13.533.  Google Scholar [21] Communications in Mathematical Physics, 99 (1985), 177-195.  Google Scholar [22] Topology and Its Applications, 23 (1986), 87-90. doi: 10.1016/0166-8641(86)90019-2.  Google Scholar [23] Australian Mathematical Society Journal, Series A, 61 (1996), 57-72.  Google Scholar [24] Transactions of the American Mathematical Society, 122 (1966), 368-378.  Google Scholar [25] Springer-Verlag, Berlin, 1999.  Google Scholar [26] Cambridge University Press, Cambridge, 2010.  Google Scholar [27] International Journal of Mathematics and Mathematical Sciences, 53-56 (2004), 2907-2910. doi: 10.1155/S0161171204312184.  Google Scholar [28] Topology and Its Applications, 131 (2003), 15-31. doi: 10.1016/S0166-8641(02)00260-2.  Google Scholar [29] Ukrainskiĭ Matematicheskiĭ Zhurnal, 18 (1966), 127-130.  Google Scholar [30] Doklady Akademii Nauk SSSR, 160 (1965), 1036-1038.  Google Scholar [31] in "The Structure of Attractors in Dynamical Systems," Springer, Berlin, (1978).  Google Scholar [32] Northeastern Mathematical Journal, 17 (2001), 120-126.  Google Scholar
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