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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 |
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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