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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 (): 483.
|
[2] |
N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems,", North-Holland Publishing Co., (1994).
|
[3] |
F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps,, Acta Mathematica Hungarica, 88 (2000), 291.
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.
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.
doi: 10.1017/S0143385708001089. |
[6] |
L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Springer-Verlag, (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.
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.
|
[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.
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.
doi: 10.2307/2048466. |
[11] |
P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems,", Birkhäuser, (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.
doi: 10.2307/2000960. |
[13] |
W. De Melo and S. van Strien, "One-Dimensional Dynamics,", Springer-Verlag, (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.
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.
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.
doi: 10.1023/A:1009044515567. |
[17] |
A. Kazda, The chain relation in sofic subshifts,, Fundamenta Informaticae, 84 (2008), 375.
|
[18] |
M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property,, Fundamenta Mathematicae, 212 (2011), 35.
doi: 10.4064/fm212-1-3. |
[19] |
P. Kurka, "Topological and Symbolic Dynamics,", Société Mathématique de France, (2003).
|
[20] |
K. Lee and K. Sakai, Various shadowing properties and their equivalence,, Discrete and Continuous Dynamical Systems. Series A, 13 (2005), 533.
doi: 10.3934/dcds.2005.13.533. |
[21] |
J. Milnor, On the concept of attractor,, Communications in Mathematical Physics, 99 (1985), 177.
|
[22] |
J. Ombach, Equivalent conditions for hyperbolic coordinates,, Topology and Its Applications, 23 (1986), 87.
doi: 10.1016/0166-8641(86)90019-2. |
[23] |
J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms,, Australian Mathematical Society Journal, 61 (1996), 57.
|
[24] |
W. Parry, Symbolic dynamics and transformations of the unit interval,, Transactions of the American Mathematical Society, 122 (1966), 368.
|
[25] |
S. Y. Pilyugin, "Shadowing in Dynamical Systems,", Springer-Verlag, (1999).
|
[26] |
F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods,", Cambridge University Press, (2010).
|
[27] |
D. Richeson and J. Wiseman, Positively expansive homeomorphisms of compact spaces,, International Journal of Mathematics and Mathematical Sciences, 53-56 (2004), 53.
doi: 10.1155/S0161171204312184. |
[28] |
K. Sakai, Various shadowing properties for positively expansive maps,, Topology and Its Applications, 131 (2003), 15.
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.
|
[30] |
O. M. Šarkovskiĭ, On attracting and attracted sets,, Doklady Akademii Nauk SSSR, 160 (1965), 1036.
|
[31] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability,, in, (1978).
|
[32] |
R. Yang, Topological Anosov maps of non-compact metric spaces,, Northeastern Mathematical Journal, 17 (2001), 120.
|
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.
|
[2] |
N. Aoki and K. Hiraide, "Topological Theory of Dynamical Systems,", North-Holland Publishing Co., (1994).
|
[3] |
F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps,, Acta Mathematica Hungarica, 88 (2000), 291.
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.
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.
doi: 10.1017/S0143385708001089. |
[6] |
L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Springer-Verlag, (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.
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.
|
[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.
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.
doi: 10.2307/2048466. |
[11] |
P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems,", Birkhäuser, (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.
doi: 10.2307/2000960. |
[13] |
W. De Melo and S. van Strien, "One-Dimensional Dynamics,", Springer-Verlag, (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.
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.
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.
doi: 10.1023/A:1009044515567. |
[17] |
A. Kazda, The chain relation in sofic subshifts,, Fundamenta Informaticae, 84 (2008), 375.
|
[18] |
M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property,, Fundamenta Mathematicae, 212 (2011), 35.
doi: 10.4064/fm212-1-3. |
[19] |
P. Kurka, "Topological and Symbolic Dynamics,", Société Mathématique de France, (2003).
|
[20] |
K. Lee and K. Sakai, Various shadowing properties and their equivalence,, Discrete and Continuous Dynamical Systems. Series A, 13 (2005), 533.
doi: 10.3934/dcds.2005.13.533. |
[21] |
J. Milnor, On the concept of attractor,, Communications in Mathematical Physics, 99 (1985), 177.
|
[22] |
J. Ombach, Equivalent conditions for hyperbolic coordinates,, Topology and Its Applications, 23 (1986), 87.
doi: 10.1016/0166-8641(86)90019-2. |
[23] |
J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms,, Australian Mathematical Society Journal, 61 (1996), 57.
|
[24] |
W. Parry, Symbolic dynamics and transformations of the unit interval,, Transactions of the American Mathematical Society, 122 (1966), 368.
|
[25] |
S. Y. Pilyugin, "Shadowing in Dynamical Systems,", Springer-Verlag, (1999).
|
[26] |
F. Przytycki and M. Urbański, "Conformal Fractals: Ergodic Theory Methods,", Cambridge University Press, (2010).
|
[27] |
D. Richeson and J. Wiseman, Positively expansive homeomorphisms of compact spaces,, International Journal of Mathematics and Mathematical Sciences, 53-56 (2004), 53.
doi: 10.1155/S0161171204312184. |
[28] |
K. Sakai, Various shadowing properties for positively expansive maps,, Topology and Its Applications, 131 (2003), 15.
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.
|
[30] |
O. M. Šarkovskiĭ, On attracting and attracted sets,, Doklady Akademii Nauk SSSR, 160 (1965), 1036.
|
[31] |
P. Walters, On the pseudo-orbit tracing property and its relationship to stability,, in, (1978).
|
[32] |
R. Yang, Topological Anosov maps of non-compact metric spaces,, Northeastern Mathematical Journal, 17 (2001), 120.
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