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On the structure of $ \alpha $-limit sets of backward trajectories for graph maps
| 1. | AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland |
| 2. | Mathematical Institute of the Silesian University in Opava, Na Rybníčku 1, 74601, Opava, Czech Republic |
| 3. | Centre of Excellence IT4Innovations - Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic |
In the paper we study what sets can be obtained as $ \alpha $-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $ \alpha $-limit sets are $ \omega $-limit sets and for all but finitely many points $ x $, we can obtain every $ \omega $-limits set as the $ \alpha $-limit set of a backward trajectory starting in $ x $. For zero entropy maps, every $ \alpha $-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.
Mathematics Subject Classification:
Primary: 37E25;Secondary: 37B20, 37B40.
Citation:
Magdalena Foryś-Krawiec, Jana Hantáková, Piotr Oprocha.
On the structure of $ \alpha $-limit sets of backward trajectories for graph maps. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021159
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E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Ergodic Theory and Probability, (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40. |
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E. Akin and E. Glasner,
Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.
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J. Auslander and J. A. Yorke,
Interval maps, factors of maps, and chaos, Tohoku Math. J., 32 (1980), 177-188.
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F. Balibrea, G. Dvorníková, M. Lampart and P. Oprocha,
On negative limit sets for one-dimensional dynamics, Nonlinear Anal., 75 (2012), 3262-3267.
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A. Barwell, C. Good, R. Knight and B. Raines,
A characterization of $\omega$-limit sets in shift spaces, Ergodic Theory Dynam. Systems, 30 (2010), 21-31.
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A. Barwell, C. Good, P. Oprocha and B. Raines,
Characterizations of $\omega$-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst., 33 (2013), 1819-1833.
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A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅰ, J. Soviet Math., 48 (1990), 500-508.
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A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅱ, J. Soviet Math., 48 (1990), 668-674.
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A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅲ, J. Soviet Math., 49 (1990), 875-883.
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A. Blokh, A. M. Bruckner, P. D. Humke and J. Smítal,
The space of $\omega$-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc., 348 (1996), 1357-1372.
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R. Bowen,
$\omega$-limit sets for Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
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J. Chudziak, J. L. G. Guirao, L. Snoha and V. Špitalský,
Universality with respect to $\omega$-limit sets, Nonlinear Anal., 71 (2009), 1485-1495.
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E. Coven and Z. Nitecki,
Non-wandering sets of the powers of maps of the interval, Ergodic Theory Dynam. Systems, 1 (1981), 9-31.
doi: 10.1017/S0143385700001139. |
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H. Cui and Y. Ding,
The $\alpha$-limit sets of a unimodal map without homtervals, Topology Appl., 157 (2010), 22-28.
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Y. Dowker and F. Frielander,
On limit sets in dynamical systems, Proc. London Math. Soc., 4 (1954), 168-176.
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J. Hantáková and S. Roth,
On backward attractors of interval maps, Nonlinearity, 34 (2021), 7415-7445.
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G. Harańczyk, D. Kwietniak and P. Oprocha,
Topological structure and entropy of mixing graph maps, Ergodic Theory Dynam. Systems, 34 (2014), 1587-1614.
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G. Haranczyk, D. Kwietniak and P. Oprocha,
A note on transitivity, sensitivity and chaos for graph maps, J. Difference Equ. Appl., 17 (2011), 1549-1553.
doi: 10.1080/10236191003657253. |
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M. Hero,
Special $\alpha$-limit points for maps of the interval, Proc. Amer. Math. Soc., 116 (1992), 1015-1022.
doi: 10.2307/2159483. |
| [20] |
M. W. Hirsch, H. L. Smith and X. Zhao,
Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
| [21] |
R. Hric and M. Málek,
Omega limit sets and distributional chaos on graphs, Topology Appl., 153 (2006), 2469-2475.
doi: 10.1016/j.topol.2005.09.007. |
| [22] |
S. Jackson, B. Mance and S. Roth, A non-Borel special alpha-limit set in the square, Ergodic Theory Dynam. Systems, (2021), 1–11.
doi: 10.1017/etds. 2021.68. |
| [23] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
| [24] |
Z. Kočan, M. Málek and V. Kurková,
On the centre and the set of $\omega$-limit points of continuous maps on dendrites, Topology Appl., 156 (2009), 2923-2931.
doi: 10.1016/j.topol.2009.02.008. |
| [25] |
S. Kolyada, M. Misiurewicz and L. Snoha, Special $\alpha$-limit sets, Dynamics: Topology and Numbers, Contemp. Math., Amer. Math. Soc., Providence, RI, 744 (2020), 157–173.
doi: 10.1090/conm/744/14976. |
| [26] |
J. H. Mai and S. Shao,
Spaces of $\omega$-limit sets of graph maps, Fund. Math., 196 (2007), 91-100.
doi: 10.4064/fm196-1-2. |
| [27] |
J. Mai and S. Shao,
The structure of graph maps without periodic points, Topology Appl., 154 (2007), 2714-2728.
doi: 10.1016/j.topol.2007.05.005. |
| [28] |
J. Mai and T. Sun,
Non-wandering points and the depth for graph maps, Sci. China Ser. A Math., 50 (2007), 1818-1824.
doi: 10.1007/s11425-007-0139-8. |
| [29] |
J. Mai, T. Sun and G. Zhang,
Recurrent points and non–wandering points of graph maps, J. Math. Anal. Appl., 383 (2011), 553-559.
doi: 10.1016/j.jmaa.2011.05.052. |
| [30] |
J. Munkres, Topology, 2$^nd$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [31] |
A. N. Sharkovsky, Continuous maps on the set of limit points of an iterated sequence, Ukr. Math. J., 18 (1966), 127-130. Google Scholar |
| [32] |
S. Ruette and L. Snoha,
For graph maps, one scrambled pair implies Li-Yorke chaos, Proc. Amer. Math. Soc., 142 (2014), 2087-2100.
doi: 10.1090/S0002-9939-2014-11937-X. |
| [33] |
T. Sun, Y. Tang, G. Su, H. Xi and B. Qin,
Special $\alpha$-limit points and $\gamma$-limit points of a dendrite map, Qual. Theory Dyn. Syst., 17 (2018), 245-257.
doi: 10.1007/s12346-017-0225-4. |
| [34] |
T. Sun, H. Xi and H. Liang,
Special $\alpha$-limit points and unilateral $\gamma$ limit points for graph maps, Sci China Math., 54 (2011), 2013-2018.
doi: 10.1007/s11425-011-4254-1. |
show all references
References:
| [1] |
E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Ergodic Theory and Probability, (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40. |
| [2] |
E. Akin and E. Glasner,
Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.
doi: 10.1007/BF02788112. |
| [3] |
J. Auslander and J. A. Yorke,
Interval maps, factors of maps, and chaos, Tohoku Math. J., 32 (1980), 177-188.
doi: 10.2748/tmj/1178229634. |
| [4] |
F. Balibrea, G. Dvorníková, M. Lampart and P. Oprocha,
On negative limit sets for one-dimensional dynamics, Nonlinear Anal., 75 (2012), 3262-3267.
doi: 10.1016/j.na.2011.12.030. |
| [5] |
A. Barwell, C. Good, R. Knight and B. Raines,
A characterization of $\omega$-limit sets in shift spaces, Ergodic Theory Dynam. Systems, 30 (2010), 21-31.
doi: 10.1017/S0143385708001089. |
| [6] |
A. Barwell, C. Good, P. Oprocha and B. Raines,
Characterizations of $\omega$-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst., 33 (2013), 1819-1833.
doi: 10.3934/dcds.2013.33.1819. |
| [7] |
A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅰ, J. Soviet Math., 48 (1990), 500-508.
doi: 10.1007/BF01095616. |
| [8] |
A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅱ, J. Soviet Math., 48 (1990), 668-674.
doi: 10.1007/BF01094721. |
| [9] |
A. M. Blokh,
Dynamical systems on one–dimensional branched manifolds Ⅲ, J. Soviet Math., 49 (1990), 875-883.
doi: 10.1007/BF02205632. |
| [10] |
A. Blokh, A. M. Bruckner, P. D. Humke and J. Smítal,
The space of $\omega$-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc., 348 (1996), 1357-1372.
doi: 10.1090/S0002-9947-96-01600-5. |
| [11] |
R. Bowen,
$\omega$-limit sets for Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
| [12] |
J. Chudziak, J. L. G. Guirao, L. Snoha and V. Špitalský,
Universality with respect to $\omega$-limit sets, Nonlinear Anal., 71 (2009), 1485-1495.
doi: 10.1016/j.na.2008.12.034. |
| [13] |
E. Coven and Z. Nitecki,
Non-wandering sets of the powers of maps of the interval, Ergodic Theory Dynam. Systems, 1 (1981), 9-31.
doi: 10.1017/S0143385700001139. |
| [14] |
H. Cui and Y. Ding,
The $\alpha$-limit sets of a unimodal map without homtervals, Topology Appl., 157 (2010), 22-28.
doi: 10.1016/j.topol.2009.04.054. |
| [15] |
Y. Dowker and F. Frielander,
On limit sets in dynamical systems, Proc. London Math. Soc., 4 (1954), 168-176.
doi: 10.1112/plms/s3-4.1.168. |
| [16] |
J. Hantáková and S. Roth,
On backward attractors of interval maps, Nonlinearity, 34 (2021), 7415-7445.
doi: 10.1088/1361-6544/ac23b6. |
| [17] |
G. Harańczyk, D. Kwietniak and P. Oprocha,
Topological structure and entropy of mixing graph maps, Ergodic Theory Dynam. Systems, 34 (2014), 1587-1614.
doi: 10.1017/etds.2013.6. |
| [18] |
G. Haranczyk, D. Kwietniak and P. Oprocha,
A note on transitivity, sensitivity and chaos for graph maps, J. Difference Equ. Appl., 17 (2011), 1549-1553.
doi: 10.1080/10236191003657253. |
| [19] |
M. Hero,
Special $\alpha$-limit points for maps of the interval, Proc. Amer. Math. Soc., 116 (1992), 1015-1022.
doi: 10.2307/2159483. |
| [20] |
M. W. Hirsch, H. L. Smith and X. Zhao,
Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
| [21] |
R. Hric and M. Málek,
Omega limit sets and distributional chaos on graphs, Topology Appl., 153 (2006), 2469-2475.
doi: 10.1016/j.topol.2005.09.007. |
| [22] |
S. Jackson, B. Mance and S. Roth, A non-Borel special alpha-limit set in the square, Ergodic Theory Dynam. Systems, (2021), 1–11.
doi: 10.1017/etds. 2021.68. |
| [23] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
| [24] |
Z. Kočan, M. Málek and V. Kurková,
On the centre and the set of $\omega$-limit points of continuous maps on dendrites, Topology Appl., 156 (2009), 2923-2931.
doi: 10.1016/j.topol.2009.02.008. |
| [25] |
S. Kolyada, M. Misiurewicz and L. Snoha, Special $\alpha$-limit sets, Dynamics: Topology and Numbers, Contemp. Math., Amer. Math. Soc., Providence, RI, 744 (2020), 157–173.
doi: 10.1090/conm/744/14976. |
| [26] |
J. H. Mai and S. Shao,
Spaces of $\omega$-limit sets of graph maps, Fund. Math., 196 (2007), 91-100.
doi: 10.4064/fm196-1-2. |
| [27] |
J. Mai and S. Shao,
The structure of graph maps without periodic points, Topology Appl., 154 (2007), 2714-2728.
doi: 10.1016/j.topol.2007.05.005. |
| [28] |
J. Mai and T. Sun,
Non-wandering points and the depth for graph maps, Sci. China Ser. A Math., 50 (2007), 1818-1824.
doi: 10.1007/s11425-007-0139-8. |
| [29] |
J. Mai, T. Sun and G. Zhang,
Recurrent points and non–wandering points of graph maps, J. Math. Anal. Appl., 383 (2011), 553-559.
doi: 10.1016/j.jmaa.2011.05.052. |
| [30] |
J. Munkres, Topology, 2$^nd$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [31] |
A. N. Sharkovsky, Continuous maps on the set of limit points of an iterated sequence, Ukr. Math. J., 18 (1966), 127-130. Google Scholar |
| [32] |
S. Ruette and L. Snoha,
For graph maps, one scrambled pair implies Li-Yorke chaos, Proc. Amer. Math. Soc., 142 (2014), 2087-2100.
doi: 10.1090/S0002-9939-2014-11937-X. |
| [33] |
T. Sun, Y. Tang, G. Su, H. Xi and B. Qin,
Special $\alpha$-limit points and $\gamma$-limit points of a dendrite map, Qual. Theory Dyn. Syst., 17 (2018), 245-257.
doi: 10.1007/s12346-017-0225-4. |
| [34] |
T. Sun, H. Xi and H. Liang,
Special $\alpha$-limit points and unilateral $\gamma$ limit points for graph maps, Sci China Math., 54 (2011), 2013-2018.
doi: 10.1007/s11425-011-4254-1. |
Figure 2.
A map mixing interval map $ g $ with a fixed point $ p $ such that every backward branch $ \{z_j\}_{j\leq 0} $ with $ \{p\} = \alpha_g(\{z_j\}_{j\leq 0}) $ converges to $ p $ only from the right side. Every backward branch $ \{z_j\}_{j\leq 0} $ converging to $ p $ from the left side or from both sides has $ \alpha_g(\{z_j\}_{j\leq 0})\cap \{q, r\}\neq\emptyset $ , where $ q, r $ are preimages of $ p $
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