# American Institute of Mathematical Sciences

March  2022, 42(3): 1435-1463. doi: 10.3934/dcds.2021159

## On the structure of α-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

* Corresponding author: Jana Hantáková

Received  July 2021 Revised  September 2021 Published  March 2022 Early access  November 2021

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.

Citation: Magdalena Foryś-Krawiec, Jana Hantáková, Piotr Oprocha. On the structure of α-limit sets of backward trajectories for graph maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1435-1463. doi: 10.3934/dcds.2021159
##### 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. [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.

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##### 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. [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.
A map where the family $\mathcal{A}(x)$ is uncountable
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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