• Previous Article
    Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification
  • DCDS Home
  • This Issue
  • Next Article
    Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth
doi: 10.3934/dcds.2021159
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

* Corresponding author: Jana Hantáková

Received  July 2021 Revised  September 2021 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 $ \alpha $-limit sets of backward trajectories for graph maps. Discrete & Continuous Dynamical Systems, 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.  Google Scholar

[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.  Google Scholar

[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.  Google Scholar

[4]

F. BalibreaG. DvorníkováM. 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.  Google Scholar

[5]

A. BarwellC. GoodR. 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.  Google Scholar

[6]

A. BarwellC. GoodP. 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.  Google Scholar

[7]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅰ, J. Soviet Math., 48 (1990), 500-508.  doi: 10.1007/BF01095616.  Google Scholar

[8]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅱ, J. Soviet Math., 48 (1990), 668-674.  doi: 10.1007/BF01094721.  Google Scholar

[9]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅲ, J. Soviet Math., 49 (1990), 875-883.  doi: 10.1007/BF02205632.  Google Scholar

[10]

A. BlokhA. M. BrucknerP. 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.  Google Scholar

[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.  Google Scholar

[12]

J. ChudziakJ. L. G. GuiraoL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Hantáková and S. Roth, On backward attractors of interval maps, Nonlinearity, 34 (2021), 7415-7445.  doi: 10.1088/1361-6544/ac23b6.  Google Scholar

[17]

G. HarańczykD. 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.  Google Scholar

[18]

G. HaranczykD. 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.  Google Scholar

[19]

M. Hero, Special $\alpha$-limit points for maps of the interval, Proc. Amer. Math. Soc., 116 (1992), 1015-1022.  doi: 10.2307/2159483.  Google Scholar

[20]

M. W. HirschH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. KočanM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. MaiT. 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.  Google Scholar

[30]

J. Munkres, Topology, 2$^nd$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.  Google Scholar

[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.  Google Scholar

[33]

T. SunY. TangG. SuH. 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.  Google Scholar

[34]

T. SunH. 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.  Google Scholar

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.  Google Scholar

[2]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.  Google Scholar

[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.  Google Scholar

[4]

F. BalibreaG. DvorníkováM. 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.  Google Scholar

[5]

A. BarwellC. GoodR. 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.  Google Scholar

[6]

A. BarwellC. GoodP. 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.  Google Scholar

[7]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅰ, J. Soviet Math., 48 (1990), 500-508.  doi: 10.1007/BF01095616.  Google Scholar

[8]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅱ, J. Soviet Math., 48 (1990), 668-674.  doi: 10.1007/BF01094721.  Google Scholar

[9]

A. M. Blokh, Dynamical systems on one–dimensional branched manifolds Ⅲ, J. Soviet Math., 49 (1990), 875-883.  doi: 10.1007/BF02205632.  Google Scholar

[10]

A. BlokhA. M. BrucknerP. 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.  Google Scholar

[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.  Google Scholar

[12]

J. ChudziakJ. L. G. GuiraoL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Hantáková and S. Roth, On backward attractors of interval maps, Nonlinearity, 34 (2021), 7415-7445.  doi: 10.1088/1361-6544/ac23b6.  Google Scholar

[17]

G. HarańczykD. 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.  Google Scholar

[18]

G. HaranczykD. 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.  Google Scholar

[19]

M. Hero, Special $\alpha$-limit points for maps of the interval, Proc. Amer. Math. Soc., 116 (1992), 1015-1022.  doi: 10.2307/2159483.  Google Scholar

[20]

M. W. HirschH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. KočanM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. MaiT. 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.  Google Scholar

[30]

J. Munkres, Topology, 2$^nd$ edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.  Google Scholar

[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.  Google Scholar

[33]

T. SunY. TangG. SuH. 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.  Google Scholar

[34]

T. SunH. 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.  Google Scholar

Figure 1.  A map where the family $ \mathcal{A}(x) $ is uncountable
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 $
[1]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[2]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[3]

Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075

[4]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

[5]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[6]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[7]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[8]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275

[9]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127

[10]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555

[11]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363

[12]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

[13]

Yang Cao, Song Shao. Topological mild mixing of all orders along polynomials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021150

[14]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[15]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

[16]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

[17]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[18]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201

[19]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827

[20]

Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021131

2020 Impact Factor: 1.392

Article outline

Figures and Tables

[Back to Top]