Document Type : Research Paper
Author
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València
Abstract
Given a dynamical system $(X,f)$ we investigate several dynamical properties for its Zadeh extension $(F(X),\hat{f})$ endowed with the endograph metric $d_E$. In particular, we prove that for topological A-transitivity, topological (ℓ,A)-recurrence, Devaney chaos, and the specification property, the endograph metric behaves similarly to the supremum metric $d_∞$, the Skorokhod metric $d_0$ and the sendograph metric $d_S$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-A-transitivity.
Keywords
Main Subjects
[1] E. Abakumov, A. Abbar, On several dynamical properties of shifts acting on directed trees, Canadian Journal of
Mathematics, 78(2) (2026), 375-404. https://doi.org/10.4153/S0008414X24000981
Mathematics, 78(2) (2026), 375-404. https://doi.org/10.4153/S0008414X24000981
[2] I. ´Alvarez, A. L´opez-Mart´ınez, Li-Yorke chaos on fuzzy dynamical systems, Information Sciences, 749 (2026), 26
pages. https://doi.org/10.1016/j.ins.2026.123527
pages. https://doi.org/10.1016/j.ins.2026.123527
[3] I. ´Alvarez, A. L´opez-Mart´ınez, A. Peris, Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical
systems, Fuzzy Sets and Systems, 506(109296) (2025), 1-13. https://doi.org/10.1016/j.fss.2025.109296
systems, Fuzzy Sets and Systems, 506(109296) (2025), 1-13. https://doi.org/10.1016/j.fss.2025.109296
[4] J. Banks, Chaos for induced hyperspace maps, Chaos Solitons and Fractals, 25 (2005), 681-685. https://doi.
org/10.1016/j.chaos.2004.11.089
org/10.1016/j.chaos.2004.11.089
[5] S. Bartoll, F. Mart´ınez-Gim´enez, A. Peris, F. Rodenas, Orbit tracing properties on hyperspaces and fuzzy dynamical
systems, Axioms, 11(12) (2022), 733. https://doi.org/10.3390/axioms11120733
systems, Axioms, 11(12) (2022), 733. https://doi.org/10.3390/axioms11120733
[6] W. Bauer, K. Sigmund, Topological dynamics of transformations induced on the space of probability measures,
Monatshefte fur Mathematik, 79 (1975), 81-92. https://doi.org/10.1007/BF01585664
Monatshefte fur Mathematik, 79 (1975), 81-92. https://doi.org/10.1007/BF01585664
[7] F. Bayart, ´E. Matheron, Dynamics of linear operators, Cambridge University Press, 2009. https://doi.org/10.
1017/CBO9780511581113
1017/CBO9780511581113
[8] N. C. Bernardes, A. Peris, F. Rodenas, Set-valued chaos in linear dynamics, Integral Equations and Operator
Theory, 88 (2017), 451-463. https://doi.org/10.1007/s00020-017-2394-6
Theory, 88 (2017), 451-463. https://doi.org/10.1007/s00020-017-2394-6
[9] J. B`es, Q. Menet, A. Peris, Y. Puig, Recurrence properties of hypercyclic operators, Mathematische Annalen,
366(1-2) (2016), 545-572. https://doi.org/10.1007/s00208-015-1336-3
366(1-2) (2016), 545-572. https://doi.org/10.1007/s00208-015-1336-3
[10] P. Billingsley, Convergence of probability measures, Wiley, New York, 1968.
[11] A. Bonilla, K. G. Grosse-Erdmann, A. L´opez-Mart´ınez, A. Peris, Frequently recurrent operators, Journal of
Functional Analysis, 283(12) (2022), 36 pages. https://doi.org/10.1016/j.jfa.2022.109713
Functional Analysis, 283(12) (2022), 36 pages. https://doi.org/10.1016/j.jfa.2022.109713
[12] R. Cardeccia, S. Muro, Frequently recurrence properties and block families, Rev. Real Acad. Cienc. Exactas Fis.
Nat. Ser. A-Mat., 119(60) (2025), 27 pages. https://doi.org/10.1007/s13398-025-01724-1
Nat. Ser. A-Mat., 119(60) (2025), 27 pages. https://doi.org/10.1007/s13398-025-01724-1
[13] Q. Chen, Fuzzy values and their applications to fuzzy reasoning, Beijing Normal University Press, Beijing, 2000.
[14] G. Costakis, A. Manoussos, I. Parissis, Recurrent linear operators, Complex Analysis and Operator Theory, 8
(2014), 1601-1643. https://doi.org/10.1007/s11785-013-0348-9
(2014), 1601-1643. https://doi.org/10.1007/s11785-013-0348-9
[15] H. Fu, Z. Xing, Mixing properties of set-valued maps on hyperspaces via Furstenberg families, Chaos Solitons and
Fractals, 45(4) (2012), 439-443. https://doi.org/10.1016/j.chaos.2012.01.003
Fractals, 45(4) (2012), 439-443. https://doi.org/10.1016/j.chaos.2012.01.003
[16] J. L. Garc´ıa-Guirao, D. Kwietniak, M. Lampart, P. Oprocha, A. Peris, Chaos on hyperspaces, Nonlinear Analysis,
71 (2009), 1-8. https://doi.org/10.1016/j.na.2008.10.055
71 (2009), 1-8. https://doi.org/10.1016/j.na.2008.10.055
[17] E. Glasner, Classifying dynamical systems by their recurrence properties, Topological Methods Nonlinear Analysis,
24(1) (2004), 21-40. https://mathscinet.ams.org/mathscinet/relay-station?mr=MR2111980
24(1) (2004), 21-40. https://mathscinet.ams.org/mathscinet/relay-station?mr=MR2111980
[18] G. Greco, M. Moschen, E. Rezende, Q. F. E. Quelho, On the variational convergence of fuzzy sets in metric
spaces, Annals of the University of Ferrara Sez., 44 (1998), 27-39. https://doi.org/10.1007/BF02828014
spaces, Annals of the University of Ferrara Sez., 44 (1998), 27-39. https://doi.org/10.1007/BF02828014
[19] S. Grivaux, A. L´opez-Mart´ınez, Recurrence properties for linear dynamical systems: An approach via invariant
measures, Journal de Math´ematiques Pures et Appliqu´ees, 169 (2023), 155-188. https://doi.org/10.1016/j.
matpur.2022.11.011
measures, Journal de Math´ematiques Pures et Appliqu´ees, 169 (2023), 155-188. https://doi.org/10.1016/j.
matpur.2022.11.011
[20] S. Grivaux, A. L´opez-Mart´ınez, A. Peris, Questions in linear recurrence I: The T ⊕T-recurrence problem, Analysis
and Mathematical Physics, 15(1) (2025), 26 pages. https://doi.org/10.1007/s13324-024-00999-8
and Mathematical Physics, 15(1) (2025), 26 pages. https://doi.org/10.1007/s13324-024-00999-8
[21] S. Grivaux, A. L´opez-Mart´ınez, A. Peris, Questions in linear recurrence II: Lineability properties, Banach Journal
of Mathematical Analysis, 19(61) (2025), 28 pages. https://doi.org/10.1007/s43037-025-00448-z
of Mathematical Analysis, 19(61) (2025), 28 pages. https://doi.org/10.1007/s43037-025-00448-z
[22] K. G. Grosse-Erdmann, A. Peris, Linear Chaos. Springer, London, 2011. https://doi.org/10.1007/
978-1-4471-2170-1
978-1-4471-2170-1
[23] H. Huang, Characterizations of endograph metric and Γ-convergence on fuzzy sets, Fuzzy Sets and Systems, 350
(2018), 55-84. https://doi.org/10.1016/j.fss.2018.02.014
(2018), 55-84. https://doi.org/10.1016/j.fss.2018.02.014
[24] H. Huang, Some properties of Skorokhod metric on fuzzy sets, Fuzzy Sets and Systems, 437 (2022), 35-52.
https://doi.org/10.1016/j.fss.2021.04.004
https://doi.org/10.1016/j.fss.2021.04.004
[25] W. Huang, H. Li, X. Ye, Family independence for topological and measurable dynamics, Transactions of the American
Mathematical Society, 364(10) (2012), 5209-5242. https://doi.org/10.1090/S0002-9947-2012-05493-6
Mathematical Society, 364(10) (2012), 5209-5242. https://doi.org/10.1090/S0002-9947-2012-05493-6
[26] A. Illanes, S. B. Nadler Jr, Hyperspaces: Fundamentals and recent advances, 1st ed., Marcel Dekker, Inc.: New
York, NY, USA, 1999.
York, NY, USA, 1999.
[27] D. Jard´on, I. S´anchez, Expansive properties of induced dynamical systems, Fuzzy Sets and Systems, 425 (2021),
48-61. https://doi.org/10.1016/j.fss.2020.11.023
48-61. https://doi.org/10.1016/j.fss.2020.11.023
[28] D. Jard´on, I. S´anchez, Sensitivity and strong sensitivity on induced dynamical systems, Iranian Journal of Fuzzy
Systems, 18(4) (2021), 69-78. https://ijfs.usb.ac.ir/article_6177_fb318d654462c7dc4d32cb53d08091ee.
Systems, 18(4) (2021), 69-78. https://ijfs.usb.ac.ir/article_6177_fb318d654462c7dc4d32cb53d08091ee.
[29] D. Jard´on, I. S´anchez, M. Sanchis, Some questions about Zadeh’s extension on metric spaces, Fuzzy Sets and
Systems, 379 (2020), 115-124. https://doi.org/10.1016/j.fss.2018.10.019
Systems, 379 (2020), 115-124. https://doi.org/10.1016/j.fss.2018.10.019
[30] D. Jard´on, I. S´anchez, M. Sanchis, Transitivity in fuzzy hyperspaces, Mathematics, 8 (2020), 1862. https://doi.
org/10.3390/math8111862
org/10.3390/math8111862
[31] D. Jard´on, I. S´anchez, M. Sanchis, Fuzzy sets on uniform spaces, Iranian Journal of Fuzzy Systems, 20(6) (2023),
123-135. https://doi.org/10.22111/IJFS.2023.43774.7700
123-135. https://doi.org/10.22111/IJFS.2023.43774.7700
[32] D. Jard´on, I. S´anchez, M. Sanchis, Transitivity of some uniformities on fuzzy sets, Topology and its Applications,
377 (2026), 12 pages. https://doi.org/10.1016/j.topol.2025.109660
377 (2026), 12 pages. https://doi.org/10.1016/j.topol.2025.109660
[33] S. Y. Joo, Y. K. Kim, The Skorokhod topology on space of fuzzy numbers, Fuzzy Sets and Systems, 111(3) (2000),
497-501. https://doi.org/10.1016/S0165-0114(98)00185-7
497-501. https://doi.org/10.1016/S0165-0114(98)00185-7
[34] J. Kupka, On fuzzifications of discrete dynamical systems, Information Sciences, 181(13) (2011), 2858-2872.
https://doi.org/10.1016/j.ins.2011.02.024
https://doi.org/10.1016/j.ins.2011.02.024
[35] D. Kwietniak, J. Li, P. Oprocha, X. Ye, Multi-recurrence and van der waerden systems, Science China Mathematics,
60(1) (2017), 59-82. https://doi.org/10.1007/s11425-015-0860-8
60(1) (2017), 59-82. https://doi.org/10.1007/s11425-015-0860-8
[36] J. Li, Transitive points via Furstenberg family, Topology and its Applications, 158(16) (2011), 2221-2231. https:
//doi.org/10.1016/j.topol.2011.07.013
//doi.org/10.1016/j.topol.2011.07.013
[37] G. Liao, L. Wang, Y. Zhang, Transitivity, mixing and chaos for a class of set-valued mappings, Science in China,
Series A: Mathematics, 49 (2006), 1-8. https://doi.org/10.1007/s11425-004-5234-5
Series A: Mathematics, 49 (2006), 1-8. https://doi.org/10.1007/s11425-004-5234-5
[38] A. L´opez-Mart´ınez, Invariant measures from locally bounded orbits, Results in Mathematics, 79(185) (2024), 30
pages. https://doi.org/10.1007/s00025-024-02204-5
pages. https://doi.org/10.1007/s00025-024-02204-5
[39] A. L´opez-Mart´ınez, Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree,
Mediterranean Journal of Mathematics, 23(87) (2026), 1-27. https://doi.org/10.1007/s00009-026-03075-6
Mediterranean Journal of Mathematics, 23(87) (2026), 1-27. https://doi.org/10.1007/s00009-026-03075-6
[40] A. L´opez-Mart´ınez, Topological dynamics for the endograph metric II: Extremely radical properties, Journal of Inequalities and Applications, to appear. (Actual arXiv’s LINK: https://doi.org/10.48550/arXiv.2510.19337)
[41] A. L´opez-Mart´ınez, Q. Menet, Two remarks on the set of recurrent vectors, Journal of Mathematical Analysis
and Applications, 541(1) (2025), 17 pages. https://doi.org/10.1016/j.jmaa.2024.128686
and Applications, 541(1) (2025), 17 pages. https://doi.org/10.1016/j.jmaa.2024.128686
[42] F. Mart´ınez-Gim´enez, A. Peris, F. Rodenas, Chaos on fuzzy dynamical systems, Mathematics, 9(20) (2021), 2629.
https://doi.org/10.3390/math9202629
https://doi.org/10.3390/math9202629
[43] A. Peris, Set-valued discrete chaos, Chaos Solitons and Fractals, 26(1) (2005), 19-23. https://doi.org/10.1016/
j.chaos.2004.12.039
j.chaos.2004.12.039
[44] H. Rom´an-Flores, Y. Chalco-Cano, Some chaotic properties of Zadeh’s extensions, Chaos Solitons and Fractals,
35(3) (2008), 452-459. https://doi.org/10.1016/j.chaos.2006.05.036
35(3) (2008), 452-459. https://doi.org/10.1016/j.chaos.2006.05.036
[45] A. V. Skorokhod, Limit theorems for stochastic processes, Theory of Probability and its Applications, 1(3) (1956),
261-290. https://doi.org/10.1137/1101022
261-290. https://doi.org/10.1137/1101022
[46] X. Wu, X. Ding, T. Lu, J. Wang, Topological dynamics of Zadeh’s extension on upper semi-continuous fuzzy
sets, International Journal of Bifurcation and Chaos, 27(10) (2017), 13 pages. https://doi.org/10.1142/
S0218127417501656
sets, International Journal of Bifurcation and Chaos, 27(10) (2017), 13 pages. https://doi.org/10.1142/
S0218127417501656
)