{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:45:58Z","timestamp":1760244358429,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T00:00:00Z","timestamp":1671062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004837","name":"MCIN\/AEI\/10.13039\/501100011033","doi-asserted-by":"publisher","award":["PID2019-105011GB-I00","PROMETEU\/2021\/070"],"award-info":[{"award-number":["PID2019-105011GB-I00","PROMETEU\/2021\/070"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003359","name":"Generalitat Valenciana","doi-asserted-by":"publisher","award":["PID2019-105011GB-I00","PROMETEU\/2021\/070"],"award-info":[{"award-number":["PID2019-105011GB-I00","PROMETEU\/2021\/070"]}],"id":[{"id":"10.13039\/501100003359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let X be a compact metric space and a continuous map f:X\u2192X which defines a discrete dynamical system (X,f). The map f induces two natural maps, namely f\u00af:K(X)\u2192K(X) on the hyperspace K(X) of non-empty compact subspaces of X and the Zadeh\u2019s extension f^:F(X)\u2192F(X) on the space F(X) of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system (X,f) and its induced discrete dynamical systems (K(X),f\u00af) and (F(X),f^). Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator.<\/jats:p>","DOI":"10.3390\/axioms11120733","type":"journal-article","created":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T02:54:02Z","timestamp":1671159242000},"page":"733","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6395-0981","authenticated-orcid":false,"given":"Salud","family":"Bartoll","sequence":"first","affiliation":[{"name":"Institut Universitari de Matem\u00e0tica Pura i Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3369-4286","authenticated-orcid":false,"given":"F\u00e9lix","family":"Mart\u00ednez-Gim\u00e9nez","sequence":"additional","affiliation":[{"name":"Institut Universitari de Matem\u00e0tica Pura i Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1683-2373","authenticated-orcid":false,"given":"Alfred","family":"Peris","sequence":"additional","affiliation":[{"name":"Institut Universitari de Matem\u00e0tica Pura i Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4564-5171","authenticated-orcid":false,"given":"Francisco","family":"Rodenas","sequence":"additional","affiliation":[{"name":"Institut Universitari de Matem\u00e0tica Pura i Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bowen, R. (1970). Topological entropy and axiom A. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), American Mathematical Society.","DOI":"10.1090\/pspum\/014\/9986"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1090\/S0002-9947-1974-0352411-X","article-title":"On dynamical systems with the specification property","volume":"190","author":"Sigmund","year":"1974","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF01585664","article-title":"Topological dynamics of transformations induced on the space of probability measures","volume":"79","author":"Bauer","year":"1975","journal-title":"Monatsh. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1016\/j.chaos.2004.11.089","article-title":"Chaos for induced hyperspace maps","volume":"25","author":"Banks","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.chaos.2004.12.039","article-title":"Set-valued discrete chaos","volume":"26","author":"Peris","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.na.2008.10.055","article-title":"Chaos on hyperspaces","volume":"71","author":"Guirao","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/s00020-017-2394-6","article-title":"Set-valued chaos in linear dynamics","volume":"88","author":"Bernardes","year":"2017","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.topol.2007.09.003","article-title":"Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems","volume":"155","author":"Wang","year":"2007","journal-title":"Topol. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1016\/j.topol.2008.10.014","article-title":"Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems","volume":"156","author":"Wang","year":"2009","journal-title":"Topol. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jard\u00f3n, D., S\u00e1nchez, I., and Sanchis, M. (2020). Transitivity in Fuzzy Hyperspaces. Mathematics, 8.","DOI":"10.3390\/math8111862"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez-Gim\u00e9nez, F., Peris, A., and Rodenas, F. (2021). Chaos on Fuzzy Dynamical Systems. Mathematics, 9.","DOI":"10.3390\/math9202629"},{"key":"ref_12","unstructured":"Devaney, R.L. (1989). An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company Advanced Book Program. [2nd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1080\/00029890.1975.11994008","article-title":"Period three implies chaos","volume":"82","author":"Li","year":"1975","journal-title":"Am. Math. Mon."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1007\/s11253-005-0055-4","article-title":"Li-Yorke sensitivity and other concepts of chaos","volume":"56","author":"Kolyada","year":"2004","journal-title":"Ukr. Math. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1090\/S0002-9947-1994-1227094-X","article-title":"Measures of chaos and a spectral decomposition of dynamical systems on the interval","volume":"344","author":"Schweizer","year":"1994","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00010-005-2771-0","article-title":"Various notions of chaos for discrete dynamical systems. A brief survey","volume":"70","author":"Forti","year":"2005","journal-title":"Aequ. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.fss.2011.04.006","article-title":"On Devaney chaotic induced fuzzy and set-valued dynamical systems","volume":"177","author":"Kupka","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1750165","DOI":"10.1142\/S0218127417501656","article-title":"Topological dynamics of Zadeh\u2019s extension on upper semi-continuous fuzzy sets","volume":"27","author":"Wu","year":"2017","journal-title":"Int. J. Bifur. Chaos Appl. Sci. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.ins.2017.02.042","article-title":"Sensitivity and transitivity of fuzzified dynamical systems","volume":"396","author":"Wu","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1850268","DOI":"10.1142\/S0217984918502688","article-title":"Sensitivity and specification property of fuzzified dynamical systems","volume":"32","author":"Li","year":"2018","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_21","first-page":"59","article-title":"An effective computational attempt for solving fully fuzzy linear programming using MOLP problem","volume":"36","author":"Ebrahimnejad","year":"2019","journal-title":"J. Ind. Prod. Eng."},{"key":"ref_22","unstructured":"Illanes, A., and Nadler, S.B. (1999). Hyperspaces\u2014Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc.. Fundamentals and Recent Advances."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.chaos.2006.05.036","article-title":"Some chaotic properties of Zadeh\u2019s extensions","volume":"35","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.fss.2018.10.019","article-title":"Some questions about Zadeh\u2019s extension on metric spaces","volume":"379","author":"Sanchis","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1090\/conm\/669\/13428","article-title":"A panorama of specification-like properties and their consequences","volume":"Volume 669","author":"Kwietniak","year":"2016","journal-title":"Dynamics and Numbers"},{"key":"ref_26","unstructured":"Rudin, W. (1991). Functional Analysis, McGraw-Hill, Inc.. [2nd ed.]. International Series in Pure and Applied Mathematics."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.jmaa.2015.12.004","article-title":"Operators with the specification property","volume":"436","author":"Bartoll","year":"2016","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.3934\/dcds.2013.33.1819","article-title":"Characterizations of \u03c9-limit sets in topologically hyperbolic systems","volume":"33","author":"Barwell","year":"2013","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"277","DOI":"10.4064\/fm136-2-2016","article-title":"Shadowing for induced maps of hyperspaces","volume":"235","author":"Good","year":"2016","journal-title":"Fund. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"123767","DOI":"10.1016\/j.jmaa.2019.123767","article-title":"Preservation of shadowing in discrete dynamical systems","volume":"485","author":"Good","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","first-page":"193","article-title":"The role of real characters in the Pontryagin duality of topological abelian groups","volume":"18","author":"Chasco","year":"2008","journal-title":"J. Lie Theory"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/733\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:55Z","timestamp":1760146915000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/733"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,15]]},"references-count":31,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120733"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120733","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,15]]}}}