{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T13:50:04Z","timestamp":1762005004367,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,12]],"date-time":"2018-03-12T00:00:00Z","timestamp":1520812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper is concerned with invariance     (  F 1  ,  F 2  )    -scrambled sets under iterations. The main results are an extension of the compound invariance of Li\u2013Yorke chaos and distributional chaos. New definitions of     (  F 1  ,  F 2  )    -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties     P ( k )     and     Q ( k )     of Furstenberg families are introduced. It is shown that, for any positive integer k, for any     s \u2208 [ 0 , 1 ]    , Furstenberg family      M \u00af   ( s )      has properties     P ( k )     and     Q ( k )    , where      M \u00af   ( s )      denotes the family of all infinite subsets of     Z +     whose upper density is not less than s. Then, the following conclusion is obtained. D is an     (  M \u00af   ( s )  ,  M \u00af   ( t )  )    -scrambled set of     ( X ,  f  1 , \u221e   )     if and only if D is an     (  M \u00af   ( s )  ,  M \u00af   ( t )  )    -scrambled set of     ( X ,  f  1 , \u221e   [ m ]   )    .<\/jats:p>","DOI":"10.3390\/e20030188","type":"journal-article","created":{"date-parts":[[2018,3,12]],"date-time":"2018-03-12T13:13:48Z","timestamp":1520860428000},"page":"188","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Some Iterative Properties of                                                         (                                    F                   1                                  ,                                    F                   2                                  )                                                -Chaos in Non-Autonomous Discrete Systems"],"prefix":"10.3390","volume":"20","author":[{"given":"Xiao","family":"Tang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1381-7418","authenticated-orcid":false,"given":"Guanrong","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR 999077, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianxiu","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1080\/00029890.1975.11994008","article-title":"Period 3 implies chaos","volume":"82","author":"Li","year":"1975","journal-title":"Am. 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