{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:53:38Z","timestamp":1760234018568,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,3,20]],"date-time":"2021-03-20T00:00:00Z","timestamp":1616198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, it is proved that every diffeomorphism possessing the filtrated pseudo-orbit shadowing property admits an approximately shadowable Lebesgue measure. Furthermore, the C1-interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. As a corollary, it is proved that there exists a C1-open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.<\/jats:p>","DOI":"10.3390\/axioms10010038","type":"journal-article","created":{"date-parts":[[2021,3,21]],"date-time":"2021-03-21T22:00:37Z","timestamp":1616364037000},"page":"38","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures"],"prefix":"10.3390","volume":"10","author":[{"given":"Kazuhiro","family":"Sakai","sequence":"first","affiliation":[{"name":"Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505, Japan"}]},{"given":"Naoya","family":"Sumi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kumamoto University, Kumamoto 860-8555, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,20]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Moriyasu, K., Sakai, K., and Sumi, N. (2018). Diffeomorphisms with shadowable measures. Axioms, 7.","key":"ref_1","DOI":"10.3390\/axioms7040093"},{"doi-asserted-by":"crossref","unstructured":"Pilyugin, S.Y., and Sakai, K. (2017). Shadowing and Hyperbolicity, Springer. Lecture Notes in Mathematics.","key":"ref_2","DOI":"10.1007\/978-3-319-65184-2"},{"unstructured":"Robinson, C. (1999). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press. [2nd ed.]. Studies in Advanced Mathematics.","key":"ref_3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0022-0396(77)90096-1","article-title":"Hyperbolicity and chain recurrence","volume":"26","author":"Franke","year":"1977","journal-title":"J. Differ. Equ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1090\/S0002-9947-1977-0482849-4","article-title":"Quasi-Anosov diffeomorphisms and hyperbolic manifolds","volume":"229","year":"1977","journal-title":"Trans. Am. Math. 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