{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:45:32Z","timestamp":1760237132381,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,27]],"date-time":"2020-02-27T00:00:00Z","timestamp":1582761600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>It is presented and proved a version of Livschitz Theorem for hyperbolic flows pragmatically oriented to the cohomological context. Previously, it is introduced the concept of cocycle and a natural notion of symmetry for cocycles. It is discussed the fundamental relationship between the existence of solutions of cohomological equations and the behavior of the cocycles along periodic orbits. The generalization of this theorem to a class of suspension flows is also discussed and proved. This generalization allows giving a different proof of the Livschitz Theorem for flows based on the construction of Markov systems for hyperbolic flows.<\/jats:p>","DOI":"10.3390\/sym12030338","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T07:50:53Z","timestamp":1583135453000},"page":"338","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2669-2581","authenticated-orcid":false,"given":"Ros\u00e1rio D.","family":"Laureano","sequence":"first","affiliation":[{"name":"Department of Mathematics, ISCTE-IUL Instituto Universit\u00e1rio de Lisboa, Av. das For\u00e7as Armadas, 1649-026 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,27]]},"reference":[{"key":"ref_1","first-page":"758","article-title":"Some homology properties of Y-systems","volume":"10","year":"1971","journal-title":"Math. Notes U.S.S.R. Acad. Sci."},{"key":"ref_2","first-page":"1278","article-title":"Cohomology of dynamical systems","volume":"6","year":"1972","journal-title":"Math. U.S.S.R.-Izv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/BF02771776","article-title":"Markov partitions for Anosov flows on n-dimensional manifolds","volume":"15","author":"Ratner","year":"1973","journal-title":"Israel J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"429","DOI":"10.2307\/2373793","article-title":"Symbolic dynamics for hyperbolic flows","volume":"95","author":"Bowen","year":"1973","journal-title":"Am. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Krantz, S., and Parks, H. (2003). The Implicit Function Theorem. Modern Birkhauser Classics, Birkhauser.","DOI":"10.1007\/978-1-4612-0059-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1017\/S0143385701001511","article-title":"Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces","volume":"21","author":"Foth","year":"2001","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.physd.2007.07.001","article-title":"Jacobi fields on statistical manifolds of negative curvature","volume":"234","author":"Cafaro","year":"2007","journal-title":"Phys. D"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/0022-0396(72)90013-7","article-title":"Expansive one-parameter flows","volume":"12","author":"Bowen","year":"1972","journal-title":"J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s002200000268","article-title":"Multifractal analysis of hyperbolic flows","volume":"214","author":"Barreira","year":"2000","journal-title":"Comm. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/338\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:02:21Z","timestamp":1760173341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/338"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,27]]},"references-count":9,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030338"],"URL":"https:\/\/doi.org\/10.3390\/sym12030338","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,2,27]]}}}