{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T07:05:51Z","timestamp":1774595151637,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,5,2]],"date-time":"2021-05-02T00:00:00Z","timestamp":1619913600000},"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>We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.<\/jats:p>","DOI":"10.3390\/axioms10020080","type":"journal-article","created":{"date-parts":[[2021,5,2]],"date-time":"2021-05-02T08:05:21Z","timestamp":1619942721000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1253-1015","authenticated-orcid":false,"given":"Sergey","family":"Kryzhevich","sequence":"first","affiliation":[{"name":"Mathematics and Mechanics Faculty, Saint Petersburg State University, 199034 St. Petersburg, Russia"}]},{"given":"Viktor","family":"Avrutin","sequence":"additional","affiliation":[{"name":"Institute for Systems Theory and Automatic Control, University of Stuttgart, 70174 Stuttgart, Germany"}]},{"given":"Nikita","family":"Begun","sequence":"additional","affiliation":[{"name":"Mathematics and Mechanics Faculty, Saint Petersburg State University, 199034 St. Petersburg, Russia"}]},{"given":"Dmitrii","family":"Rachinskii","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, USA"}]},{"given":"Khosro","family":"Tajbakhsh","sequence":"additional","affiliation":[{"name":"Faculty of Mathematical Sciences, Tarbiat Modares University, 14115-134 Tehran, Iran"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1051\/mmnp\/2019041","article-title":"Invariant measures for interval translations and some other piecewise continuous maps","volume":"50","author":"Kryzhevich","year":"2020","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1017\/S0143385700009652","article-title":"Interval translation maps","volume":"15","author":"Boshernitzan","year":"1995","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_3","unstructured":"Katok, A., and Hasselblatt, B. (1997). Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press."},{"key":"ref_4","unstructured":"Baron, P.G. (2021, May 01). Spectral and Mixing Properties of Interval Exchange Transformations with Flips. Available online: https:\/\/baronpolina1.wixsite.com\/website."},{"key":"ref_5","unstructured":"Yoccoz, J.C. (2021, May 01). Echange d\u2019intervalles. Available online: https:\/\/www.college-de-france.fr\/media\/jean-christophe-yoccoz\/UPL8726_yoccoz05.pdf."},{"key":"ref_6","unstructured":"Schmeling, J., and Troubetzkoy, S. (1998). Interval Translation maps. Dynamical Systems (Luminy-Marseille), World Scientific."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1080\/14689360601028084","article-title":"Renormalization in a class of interval translation maps of d branches","volume":"22","author":"Bruin","year":"2007","journal-title":"Dyn. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/BF02785958","article-title":"The Gauss map on a class of interval translation maps","volume":"137","author":"Bruin","year":"2003","journal-title":"Isr. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4133","DOI":"10.3934\/dcds.2012.32.4133","article-title":"Inducing and unique ergodicity of double rotations","volume":"32","author":"Bruin","year":"2012","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Artigiani, M., Fougeron, C., Hubert, P., and Skripchenko, A. (2021). A note on double rotations of infinite type. arXiv.","DOI":"10.1090\/mosc\/311"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"515","DOI":"10.3934\/dcds.2005.13.515","article-title":"Double rotations","volume":"13","author":"Suzuki","year":"2005","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1070\/IM2009v073n01ABEH002439","article-title":"Two-color rotations of the unit circle","volume":"73","author":"Zhuravlev","year":"2009","journal-title":"Izv. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1017\/S0143385703000488","article-title":"Piecewise monotone maps without periodic points: Rigidity, measures and complexity","volume":"24","author":"Buzzi","year":"2004","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.3934\/dcds.2014.34.2307","article-title":"Almost every interval translation map of three intervals is finite type","volume":"34","author":"Volk","year":"2014","journal-title":"Discret. Contin. Dyn. Syst. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Pires, B. (2016). Invariant measures for piecewise continuous maps. arXiv.","DOI":"10.1016\/j.crma.2016.05.002"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1017\/S0143385701001651","article-title":"Piecewise isometries have zero topological entropy","volume":"21","author":"Buzzi","year":"2001","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1017\/S0143385799151964","article-title":"Sofic subshifts and piecewise isometric systems","volume":"19","author":"Goetz","year":"1999","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_18","first-page":"465","article-title":"Dynamics of piecewise isometries","volume":"44","author":"Goetz","year":"2000","journal-title":"Ill. J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1002\/zamm.19280080202","article-title":"Ein Gedankenmodell zur kinetischen Theorie der festen K\u00f6rper","volume":"8","author":"Prandtl","year":"1928","journal-title":"ZAMM J. Appl. Math. Mech."},{"key":"ref_20","unstructured":"Krej\u010d\u00ec, P. (1996). Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho."},{"key":"ref_21","unstructured":"Krasnosel\u2019skii, M.A., and Pokrovskii, A.V. (1989). Systems with Hysteresis, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mayergoyz, I.D. (2003). Mathematical Models of Hysteresis and Their Applications, Elsevier Science.","DOI":"10.1016\/B978-012480873-7\/50005-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1140\/epjb\/e2007-00108-5","article-title":"Stylized facts from a threshold-based heterogeneous agent model","volume":"57","author":"Cross","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.physb.2007.08.017","article-title":"A new paradigm for modelling hysteresis in macroeconomic flows","volume":"403","author":"Cross","year":"2008","journal-title":"Phys. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"032822","DOI":"10.1103\/PhysRevE.90.032822","article-title":"Analytical solution for a class of network dynamics with mechanical and financial applications","volume":"90","author":"Lamba","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1137\/16M1073522","article-title":"Dynamics of Discrete Time Systems with a Hysteresis Stop Operator","volume":"16","author":"Arnold","year":"2017","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1930005","DOI":"10.1142\/S0218127419300052","article-title":"Chaos in the saw map","volume":"29","author":"Begun","year":"2019","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_28","unstructured":"Pilyugin, S.Y., and Sabirova, D.Z. Dynamics of one continual sociological model. Vestnik Sankt-Peterburgskogo Universiteta, Saint Petersburg State University. in press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1088\/0951-7715\/21\/5\/010","article-title":"On the Fully Developed Bandcount Adding Scenario","volume":"21","author":"Avrutin","year":"2008","journal-title":"Nonlinearity"},{"key":"ref_30","unstructured":"Avrutin, V., Gardini, L., Sushko, I., and Tramontana, F. (2019). Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, World Scientific."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/2\/80\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:56:46Z","timestamp":1760162206000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/2\/80"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,2]]},"references-count":30,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["axioms10020080"],"URL":"https:\/\/doi.org\/10.3390\/axioms10020080","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,2]]}}}