{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,21]],"date-time":"2024-08-21T23:35:42Z","timestamp":1724283342535},"reference-count":19,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Commun. Contemp. Math."],"published-print":{"date-parts":[[2011,4]]},"abstract":"<jats:p>We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure \u03bc is unique and the support of \u03bc is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus \ud835\udd4b<jats:sup>N<\/jats:sup>and v\u2208\u211d<jats:sup>N<\/jats:sup>, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of \u03f5 and h, the entropy penalized Mather problem [Formula: see text] where the entropy S is given by [Formula: see text], and the minimization is performed over the space of probability densities \u03bc(x, v) on \ud835\udd4b<jats:sup>N<\/jats:sup>\u00d7\u211d<jats:sup>N<\/jats:sup>that satisfy the discrete holonomy constraint \u222b<jats:sub>\ud835\udd4b<jats:sup>N<\/jats:sup>\u00d7\u211d<jats:sup>N<\/jats:sup><\/jats:sub>\u03c6(x + hv) - \u03c6(x) d\u03bc = 0. It is known [17] that there exists a unique minimizing measure \u03bc<jats:sub>\u03f5, h<\/jats:sub>which converges to a Mather measure \u03bc, as \u03f5, h\u21920. In the case in which the Mather measure \u03bc is unique we prove a Large Deviation Principle for the limit lim<jats:sub>\u03f5, h\u21920<\/jats:sub>\u03f5 ln \u03bc<jats:sub>\u03f5, h<\/jats:sub>(A), where A \u2282 \ud835\udd4b<jats:sup>N<\/jats:sup>\u00d7\u211d<jats:sup>N<\/jats:sup>. In particular, we prove that the deviation function I can be written as [Formula: see text], where \u03d5<jats:sub>0<\/jats:sub>is the unique viscosity solution of the Hamilton \u2013 Jacobi equation, [Formula: see text]. We also prove a large deviation principle for the limit \u03f5\u2192 0 with fixed h.<\/jats:p><jats:p>Finally, in the last section, we study some dynamical properties of the discrete time Aubry\u2013Mather problem, and present a proof of the existence of a separating subaction.<\/jats:p>","DOI":"10.1142\/s021919971100421x","type":"journal-article","created":{"date-parts":[[2011,4,27]],"date-time":"2011-04-27T09:52:53Z","timestamp":1303897973000},"page":"235-268","source":"Crossref","is-referenced-by-count":7,"title":["THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD"],"prefix":"10.1142","volume":"13","author":[{"given":"D. A.","family":"GOMES","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico, Departamento de Matem\u00e1tica Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"given":"A. O.","family":"LOPES","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica, Universidade Federal do Rio Grande do Sul Av. Bento Gon\u00e7alves, 9500, 91509-900 Porto Alegre, RS, Brazil"}]},{"given":"J.","family":"MOHR","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica, Universidade Federal do Rio Grande do Sul Av. Bento Gon\u00e7alves, 9500, 91509-900 Porto Alegre, RS, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"crossref","first-page":"207","DOI":"10.4171\/jems\/9","volume":"6","author":"Anantharaman N.","journal-title":"J. Eur. Math. Soc."},{"key":"rf2","first-page":"353","volume":"23","author":"Anantharaman N.","journal-title":"Ergodic Theons Dynam. 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