{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T18:59:22Z","timestamp":1757703562324,"version":"3.40.5"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Ergod. Th. Dynam. Sys."],"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider continuous free semigroup actions generated by a family \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline1.png\"\/><jats:tex-math>\n$(g_y)_{y \\,\\in \\, Y}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of continuous endomorphisms of a compact metric space \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline2.png\"\/><jats:tex-math>\n$(X,d)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, subject to a random walk \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline3.png\"\/><jats:tex-math>\n$\\mathbb P_\\nu =\\nu ^{\\mathbb N}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> defined on a shift space \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline4.png\"\/><jats:tex-math>\n$Y^{\\mathbb N}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline5.png\"\/><jats:tex-math>\n$(Y, d_Y)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a compact metric space with finite upper box dimension and \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline6.png\"\/><jats:tex-math>\n$\\nu $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a Borel probability measure on <jats:italic>Y<\/jats:italic>. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline7.png\"\/><jats:tex-math>\n$\\sigma $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline8.png\"\/><jats:tex-math>\n$Y^{\\mathbb {N}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and compare them with the upper box dimension of <jats:italic>Y<\/jats:italic>. In particular, we obtain exact formulas whenever \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline9.png\"\/><jats:tex-math>\n$\\nu $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385720001431_inline10.png\"\/><jats:tex-math>\n$\\nu $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and to test the scope of our results.\n<\/jats:p>","DOI":"10.1017\/etds.2020.143","type":"journal-article","created":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T08:56:52Z","timestamp":1612169812000},"page":"65-85","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["A variational principle for the metric mean dimension of free semigroup actions"],"prefix":"10.1017","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6929-6442","authenticated-orcid":false,"given":"MARIA","family":"CARVALHO","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7596-8214","authenticated-orcid":false,"given":"FAGNER B.","family":"RODRIGUES","sequence":"additional","affiliation":[]},{"given":"PAULO","family":"VARANDAS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,2,1]]},"reference":[{"key":"S0143385720001431_r8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2018.06.010"},{"key":"S0143385720001431_r6","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-016-1697-3"},{"key":"S0143385720001431_r2","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7931-5"},{"key":"S0143385720001431_r22","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2019.106935"},{"key":"S0143385720001431_r3","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2778"},{"key":"S0143385720001431_r20","doi-asserted-by":"publisher","DOI":"10.2307\/1996437"},{"key":"S0143385720001431_r21","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-018-1813-y"},{"key":"S0143385720001431_r13","doi-asserted-by":"publisher","DOI":"10.1017\/S014338571100109X"},{"key":"S0143385720001431_r9","doi-asserted-by":"publisher","DOI":"10.1017\/etds.2020.130"},{"key":"S0143385720001431_r11","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009841100168"},{"key":"S0143385720001431_r14","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-16.3.568"},{"key":"S0143385720001431_r5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021796818247"},{"key":"S0143385720001431_r16","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-019-00501-8"},{"key":"S0143385720001431_r19","doi-asserted-by":"publisher","DOI":"10.1063\/1.4950928"},{"key":"S0143385720001431_r18","doi-asserted-by":"publisher","DOI":"10.7208\/chicago\/9780226662237.001.0001"},{"key":"S0143385720001431_r7","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/aa999f"},{"volume-title":"Dynamics Beyond Uniform Hyperbolicity","year":"2005","author":"Bonatti","key":"S0143385720001431_r4"},{"key":"S0143385720001431_r23","unstructured":"[20] Velozo, A. and Velozo, R. . 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