{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:32:30Z","timestamp":1772368350661,"version":"3.50.1"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2019,12]]},"abstract":"<jats:p> Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609\u2013641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupi\u0144ski, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863\u2013932]) from the Ellis group of the flow [Formula: see text] to the Kim\u2013Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061319500120","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T06:36:51Z","timestamp":1558075011000},"page":"1950012","source":"Crossref","is-referenced-by-count":5,"title":["Boundedness and absoluteness of some dynamical invariants in model theory"],"prefix":"10.1142","volume":"19","author":[{"given":"Krzysztof","family":"Krupi\u0144ski","sequence":"first","affiliation":[{"name":"Instytut Matematyczny, Uniwersytet Wroc\u0142awski, pl. Grunwaldzki 2\/4, 50-384 Wroc\u0142aw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ludomir","family":"Newelski","sequence":"additional","affiliation":[{"name":"Instytut Matematyczny, Uniwersytet Wroc\u0142awski, pl. Grunwaldzki 2\/4, 50-384 Wroc\u0142aw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"Simon","sequence":"additional","affiliation":[{"name":"Mathematics Department, University of California, Berkeley, Evans Hall, Berkeley, CA 94720-3840, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2019,10,2]]},"reference":[{"key":"S0219061319500120BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061301000119"},{"key":"S0219061319500120BIB002","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1327068688"},{"key":"S0219061319500120BIB003","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/896"},{"key":"S0219061319500120BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2017.06.006"},{"key":"S0219061319500120BIB005","doi-asserted-by":"publisher","DOI":"10.4064\/fm229-2-2"},{"key":"S0219061319500120BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0080139"},{"key":"S0219061319500120BIB007","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/274"},{"key":"S0219061319500120BIB008","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004116000967"},{"key":"S0219061319500120BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2019.01.033"},{"key":"S0219061319500120BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-018-1780-3"},{"key":"S0219061319500120BIB011","doi-asserted-by":"publisher","DOI":"10.2307\/2694914"},{"key":"S0219061319500120BIB012","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1231082302"},{"key":"S0219061319500120BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-011-0202-6"},{"key":"S0219061319500120BIB014","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdr075"},{"key":"S0219061319500120BIB015","doi-asserted-by":"publisher","DOI":"10.4064\/fm191-3-2"},{"key":"S0219061319500120BIB017","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511549786.013"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061319500120","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,3]],"date-time":"2019-10-03T06:27:02Z","timestamp":1570084022000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219061319500120"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,2]]},"references-count":16,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["10.1142\/S0219061319500120"],"URL":"https:\/\/doi.org\/10.1142\/s0219061319500120","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,2]]}}}