{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T22:31:32Z","timestamp":1781562692119,"version":"3.54.5"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>In this note we show:<\/jats:p><jats:p>Theorem 1.1. <jats:italic>Let G be a Polish group and X a Polish G-space with the induced orbit equivalence relation E<jats:sub>G<\/jats:sub> Borel as a subset of X \u00d7 X. Then exactly one of the following<\/jats:italic>:<\/jats:p><jats:p>(I) <jats:italic>There is a countable language<\/jats:italic><jats:italic>\u2112<\/jats:italic><jats:italic>and a Borel function<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009191_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p><jats:italic>such that for all x<jats:sub>1<\/jats:sub>, x<jats:sub>2<\/jats:sub> \u2208 X<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009191_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p><jats:italic>or<\/jats:italic><\/jats:p><jats:p>(II) <jats:italic>there is a turbulent Polish G-space Y and a continuous G-embedding<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009191_eqnU3\"\/><\/jats:disp-formula><\/jats:p><jats:p>There are various bows and ribbons which can be woven into these statements. We can strengthen (I) by asking that \u03b8 also admit a Borel orbit inverse, that is to say some Borel function<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009191_eqnU4\"\/><\/jats:disp-formula><\/jats:p><jats:p>for some Borel set <jats:italic>B<\/jats:italic> \u2282 Mod(<jats:italic>\u2112<\/jats:italic>), such that for all <jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009191_eqnU5\"\/><\/jats:disp-formula><\/jats:p><jats:p>and then after having passed to this strengthened version of (I) we still obtain the exact same dichotomy theorem, and hence the conclusion that the two competing versions of (I) are equivalent. Similarly (II) can be relaxed to just asking that \u03c4 be a Borel <jats:italic>G<\/jats:italic>-embedding, or even simply a Borel reduction of the relevant orbit equivalence relations. It is in fact a consequence of 1.1 that all the plausible weakenings and strengthenings of (I) and (II) are respectively equivalent to one another.<\/jats:p><jats:p>I will not closely examine these possible variations here. The equivalences alluded to above follow from our main theorem and the results of [3]. That monograph had previously shown that (I) and (II) are incompatible, and proved a barbaric forerunner of 1.1, and gone on to conjecture the dichotomy result above.<\/jats:p>","DOI":"10.2178\/jsl\/1190150297","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:16:54Z","timestamp":1197573414000},"page":"1520-1540","source":"Crossref","is-referenced-by-count":5,"title":["A dichotomy theorem for turbulence"],"prefix":"10.1017","volume":"67","author":[{"given":"Greg","family":"Hjorth","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009191_ref003","volume-title":"Classification and orbit equivalence relations","author":"Hjorth","year":"2000"},{"key":"S0022481200009191_ref001","unstructured":"Becker H. and Kechris A. S. , The descriptive set theory of Polish group actions , London Mathematical Society."},{"key":"S0022481200009191_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1988-0965754-3"},{"key":"S0022481200009191_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-97-00221-X"},{"key":"S0022481200009191_ref005","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009191","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:18:06Z","timestamp":1557188286000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009191\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S0022481200009191"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150297","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}