{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T09:03:33Z","timestamp":1718874213791},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T00:00:00Z","timestamp":1375747200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Inst. Math. Jussieu"],"published-print":{"date-parts":[[2014,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider a pseudogroup on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1474748013000248_inline3\" \/><jats:tex-math>$( \\mathbb{C} , 0)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> generated by two local diffeomorphisms having analytic conjugacy\n               classes <jats:italic>a priori<\/jats:italic> fixed in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1474748013000248_inline4\" \/><jats:tex-math>$\\mathrm{Diff} \\hspace{0.167em} ( \\mathbb{C} , 0)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We show that a generic pseudogroup as above is such that every\n               point has a (possibly trivial) cyclic stabilizer. It also follows that these generic\n               groups possess infinitely many hyperbolic orbits. This result possesses several\n               applications to the topology of leaves of foliations, and we shall explicitly\n               describe the case of nilpotent foliations associated to Arnold\u2019s singularities of\n               type <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1474748013000248_inline5\" \/><jats:tex-math>${A}^{2n+ 1} $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s1474748013000248","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T10:41:29Z","timestamp":1375785689000},"page":"413-446","source":"Crossref","is-referenced-by-count":2,"title":["Cyclic stabilizers and infinitely many hyperbolic orbits for pseudogroups\n               on"],"prefix":"10.1017","volume":"13","author":[{"given":"Julio C.","family":"Rebelo","sequence":"first","affiliation":[]},{"given":"Helena","family":"Reis","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,8,6]]},"reference":[{"key":"S1474748013000248_r15","first-page":"10","article-title":"On the density of an orbit of a pseudogroup of conformal\n                  mappings and a generalization of the Hudai-Verenov theorem","volume":"31","author":"Shcherbakov","year":"1982","journal-title":"Vestnik Moskov. Univ. Ser. I. Mat."},{"key":"S1474748013000248_r11","doi-asserted-by":"publisher","DOI":"10.24033\/asens.2083"},{"key":"S1474748013000248_r8","volume-title":"Introduction to the modern theory of dynamical systems","volume":"54","author":"Katok","year":"1997"},{"key":"S1474748013000248_r17","first-page":"288","volume-title":"Algebraic and differential topology \u2013\n                  global differential geometry","volume":"70","author":"Takens","year":"1984"},{"key":"S1474748013000248_r10","unstructured":"10. Loray F. , Pseudo-groupe d\u2019une singularit\u00e9 de feuilletage holomorphe en dimension deux (available from hal.archives-ouvertes.fr\/hal-00016434) (2005)."},{"key":"S1474748013000248_r4","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1997.2202"},{"key":"S1474748013000248_r5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01236994"},{"key":"S1474748013000248_r12","doi-asserted-by":"crossref","unstructured":"12. Mattei J.-F. , Rebelo J. C. and Reis H. , Generic pseudogroups on $( \\mathbb{C} , 0)$ and the topology of leaves, Compositio Math., in press, published online (2013), doi: 10.1112\/S0010437X13007161.","DOI":"10.1112\/S0010437X13007161"},{"key":"S1474748013000248_r14","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1410"},{"key":"S1474748013000248_r16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021819201777"},{"key":"S1474748013000248_r9","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-9593(99)80002-0"},{"key":"S1474748013000248_r2","doi-asserted-by":"crossref","first-page":"1599","DOI":"10.1017\/S0143385701001766","article-title":"The generic differential equation \n                     \n                        \n                        $dw\/ dz= {P}_{n} (w, z)\/ {Q}_{n} (w, z)$\n                     \n                   on \n                     \n                        \n                        $ \\mathbb{C} P(2)$\n                     \n                   carries no interesting transverse structure","volume":"21","author":"Belliart","year":"2001","journal-title":"Ergod. Th. and Dynam. Sys."},{"key":"S1474748013000248_r18","first-page":"89","article-title":"Centralisateurs et conjugaison diff\u00e9rentiable des\n                  diff\u00e9omorphismes du cercle. Petits diviseurs en dimension 1","volume":"231","author":"Yoccoz","year":"1995","journal-title":"Ast\u00e9risque"},{"key":"S1474748013000248_r13","unstructured":"13. Mattei J.-F. and Salem E. , Modules formels locaux de feuilletages holomorphes, (available from arXiv:math\/0402256v1), (2004), 89 pages."},{"key":"S1474748013000248_r7","first-page":"821","volume-title":"Proc. Int. Cong. Math.\n                  Helsinki","volume":"2","author":"Il\u2019yashenko","year":"1978"},{"key":"S1474748013000248_r1","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(97)80083-3"},{"key":"S1474748013000248_r3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4364-9"},{"key":"S1474748013000248_r6","first-page":"83","article-title":"Topology of phase portraits of analytic differential\n                  equations on a complex projective plane","volume":"4","author":"Il\u2019yashenko","year":"1978","journal-title":"Tr. Semin. im. I.\n                  G. Petrovskogo."}],"container-title":["Journal of the Institute of Mathematics of Jussieu"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1474748013000248","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T20:52:43Z","timestamp":1556052763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1474748013000248\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,8,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,4]]}},"alternative-id":["S1474748013000248"],"URL":"https:\/\/doi.org\/10.1017\/s1474748013000248","relation":{},"ISSN":["1474-7480","1475-3030"],"issn-type":[{"value":"1474-7480","type":"print"},{"value":"1475-3030","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,8,6]]}}}