{"id":"https://openalex.org/W2170488951","doi":"https://doi.org/10.1017/s0305004111000119","title":"Positive topological entropy of Reeb flows on spherizations","display_name":"Positive topological entropy of Reeb flows on spherizations","publication_year":2011,"publication_date":"2011-04-08","ids":{"openalex":"https://openalex.org/W2170488951","doi":"https://doi.org/10.1017/s0305004111000119","mag":"2170488951"},"language":"en","primary_location":{"id":"doi:10.1017/s0305004111000119","doi":"https://doi.org/10.1017/s0305004111000119","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0305004111000119","pdf_url":null,"source":{"id":"https://openalex.org/S4210239678","display_name":"Mathematical Proceedings of the Cambridge Philosophical Society","issn_l":"0305-0041","issn":["0305-0041","1469-8064"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-2","norway-1","poland-100","russia-white-list-2"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Proceedings of the Cambridge Philosophical Society","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1008.4566","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055641583","display_name":"Leonardo Macarini","orcid":"https://orcid.org/0000-0002-6142-3037","observed_orcids":["https://orcid.org/0000-0002-6142-3037"]},"institutions":[{"id":"https://openalex.org/I122140584","display_name":"Universidade Federal do Rio de Janeiro","ror":"https://ror.org/03490as77","country_code":"BR","type":"education","lineage":["https://openalex.org/I122140584"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"LEONARDO MACARINI","raw_affiliation_strings":["Universidade Federal do Rio de Janeiro, Instituto de Matem\u00e1tica, Cidade Universit\u00e1ria, CEP 21941-909, Rio de Janeiro, Brazil. e-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universidade Federal do Rio de Janeiro, Instituto de Matem\u00e1tica, Cidade Universit\u00e1ria, CEP 21941-909, Rio de Janeiro, Brazil. e-mail:","institution_ids":["https://openalex.org/I122140584"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021741258","display_name":"Felix Schlenk","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I57825437","display_name":"University of Neuch\u00e2tel","ror":"https://ror.org/00vasag41","country_code":"CH","type":"education","lineage":["https://openalex.org/I57825437"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"FELIX SCHLENK","raw_affiliation_strings":["Institut de Math\u00e9matiques, Universit\u00e9 de Neuch\u00e2tel, Rue \u00c9mile Argand 11, CP 158, 2009 Neuch\u00e2tel, Switzerland. e-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de Math\u00e9matiques, Universit\u00e9 de Neuch\u00e2tel, Rue \u00c9mile Argand 11, CP 158, 2009 Neuch\u00e2tel, Switzerland. e-mail:","institution_ids":["https://openalex.org/I57825437"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":1.1165,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.75786847,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"151","issue":"1","first_page":"103","last_page":"128"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.8672000169754028,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.8672000169754028,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.10499999672174454,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.005799999926239252,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/reeb-flows","display_name":"Reeb flows","score":0.8429999947547913},{"id":"https://openalex.org/keywords/topological-entropy","display_name":"topological entropy","score":0.9390000104904175},{"id":"https://openalex.org/keywords/cotangent-bundle","display_name":"cotangent bundle","score":0.7860000133514404},{"id":"https://openalex.org/keywords/hamiltonian-flows","display_name":"Hamiltonian flows","score":0.8140000104904175},{"id":"https://openalex.org/keywords/based-loop-space","display_name":"based loop space","score":0.5920000076293945},{"id":"https://openalex.org/keywords/periodic-orbits","display_name":"periodic orbits","score":0.6669999957084656},{"id":"https://openalex.org/keywords/horseshoe","display_name":"horseshoe","score":0.8579999804496765},{"id":"https://openalex.org/keywords/floer-homology","display_name":"Floer homology","score":0.9459999799728394}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8142468929290771},{"id":"https://openalex.org/C2780350623","wikidata":"https://www.wikidata.org/wiki/Q3055355","display_name":"Topological entropy","level":2,"score":0.7060073614120483},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.5901057720184326},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.5486024022102356},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.494144469499588},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4460693299770355},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4372893273830414},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31198710203170776}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1017/s0305004111000119","doi":"https://doi.org/10.1017/s0305004111000119","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0305004111000119","pdf_url":null,"source":{"id":"https://openalex.org/S4210239678","display_name":"Mathematical Proceedings of the Cambridge Philosophical Society","issn_l":"0305-0041","issn":["0305-0041","1469-8064"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-2","norway-1","poland-100","russia-white-list-2"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Proceedings of the Cambridge Philosophical Society","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1008.4566","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1008.4566","pdf_url":"https://arxiv.org/pdf/1008.4566","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doc.rero.ch:292542","doi":null,"is_oa":false,"landing_page_url":"http://doc.rero.ch/record/292542/files/S0305004111000119.pdf","pdf_url":"http://doc.rero.ch/record/292542/files/S0305004111000119.pdf","source":{"id":"https://openalex.org/S4306400148","display_name":"reroDoc Digital Library","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.740.4564","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.740.4564","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1008.4566.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.1008.4566","doi":"https://doi.org/10.48550/arxiv.1008.4566","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1008.4566","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1008.4566","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1008.4566","pdf_url":"https://arxiv.org/pdf/1008.4566","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W1501274325","https://openalex.org/W1550122305","https://openalex.org/W1574390900","https://openalex.org/W1583799621","https://openalex.org/W1614792296","https://openalex.org/W1723182265","https://openalex.org/W1800967450","https://openalex.org/W1968892491","https://openalex.org/W1968975267","https://openalex.org/W1969907227","https://openalex.org/W1983505945","https://openalex.org/W1987903446","https://openalex.org/W1992659862","https://openalex.org/W1995135337","https://openalex.org/W2002160578","https://openalex.org/W2011140596","https://openalex.org/W2013473904","https://openalex.org/W2024297178","https://openalex.org/W2040635624","https://openalex.org/W2049001762","https://openalex.org/W2049920220","https://openalex.org/W2052760612","https://openalex.org/W2059426542","https://openalex.org/W2081817659","https://openalex.org/W2082583603","https://openalex.org/W2084216733","https://openalex.org/W2089931891","https://openalex.org/W2092192427","https://openalex.org/W2102605874","https://openalex.org/W2120703228","https://openalex.org/W2122542302","https://openalex.org/W2126247499","https://openalex.org/W2138309527","https://openalex.org/W2138645224","https://openalex.org/W2171232534","https://openalex.org/W2572250472","https://openalex.org/W2910656496","https://openalex.org/W2952345223","https://openalex.org/W4212958445","https://openalex.org/W4248641565","https://openalex.org/W4293583399","https://openalex.org/W4321258551"],"related_works":["https://openalex.org/W2132897334","https://openalex.org/W7161009564","https://openalex.org/W7128477978","https://openalex.org/W1723182265","https://openalex.org/W4392498141","https://openalex.org/W2266229386","https://openalex.org/W3134897835","https://openalex.org/W7096581112","https://openalex.org/W2295158668","https://openalex.org/W1535438347"],"abstract_inverted_index":{"Abstract":[0],"Let":[1,238],"M":[2,13,37,54,162,172,202,239],"be":[3,72,138,240],"a":[4,31,48,61,241,262],"closed":[5,233,242,271],"manifold":[6],"whose":[7,24],"based":[8],"loop":[9],"space":[10],"\u03a9":[11],"(":[12],")":[14],"is":[15,39,60,92,157],"\u201ccomplicated\u201d.":[16],"Examples":[17],"are":[18],"rationally":[19],"hyperbolic":[20],"manifolds":[21,23],"and":[22,68,109,119,181,254],"fundamental":[25],"group":[26],"has":[27,96],"exponential":[28],"growth.":[29],"Consider":[30],"hypersurface":[32],"\u03a3":[33,59,76,102,164,167,213,216],"in":[34,129,220,275],"T":[35,52,169],"*":[36,53,171],"which":[38],"fiberwise":[40,100],"starshaped":[41],"with":[42],"respect":[43],"to":[44,75,215],"the":[45,73,78,116,120,132,152,184,203,208,223,229,252,255,266],"origin.":[46],"Choose":[47],"function":[49],"H":[50,66,82],":":[51],"\u2192":[55],"\u211d":[56],"such":[57],"that":[58],"regular":[62],"energy":[63],"surface":[64,243],"of":[65,77,81,89,105,147,155,179,186,205,207,231,268],",":[67,234,248,251],"let":[69],"\u03d5":[70,90,210,259],"t":[71,91,211,260],"restriction":[74],"Hamiltonian":[79],"flow":[80,150,209],".":[83,173],"Theorem":[84,135,142,226],"1.":[85,192],"The":[86,144,174],"topological":[87,145],"entropy":[88,146],"positive.":[93,158],"This":[94],"result":[95],"been":[97],"known":[98],"for":[99,197],"convex":[101],"by":[103],"work":[104],"Dinaburg,":[106],"Gromov,":[107],"Paternain,":[108],"Paternain\u2013Petean":[110],"on":[111,151,183],"geodesic":[112],"flows.":[113],"We":[114],"use":[115],"geometric":[117],"idea":[118],"Floer":[121],"homological":[122],"technique":[123],"from":[124,212,245],"[":[125],"19":[126],"],":[127],"but":[128],"addition":[130],"apply":[131],"sandwiching":[133],"method.":[134],"1":[136,227],"can":[137],"reformulated":[139],"as":[140],"follows.":[141],"1'.":[143],"any":[148],"Reeb":[149],"spherization":[153],"SM":[154],"T*M":[156],"For":[159],"q":[160,165,170,194,214],"\u2208":[161,195,201],"abbreviate":[163],"=":[166],"\u2229":[168],"following":[175],"corollary":[176],"extends":[177],"results":[178],"Morse":[180],"Gromov":[182],"number":[185,204,267],"geodesics":[187],"between":[188],"two":[189],"points.":[190],"Corollary":[191,236],"Given":[193],"M,":[196],"almost":[198],"every":[199],"q\u2032":[200,217],"orbits":[206,272],"grows":[218,273],"exponentially":[219,274],"time.":[221,276],"In":[222,264],"lowest":[224],"dimension,":[225],"yields":[228],"existence":[230],"many":[232],"orbits.":[235],"2.":[237],"different":[244],"S":[246],"2":[247,250],"\u211dP":[249],"torus":[253],"Klein":[256],"bottle.":[257],"Then":[258],"carries":[261],"horseshoe.":[263],"particular,":[265],"geometrically":[269],"distinct":[270]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-11-03T10:50:19.383937","created_date":"2016-06-24T00:00:00"}