{"id":"https://openalex.org/W2117548458","doi":"https://doi.org/10.1142/s0218127414501648","title":"Invariant Measures for the n-Dimensional Border Collision Normal Form","display_name":"Invariant Measures for the n-Dimensional Border Collision Normal Form","publication_year":2014,"publication_date":"2014-12-01","ids":{"openalex":"https://openalex.org/W2117548458","doi":"https://doi.org/10.1142/s0218127414501648","mag":"2117548458"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127414501648","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414501648","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084524431","display_name":"Paul Glendinning","orcid":"https://orcid.org/0000-0003-1544-9161"},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Paul Glendinning","raw_affiliation_strings":["School of Mathematics, University of Manchester, Manchester M13 9PL, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Manchester, Manchester M13 9PL, UK","institution_ids":["https://openalex.org/I28407311"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5084524431"],"corresponding_institution_ids":["https://openalex.org/I28407311"],"apc_list":null,"apc_paid":null,"fwci":1.689,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.8231651,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"24","issue":"12","first_page":"1450164","last_page":"1450164"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9998999834060669,"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":0.9998999834060669,"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/T10591","display_name":"Theoretical and Computational Physics","score":0.970300018787384,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9569000005722046,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.7865041494369507},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7587310075759888},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.713826060295105},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.6709294319152832},{"id":"https://openalex.org/keywords/open-set","display_name":"Open set","score":0.6450976133346558},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6118819117546082},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5652230381965637},{"id":"https://openalex.org/keywords/hausdorff-space","display_name":"Hausdorff space","score":0.5599572658538818},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.5507550835609436},{"id":"https://openalex.org/keywords/lebesgue-covering-dimension","display_name":"Lebesgue covering dimension","score":0.5034655928611755},{"id":"https://openalex.org/keywords/fractal-dimension","display_name":"Fractal dimension","score":0.49603256583213806},{"id":"https://openalex.org/keywords/effective-dimension","display_name":"Effective dimension","score":0.4825098514556885},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.4496772289276123},{"id":"https://openalex.org/keywords/collision","display_name":"Collision","score":0.43230748176574707},{"id":"https://openalex.org/keywords/minkowski\u2013bouligand-dimension","display_name":"Minkowski\u2013Bouligand dimension","score":0.4230429530143738},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.382508784532547},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3759281635284424},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.20614752173423767},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.10985404253005981},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10583460330963135},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08369946479797363},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.07079640030860901}],"concepts":[{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.7865041494369507},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7587310075759888},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.713826060295105},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.6709294319152832},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.6450976133346558},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6118819117546082},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5652230381965637},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.5599572658538818},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.5507550835609436},{"id":"https://openalex.org/C23707678","wikidata":"https://www.wikidata.org/wiki/Q164262","display_name":"Lebesgue covering dimension","level":3,"score":0.5034655928611755},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.49603256583213806},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.4825098514556885},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.4496772289276123},{"id":"https://openalex.org/C121704057","wikidata":"https://www.wikidata.org/wiki/Q352070","display_name":"Collision","level":2,"score":0.43230748176574707},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.4230429530143738},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.382508784532547},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3759281635284424},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.20614752173423767},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.10985404253005981},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10583460330963135},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08369946479797363},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.07079640030860901},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1142/s0218127414501648","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127414501648","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2252","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/2252/1/glendinningijbcv2.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/2252/1/glendinningijbcv2.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2348","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/2348/1/bifntondimrevised.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/2348/1/bifntondimrevised.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/f4eab9c4-10a6-45b2-b70f-723f03535b5d","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/f4eab9c4-10a6-45b2-b70f-723f03535b5d","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Glendinning, P 2014, 'Invariant Measures for the n-Dimensional Border Collision Normal Form', International Journal of Bifurcation and Chaos in Applied Sciences and Engineering , vol. 24, no. 12, 1450164. https://doi.org/10.1142/S0218127414501648","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:pure.atira.dk:publications/f4eab9c4-10a6-45b2-b70f-723f03535b5d","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/invariant-measures-for-the-ndimensional-border-collision-normal-form(f4eab9c4-10a6-45b2-b70f-723f03535b5d).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Glendinning, P 2014, 'Invariant Measures for the n-Dimensional Border Collision Normal Form', International Journal of Bifurcation and Chaos in Applied Sciences and Engineering , vol. 24, no. 12, 1450164. https://doi.org/10.1142/S0218127414501648","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6297168181","display_name":null,"funder_award_id":"EP/I01912X/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W1552277675","https://openalex.org/W1591391447","https://openalex.org/W1597461812","https://openalex.org/W1602130484","https://openalex.org/W1975027618","https://openalex.org/W1997672633","https://openalex.org/W2001829487","https://openalex.org/W2009186763","https://openalex.org/W2009313082","https://openalex.org/W2010516970","https://openalex.org/W2010920083","https://openalex.org/W2011171775","https://openalex.org/W2022605731","https://openalex.org/W2027059615","https://openalex.org/W2044178030","https://openalex.org/W2058894160","https://openalex.org/W2060587651","https://openalex.org/W2062124659","https://openalex.org/W2063775788","https://openalex.org/W2064873502","https://openalex.org/W2068000468","https://openalex.org/W2094656677","https://openalex.org/W2094734627","https://openalex.org/W2096687967","https://openalex.org/W2098330591","https://openalex.org/W2113169711","https://openalex.org/W2127666064","https://openalex.org/W2143841218","https://openalex.org/W2159251357","https://openalex.org/W2170695435","https://openalex.org/W2735036401","https://openalex.org/W3022125062","https://openalex.org/W3037414779","https://openalex.org/W3099174843","https://openalex.org/W4230378422","https://openalex.org/W4240580419","https://openalex.org/W4242089755","https://openalex.org/W4292510609"],"related_works":["https://openalex.org/W2950523257","https://openalex.org/W2094411530","https://openalex.org/W4296311260","https://openalex.org/W2292272462","https://openalex.org/W2128554376","https://openalex.org/W3093896268","https://openalex.org/W4288481336","https://openalex.org/W3203184942","https://openalex.org/W2918228419","https://openalex.org/W2075848215"],"abstract_inverted_index":{"The":[0],"border":[1],"collision":[2],"normal":[3],"form":[4],"is":[5],"a":[6],"continuous":[7,28],"piecewise":[8,17],"affine":[9],"map":[10],"of":[11,35],"\u211d":[12],"n":[13,48,64,72],"with":[14],"applications":[15],"in":[16],"smooth":[18],"bifurcation":[19],"theory.":[20],"We":[21],"show":[22],"that":[23],"these":[24],"maps":[25],"have":[26,42,53,69],"absolutely":[27],"invariant":[29],"measures":[30],"for":[31],"an":[32],"open":[33,60],"set":[34],"parameter":[36],"space":[37],"and":[38,62],"hence":[39],"the":[40,51],"attractors":[41,52],"Hausdorff":[43],"(fractal)":[44],"dimension":[45,55,71],"n.":[46],"If":[47],"=":[49],"2":[50,66],"topological":[54,70],"two,":[56],"i.e.":[57],"they":[58,68],"contain":[59],"sets,":[61],"if":[63],">":[65],"then":[67],"generically.":[73]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
