{"id":"https://openalex.org/W2963519414","doi":"https://doi.org/10.1017/s0963548310000337","title":"Period Lengths for Iterated Functions","display_name":"Period Lengths for Iterated Functions","publication_year":2010,"publication_date":"2010-09-16","ids":{"openalex":"https://openalex.org/W2963519414","doi":"https://doi.org/10.1017/s0963548310000337","mag":"2963519414"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548310000337","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000337","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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":"Combinatorics, Probability and Computing","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/A5048043622","display_name":"Eric Schmutz","orcid":null},"institutions":[{"id":"https://openalex.org/I72816309","display_name":"Drexel University","ror":"https://ror.org/04bdffz58","country_code":"US","type":"education","lineage":["https://openalex.org/I72816309"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"ERIC SCHMUTZ","raw_affiliation_strings":["Mathematics Department, Drexel University, 3401 Market Street, Philadelphia, Pa., 19104, USA (e-mail:","Mathematics department, drexel university, 3401 market street, philadelphia, pa., 19104, usa (e-mail: eric.jonathan.schmutz@drexel.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department, Drexel University, 3401 Market Street, Philadelphia, Pa., 19104, USA (e-mail:","institution_ids":["https://openalex.org/I72816309"]},{"raw_affiliation_string":"Mathematics department, drexel university, 3401 market street, philadelphia, pa., 19104, usa (e-mail: eric.jonathan.schmutz@drexel.edu)#TAB#","institution_ids":["https://openalex.org/I72816309"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5048043622"],"corresponding_institution_ids":["https://openalex.org/I72816309"],"apc_list":null,"apc_paid":null,"fwci":0.3978,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.6318609,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"20","issue":"2","first_page":"289","last_page":"298"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9972000122070312,"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":0.9972000122070312,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.992900013923645,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12508","display_name":"Functional Equations Stability Results","score":0.9861000180244446,"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/iterated-function","display_name":"Iterated function","score":0.8658099174499512},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6541138291358948},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6268240213394165},{"id":"https://openalex.org/keywords/period","display_name":"Period (music)","score":0.6251319646835327},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.619563639163971},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5949029326438904},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5929879546165466},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.5404554009437561},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.5330518484115601},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.528488278388977},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4486718475818634},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.214842289686203},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.20356681942939758},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16157382726669312},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13064122200012207},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09147429466247559},{"id":"https://openalex.org/keywords/finance","display_name":"Finance","score":0.042481809854507446}],"concepts":[{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.8658099174499512},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6541138291358948},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6268240213394165},{"id":"https://openalex.org/C2781291010","wikidata":"https://www.wikidata.org/wiki/Q178580","display_name":"Period (music)","level":2,"score":0.6251319646835327},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.619563639163971},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5949029326438904},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5929879546165466},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.5404554009437561},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5330518484115601},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.528488278388977},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4486718475818634},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.214842289686203},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.20356681942939758},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16157382726669312},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13064122200012207},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09147429466247559},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.042481809854507446},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548310000337","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000337","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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":"Combinatorics, Probability and Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W186403072","https://openalex.org/W963086913","https://openalex.org/W972608077","https://openalex.org/W1489554514","https://openalex.org/W1515528048","https://openalex.org/W1967436950","https://openalex.org/W1970252670","https://openalex.org/W1978298011","https://openalex.org/W1996847651","https://openalex.org/W2011101194","https://openalex.org/W2068312480","https://openalex.org/W2070658523","https://openalex.org/W2085259775","https://openalex.org/W2087466973","https://openalex.org/W2096596497","https://openalex.org/W2103268360","https://openalex.org/W2105422170","https://openalex.org/W2115457558","https://openalex.org/W2169980960","https://openalex.org/W2226303357","https://openalex.org/W2313299119","https://openalex.org/W2340541605","https://openalex.org/W2905110430","https://openalex.org/W4230925302","https://openalex.org/W4300223101"],"related_works":["https://openalex.org/W2590211375","https://openalex.org/W3103325625","https://openalex.org/W1486898455","https://openalex.org/W2581822552","https://openalex.org/W2579643896","https://openalex.org/W3090727460","https://openalex.org/W2041905724","https://openalex.org/W2047986527","https://openalex.org/W4306313719","https://openalex.org/W2088134950"],"abstract_inverted_index":{"Let":[0,28],"\u03a9":[1],"n":[2,5,6,21],"be":[3,33],"the":[4,34,41,44,71,74,77,96],"-element":[7],"set":[8],"consisting":[9],"of":[10,36,43,59,73,76,79,99,109,129],"all":[11],"functions":[12],"that":[13,95,122],"have":[14],"{1,":[15],"2,":[16],"3,":[17],".":[18,19,56,57,58,91],".,":[20],"}":[22],"as":[23,86],"both":[24],"domain":[25],"and":[26],"codomain.":[27],"T":[29,90,110],"(":[30,67],"f":[31,37,46,48,51,54,68,80],")":[32,69],"order":[35],",":[38,40,47,50,53,81],"i.e.":[39],"period":[42],"sequence":[45],"(2)":[49],"(3)":[52],"(4)":[55],"compositional":[60],"iterates.":[61],"A":[62],"closely":[63],"related":[64],"number,":[65],"B":[66,130],"=":[70],"product":[72],"lengths":[75],"cycles":[78],"has":[82],"previously":[83],"been":[84],"used":[85],"an":[87],"approximation":[88],"for":[89],"This":[92],"paper":[93],"proves":[94],"average":[97],"values":[98],"these":[100],"two":[101],"quantities":[102],"are":[103],"quite":[104],"different.":[105],"The":[106,126],"expected":[107,127],"value":[108,128],"is":[111,115,123,131],"where":[112],"k":[113],"0":[114],"a":[116],"complicated":[117],"but":[118],"explicitly":[119],"defined":[120],"constant":[121],"approximately":[124],"3.36.":[125],"much":[132],"larger:":[133]},"counts_by_year":[{"year":2020,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2013,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}