An introduction to inverse limits with set-valued functions PDF
Preview An introduction to inverse limits with set-valued functions
SpringerBriefs in Mathematics SeriesEditors KrishnaswamiAlladi NicolaBellomo MicheleBenzi TatsienLi MatthiasNeufang OtmarScherzer DierkSchleicher BenjaminSteinberg VladasSidoravicius YuriTschinkel LoringW.Tu GeorgeYin PingZhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. Forfurthervolumes: http://www.springer.com/series/10030 W.T. Ingram An Introduction to Inverse Limits with Set-valued Functions 123 W.T.Ingram ProfessorEmeritus DepartmentofMathematicsandStatistics MissouriUniversityofScience andTechnology 1870MinerCircle Missouri,USA ISSN2191-8198 ISSN2191-8201(electronic) ISBN978-1-4614-4486-2 ISBN978-1-4614-4487-9(eBook) DOI10.1007/978-1-4614-4487-9 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012940256 MathematicsSubjectClassification(2010):54F15,37B45,54H20,37E05 ©W.T.Ingram2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To mywife, Barbara,oursons,Billand Mark,and ourdaughter,Kathie Thisworkis alsodedicatedtothememoryof BillMahavierwho gotus startedthinking aboutinverselimitswithset-valued functions andtothememoryofmyadvisor,Howard Cook, whoseinfluenceon mycareer is immeasurable. Preface Inverselimitswithset-valuedfunctionswereintroducedin2004byBillMahavier as inverse limits with closed subsets of the unit square. In the short time since then, the subject has rapidly developed into a rich topic of research, particularly among continuum theorists. This new form of inverse limit can also show up in applicationsto economicsandin dynamicalsystems. For instance,certain models ineconomics,notablyinbackwardeconomics,caninvolvetwomappings,andthe flexibility to study the effects of using either function at each stage in the model is a valuable feature of inverse limits with set-valued functions. Recent work of BenjaminMarlinontheLorenzattractorhasprovidedevidenceofthevalueofusing set-valuedfunctionsinstudyingattractorsindynamicalsystems.Thisshortbookis notspecificallyconcernedwiththeseapplicationsofset-valuedfunctions.Instead, it is intended to provide a quick introduction to the subject of inverse limits with set-valued functions. The recently published Springer book Inverse Limits: From Continua to Chaos contains most of the background a researcher might need to makeuseoftheseinverselimitsinhisorherwork.However,ashortervolumesuch asthisonededicatedtoinverselimitswithset-valuedfunctionscouldbehelpfulto someone wanting a quick introduction to this specific subject. This short book is intendedtoprovidejustsuchanintroduction.Theemphasishereisoninverselimits ontheintervalŒ0;1(cid:2)muchasthefirstchapterofInverseLimits: FromContinuato ChaosservesasanintroductiontoinverselimitsonŒ0;1(cid:2)withmappings.Webelieve thatadeeperunderstandingofinverselimitscanbeobtainedbystudyingexamples. A major featureof this bookis the inclusion of numerousexamplesand, in many instances,modelsoftheinverselimits.Severaloftheexamplesinthisvolumehave notappearedelsewhereinprint. Major differences between the theory of inverse limits with mappings and the theory with set-valued functions arise early in this study. These differences are featured prominently in this book. However, instead of viewing these differences as a negative development, we consider them as an opportunity for further study and research. Indeed, much of the research in the subject is devoted to resolving questionsarisingfortheseveryreasons.Wehaveincludedanextensive,butbyno means exhaustive, list of currently unsolved problems in the final chapter of the book. vii viii Preface A senior-level course in analysis and, perhaps, an additional one in topology should provide a sufficient introduction to the topology of metric spaces and the topologyofproductspacestomakethematerialinthisbookaccessibletoadvanced undergraduates and, certainly, to graduate students in mathematics. An alternate sourceofbackgroundmaterialistheappendixintheSpringerbookInverseLimits: FromContinuatoChaos.Insomeoftheexamples,weassumesomefamiliaritywith inverselimitswithmappings,butnodeepunderstandingofordinaryinverselimits isnecessarytoreadthisbook. Thisbookisbasedonaseriesoflecturesgivenbytheauthorataworkshopinthe summerof2011attheInstitutodeMatema´ticas,UniversidadNacionalAuto´noma de Me´xico, in Mexico City. Many thanks go to all who participated in that short course.WeareparticularlyindebtedtoVero´nicaMart´ınezdelaVegaandAlejandro IllaneswhoorganizedtheworkshopandhelpedmakeourstayinMexicoCitysuch apleasantexperience. SpringBranch,Texas TomIngram Acknowledgments TheauthorappreciatestheassistanceofVanNall,MichelSmith,andScottVaragona whoreadsignificantportionsof the manuscript.Withouttheirproofreadingskills, therewouldhavebeenmanyadditionaltypographicalerrorsinthisbook.Theauthor remainssolelyresponsiblefortheerrorsthatremain. The authorappreciatesthe assistance and supportof everyoneat Springerwho made this volume possible. Special thanks go to Springer editor, Vaishali Damle, whoinitiallysuggestedthattheshortcourseinMexicoCitycouldserveasthebasis foraSpringerBrief. ix
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