Nonlinear Systems and Complexity Volume 11 SeriesEditor AlbertC.J.Luo SouthernIllinoisUniversity Edwardsville Illinois USA Nonlinear Systems and Complexity provides a place to systematically summarize recentdevelopments,applications,andoveralladvanceinallaspectsofnonlinearity, chaos,andcomplexityaspartoftheestablishedresearchliterature,beyondthenovel andrecentfindingspublishedinprimaryjournals. Theaimsofthebookseriesare to publish theories and techniques in nonlinear systems and complexity; stimulate moreresearchinterestonnonlinearity,synchronization,andcomplexityinnonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent develop- ments, findings and progress on theories, principles, methodology, computational techniquesinnonlinearsystemsandmathematicswithengineeringapplications.The Seriesestablisheshighlyrelevantmonographsonwiderangingtopicscoveringfun- damentaladvancesandnewapplicationsinthefield.Topicalareasinclude,butare not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Compu- tational methods for nonlinear systems Stability, bifurcation, chaos and fractals in engineeringNonlinearchemicalandbiologicalphenomenaFractionaldynamicsand applicationsDiscontinuity,synchronizationandcontrol Moreinformationaboutthisseriesathttp://www.springer.com/series/11433 Hernán González-Aguilar • Edgardo Ugalde Editors Nonlinear Dynamics New Directions Theoretical Aspects 2123 Editors HernánGonzález-Aguilar EdgardoUgalde AutonomousUniversityofSanLuisPotosí AutonomousUniversityofSanLuisPotosí SanLuisPotosi SanLuisPotosi Mexico Mexico ISSN2195-9994 ISSN2196-0003(electronic) ISBN978-3-319-09866-1 ISBN978-3-319-09867-8(eBook) DOI10.1007/978-3-319-09867-8 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014951913 © SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Theideaofthisbookaroseduringthemeeting“NonlinearDynamics: NewDirec- tions”,heldinhonorofProfessorValentinAfraimovichinGuanajuatoMexicoduring May2010.Themeetingtookplaceinthe“CentrodeInvestigaciónenMatemáticas”, one of the finest places to do mathematics. During the meeting we had the oppor- tunity, not only to share our passion for nonlinear dynamics, but also to discuss itsfoundationsandtheemergingapplicationsmainlyinbiology.Itwasaswellthe occasion to share life experiences and celebrate this magnificent person, Valentin Afraimovich.Thebookcoversallscientificaspectsofthemeeting.Foranaccount- ingofitshumanaspectswehaveLevTsrimling’sphotogallery.Weareawaitinga convincingtranslationofMichaRavinovich’spoeticwork. Thistwo-volumebookcoverspartofthevastspectrumofinterestsofProfessor Afraimovich,whichrangefromfractalanalysistoveryspecificapplicationsofthe theoryofdynamicalsystemstobiology.Thefirstvolumeofthisbookisdevotedto fundamentalaspectsandincludesanumberofimportantnewcontributionsaswell assomereviewarticles,whichemphasizenewdevelopmentprospects.Thesecond volumecontainsmostlynewapplicationsofthetheoryofdynamicalsystemstoboth engineeringandbiology. Theauthorscontributingtothesetwovolumes,allofthemacademicallyrelated to ProfessorAfraimovich, are among the most prominent specialists in nonlinear dynamics.Thetopicsaddressedinthesetwovolumesincludearigoroustreatment offluctuationindynamicalsystems,topicsinfractalanalysis,studiesofthetransient dynamics in biological networks, synchronization in lasers, and control of chaotic systems,amongothers. Weareveryhappytohavefinallycompletedthiscompilationandwethankallthe contributorsfromthebottomofourhearts.Wealsothankalltheagenciesthatcon- tributedtofinancingthe2010meeting,inparticularFENOMEC-UNAM,CONACyT, CIMAT,andUASLP.Ifdespiteoureffort,somemistakesremain,weareaccountable forit. SanLuisPotosí HernánGonzález-Aguilar EdgardoUgalde v Contents ANoteontheLargeDeviationsforPiecewiseExpandingMultidimensional Maps ............................................................ 1 R.AiminoandS.Vaienti DirectionalMetricEntropyandLyapunovExponentsforDynamical SystemsGeneratedbyCellularAutomata ............................ 11 MauriceCourbageandBrunonKamin´ski OntheComplexityofSomeGeometricalObjects ..................... 29 P.Collet FluctuationsofObservablesinDynamicalSystems:FromLimitTheorems toConcentrationInequalities ....................................... 47 Jean-RenéChazottes OnFlowBarriersinDiscontinuousDynamicalSystems................ 87 AlbertC.J.Luo NonstandardAnalysisoftheBehaviorofErgodicMeansofDynamical SystemsonVeryBigFiniteProbabilitySpaces ........................ 115 E.I.Gordon,L.Yu.GlebskyandC.W.Henson OnMeasuresResistingMultifractalAnalysis ......................... 153 JörgSchmelingandStéphaneSeuret AnOverviewofComplexKleinianGroups ........................... 167 AngelCanoandJoséSeade Semigroups of Mappings and Correspondences: Characters and RepresentationsinHolomorphicDynamicalSystems .................. 195 CarlosCabrera,PeterMakienkoandPeterPlaumann vii Contributors R.Aimino UMR-6207CentredePhysiqueThéorique, CNRS,Universitésd’Aix- Marseille,UniversitéduSud,Toulon-VarandFRUMAM,FédéderationdeRecherche desUnitésdeMathématiquesdeMarseilleCPT,MarseilleCedex9,France CarlosCabrera InstitutodeMatemáticas,UNAM,Cuernavaca,MOR,Mexico AngelCano InstitutodeMatemáticas,UnidadCuernavaca,UniversidadNacional AutónomadeMéxico,Cuernavaca,México Jean-RenéChazottes CentredePhysiqueThéorique,CNRS-Écolepolytechnique, PalaiseauCedex,France P. Collet Centre de Physique Théorique, Ecole Polytechnique, Palaiseau Cedex, France Maurice Courbage Laboratoire Matière et Système Complexes and Université Paris7-DenisDiderot,ParisCedex13,France L.Yu.Glebsky IICO-UASLP,SanLuisPotosi,SLP,Mexico E.I.Gordon EasternIllinoisUniversity,Charleston,IL,USA C.W.Henson UniversityofIllinoisatUrbana-Champaign,Urbana,USA Brunon Kamin´ski Faculty of Mathematics and Computer Science, Nicolaus CopernicusUniversity,Torun´,Poland AlbertC.J.Luo DepartmentofMechanicalandIndustrialEngineering,Schoolof Engineering,SouthernIllinoisUniversityEdwardsville,Edwardsville,IL,USA PeterMakienko InstitutodeMatemáticas,UNAM,Cuernavaca,MOR,Mexico Peter Plaumann Mathematisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg,Erlangen,Germany JörgSchmeling MathematicsCentreforMathematicalSciences,LundInstituteof Technology,LundUniversity,Lund,Sweden ix x Contributors José Seade Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional AutónomadeMéxico,Cuernavaca,México StéphaneSeuret LAMA,CNRSUMR8050,UniversitéParis-EstCréteil,Créteil Cedex,France S.Vaienti UMR-7332 Centre de Physique Théorique, CNRS, Universités d’Aix- Marseille, Université du Sud, Toulon-Var andFRUMAM, Fédéderation de Recherche des Unités de Mathématiques de Marseille CPT, Marseille Cedex 9, France A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps R.AiminoandS.Vaienti Abstract We present here the large deviation principle for some systems admit- ting a spectral gap, by using the functional approach of Hennion and Hervé, with slight modification. Our main application concerns multidimensional expanding mapsintroducedbySaussol. 1 Introduction Therearedifferentwaystoestablishlargedeviationprinciples(LDP)fordynamical systems.Oneofthemistheso-calledLaplaceMethod,whichreliesonthespectral propertiesofthePerron–Frobenius,ortransfer,operator.Thisstrategyhasbeende- velopedinaverygeneralandabstractsettingbyHennionandHervéin[17].They assume that the transfer operator acts on a Banach space of measurable functions, anditisquasicompact—namelyithasaspectralgap.Theexistenceofaninvariant probabilitymeasurefollowsimmediately,andbyusingperturbationtheoryforlinear operators,theyderiveafewothersstatisticalproperties.Thisapproachcoversalot ofsystems,forinstance,expandingmapsoftheinterval[17,XII.1],Gibbsmeasures forsubshiftsoffinitetype[17,XII.2]andexpandingYoungtowers[33].Neverthe- less, this theory seems inappropriate for expanding discontinuous maps in higher dimension, like those treated by several authors [1, 4, 5, 7, 9–11, 24, 28–31]. In- deed,HennionandHervéassumethatthetransferoperatoractsonBanachspacesof measurablefunctionsdefinedeverywhere,whileforthehigherdimensionalsystems quotedabove,thefunctionalspacescurrentlyused(boundedvariation,quasi-Hölder R.Aimino((cid:2)) UMR-6207CentredePhysiqueThéorique,CNRS,Universitésd’Aix-Marseille,Universitédu Sud,Toulon-VarandFRUMAM,FédéderationdeRecherchedesUnitésdeMathématiquesde MarseilleCPT,LuminyCase907,13288MarseilleCedex9,France e-mail:[email protected] S.Vaienti UMR-7332CentredePhysiqueThéorique,CNRS,Universitésd’Aix-Marseille,UniversitéduSud, Toulon-VarandFRUMAM,FédéderationdeRecherchedesUnitésdeMathématiquesdeMarseille CPT,LuminyCase907,13288MarseilleCedex9,France e-mail:[email protected] ©SpringerInternationalPublishingSwitzerland2015 1 H.Gonzalez-Aguilar,E.Ugalde(eds.),NonlinearDynamicsNewDirections, NonlinearSystemsandComplexity11,DOI10.1007/978-3-319-09867-8_1