Non-Invertible Dynamical Systems 1: Ergodic Theory Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps

458 Pages·2021·4.818 MB·English

by  Mariusz Urbaski, Mario Roy, Sara Munday

MariuszUrbański,MarioRoy,SaraMunday
Non-InvertibleDynamicalSystems
De Gruyter Expositions in
Mathematics
|
Editedby
LevBirbrair,Fortaleza,Brazil
VictorP.Maslov,Moscow,Russia
WalterD.Neumann,NewYorkCity,NewYork,USA
MarkusJ.Pflaum,Boulder,Colorado,USA
DierkSchleicher,Bremen,Germany
KatrinWendland,Freiburg,Germany
Volume 69/1
Mariusz Urbański, Mario Roy, Sara Munday
Non-Invertible
Dynamical
Systems
|
Volume 1: Ergodic Theory – Finite and Infinite,
Thermodynamic Formalism, Symbolic Dynamics and
Distance Expanding Maps
MathematicsSubjectClassification2010
37A05,37A25,37A30,37A35,37A40,37B10,37B25,37B40,37B65,37C05,37C20,37C40,37D20,
37D35,37E05,37E10
Authors
Prof.Dr.MariuszUrbański Dr.SaraMunday
UniversityofNorthTexas JohnCabotUniversity
DepartmentofMathematics ViadellaLungara233
1155UnionCircle#311430 00165Rome
Denton,TX76203-5017 Italy
USA [email protected]
[email protected]
Prof.Dr.MarioRoy
YorkUniversity
GlendonCollege
2275BayviewAvenue
Toronto,M4N3M6
Canada
[email protected]
ISBN978-3-11-070264-4
e-ISBN(PDF)978-3-11-070268-2
e-ISBN(EPUB)978-3-11-070275-0
ISSN0938-6572
LibraryofCongressControlNumber:2021940887
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2022WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
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MariuszUrbańskidedicatesthisbooktohiswife,Irena.
ÀmesparentsThérèseetJean-Guy,àmafamilleetàmesamis,sansquicelivren’aurait
puvoirlavie...dufondducoeur,merci!Mario

Preface
Dynamicalsystemsandergodictheoryisarapidlyevolvingfieldofmathematicswith
alargevarietyofsubfields,whichuseadvancedmethodsfromvirtuallyallareasof
mathematics.Thesesubfieldscomprisebutarebynomeanslimitedto:abstracter-
godic theory, topological dynamical systems, symbolic dynamical systems, smooth
dynamical systems, holomorphic/complex dynamical systems, conformal dynam-
ical systems, one-dimensional dynamical systems, hyperbolic dynamical systems,
expandingdynamicalsystems,thermodynamicformalism,geodesicflows,Hamilto-
niansystems,KAMtheory,billiards,algebraicdynamicalsystems,iteratedfunction
systems,groupactions,andrandomdynamicalsystems.
Allofthesebranchesofdynamicalsystemsaremutuallyintertwinedinmanyin-
volvedways.Eachofthesebranchesnonethelessalsohasitsownuniquemethods
andtechniques,inparticularembracingmethodswhicharisefromthefieldsofmath-
ematicsthebranchiscloselyrelatedto.Forexample,complexdynamicsborrowsad-
vancedmethodsfromcomplexanalysis,bothofoneandseveralvariables;geodesic
flowsutilizemethodsfromdifferentialgeometry;andabstractergodictheoryandther-
modynamicformalismrelyheavilyonmeasuretheoryandfunctionalanalysis.
Indeed,itistrulyfascinatinghowlargethefieldofdynamicalsystemsisandhow
manybranchesofmathematicsitoverlapswith.Inthisbook,wefocusonsomese-
lectedsubfieldsofdynamicalsystems,primarilynoninvertibleones.
Inthefirstvolume,wegiveintroductoryaccountsoftopologicaldynamicalsys-
temsactingoncompactmetrizablespaces,offinite-statesymbolicdynamicalsystems,
andofabstractergodictheoryofmeasure-theoreticdynamicalsystemsactingonprob-
abilitymeasurespaces,thelatterincludingthemetricentropytheoryofKolmogorov
andSinai.Moreadvancedtopicsincludeinfiniteergodictheory,generalthermody-
namicformalism,andtopologicalentropyandpressure.Thisvolumealsoincludesa
treatmentofseveralclassesofdynamicalsystems,whichareinterestingontheirown
andwillbestudiedatgreaterlengthinthesecondvolume:weprovideafairlydetailed
accountofdistanceexpandingmapsanddiscussShubexpandingendomorphisms,
expansivemaps,andhomeomorphismsanddiffeomorphismsofthecircle.
Thesecondvolumeissomewhatmoreadvancedandspecialized.Itopenswith
asystematicaccountofthermodynamicformalismofHöldercontinuouspotentials
foropentransitivedistanceexpandingsystems.Onechaptercomprisesnodynamics
butratherisaconciseaccountoffractalgeometry,treatedfromthepointofviewof
dynamicalsystems.Bothoftheseaccountsarelaterusedtostudyconformalexpand-
ing repellers. Another topic exposed at length is that of thermodynamic formalism
ofcountable-statesubshiftsoffinitetype.Relyingonthislatter,thetheoryofconfor-
malgraphdirectedMarkovsystems,withtheirspecialsubclassofconformaliterated
functionsystems,isdescribed.Here,inasimilarwaytothetreatmentofconformalex-
pandingrepellers,themainfocusisonBowen’sformulafortheHausdorffdimension
https://doi.org/10.1515/9783110702682-201
VIII | Preface
ofthelimitsetandmultifractalanalysis.ArathershortexaminationofLasota–Yorke
mapsofanintervalisalsoincludedinthissecondvolume.
Thethirdvolumeisentirelydevotedtothestudyofthedynamics,ergodictheory,
thermodynamicformalism,andfractalgeometryofrationalfunctionsoftheRiemann
sphere.Wepresentafairlycompleteaccountofclassicalaswellasmoreadvanced
topologicaltheoryofFatouandJuliasets.Nevertheless,primaryemphasisisplaced
onmeasurabledynamicsgeneratedbyrationalfunctionsandfractalgeometryoftheir
Juliasets.TheseincludethethermodynamicformalismofHöldercontinuouspoten-
tialswithpressuregaps,thetheoryofSullivan’sconformalmeasures,invariantmea-
suresandtheirdimensions,entropy,andLyapunovexponents.Wefurtherexaminein
detailtheclassesofexpanding,subexpanding,andparabolicrationalfunctions.We
alsoprovide,withproofs,severalofthefundamentaltoolsfromcomplexanalysisthat
areusedincomplexdynamics.ThesecompriseMontel’sTheorem,Koebe’sDistortion
TheoremsandRiemann–Hurwitzformulas,withtheirramifications.
Invirtuallyeachchapterofthisbook,wedescribealargenumberofconcretese-
lected examples illustrating the theory and serving as examples in other chapters.
Also,eachchapterofthebookissuppliedwithanumberofexercises.Thesevaryin
difficulty,fromveryeasyonesaskingtoverifyfairlystraightforwardlogicalstepsto
moreadvancedonesenhancinglargelythetheorydevelopedinthechapter.
ThisbookoriginatedfromthegraduatelecturesMariuszUrbańskideliveredatthe
UniversityofNorthTexasintheyears2005–2010andthatSaraMundaytooknotesof.
WiththeinvolvementofMarioRoy,thebookevolvedandgrewovermanyyears.The
last2years(2020and2021)ofitswritingweremostdramaticandchallengingbecause
oftheCOVID-19pandemic.Ourbookborrowswidelyfrommanysourcesincludingthe
books[41,47,57].Weneverthelesstriedtokeepitasself-containedaspossible,avoid-
ingtoreferthereadertoooftentospecificresultsfromspecialpapersorbooks.Toward
thisend,anappendixcomprisingclassicalresults,mostlyfrommeasuretheory,func-
tionalanalysisandcomplexanalysis,isincluded.Thebookcoversquiteamanytopics
treatedwithvariousdegreesofcompleteness,noneofwhicharefullyexhaustedbe-
causeoftheirsheerlargenessandtheircontinuousdynamicalgrowth.
List of Figures
:[ , ]→[ , ]
Figure1.1 ThetentmapT 0 1 0 1.|2
:[ , ]→[ , ] =
Figure1.2 ThemapTm 0 1 0 1,wherem 5.|3
:[ , ]→[ , ]
Figure1.3 TheFareymapF 0 1 0 1.|5
:[ , ]→[ , ]
Figure1.4 Minkowski’squestion-markfunctionQ 0 1 0 1.|6
Figure1.5 ThemapTisanorientation-preservinghomeomorphismoftheunitintervalthatfixes
onlytheendpoints.Theverticaldottedlinesindicatetheintervals(Tn(△))n∈ℤ,where
△:=[ , ( )]
a T a .|29
Figure2.1 Ontheleft,Subcase2.1:(fj−n(x0),fj(x0))⊆(fk−n(x0),fk(x0)).Ontheright,
Subcase2.2:fk−n(x0)∈(fj−n(x0),fj(x0)).|57
,...,
Figure4.1 Theactionofthefourcontractingsimilaritiesφ0 φ3ontheclosedunitsquare
I2.|108
Figure4.2 IllustrationoftheballB(T(x),ξ)mappedundertheinversebranchT−1insidetheball
x
( , )
Bx δ .|110
Figure4.3 ThemapTT−n1−1(x)sendstheballB(Tn(x),ξ)intotheballB(Tn−1(x),ξ),whichisinturn
mapped(in,to)theballB(Tn−2(x),ξ)byTT−n1−2(x)andsoon,untilfinallyTx−1sendsusback
insideBx ξ .|111
Figure8.1 Anexampleofanintervalexchangetransformation.|255
( )=−
Figure9.1 Thefunctionunderlyingentropy:k t tlogt.|281