MINICOURSEONHAUSDORFFDIMENSIONConnectionwithClassical, RelativizedandConditionalVariationalPrinciplesinErgodicTheorynunoluzia30may2007Lecture1: 1-dimensionaldynamics DenitionofHausdordimension. How to compute Hausdor dimension; Volume Lemma; Example: self-aneCantorsets;Moranformula. NonlinearCantorsets; Variational PrinciplefortheTopological Pressure;Bowensequation;Measureofmaximaldimension.References[Bo1] R.Bowen.EquilibriumstatesandtheergodictheoryofAnosovdieomorphisms(LectureNotesinMathematics,470).Springer,1975.[Bo2] R.Bowen.Hausdordimensionofquasi-circles.Publ.Math.I.H.E.S. 50(1979),11-26.[F] K.Falconer,Fractal geometry,Mathematical foundationsandapplications.Wiley,1990.[M] R.Ma.Ergodictheoryanddierentiabledynamics.Springer,1987.[PT] J.PalisandF.Takens.HyperbolicityandSensitiveChaoticDynamicsatHomoclinicBi-furcations.CambridgeUniversityPress,1993.[P] Ya. Pesin. DimensionTheoryinDynamical Systems: ContemporaryViewsandApplica-tions.ChicagoUniversityPress,1997.[R1] D.Ruelle.Repellersforrealanalyticmaps.Ergod.Th.andDynam.Sys.2(1982),99-107.Lecture2: 2-dimensionaldynamics Hausdordimensionof general Sierpinski carpetsandextensiontoskew-producttransformations; Connectionwiththerelativizedvariationalprinciple;GaugefunctionsandrelativizedGibbsmeasures.References[Be] T.Bedford.Crinklycurves,Markovpartitionsandboxdimensionofselfsimilarsets.PhDThesis,UniversityofWarwick,1984.[DG] M. Denker andM. Gordin. Gibbs measures for bredsystems. Adv. Math. 148(1999),161-192.[DGH] M.Denker,M.GordinandS.Heinemann.Ontherelativevariationalprincipleforbredexpandingmaps.Ergod.Th.&Dynam.Sys. 22(2002),757-782.[GL] D. GatzourasandP. Lalley. Hausdorandboxdimensionsof certainself-anefractals.IndianaUniv.Math.J. 41(1992),533-568.[GP1] D. Gatzouras and Y. Peres. Invariant measures of full dimension for some expanding maps.Ergod.Th.andDynam.Sys. 17(1997),147-167.[HL] I.HeuterandS.Lalley.FalconersformulafortheHausdordimensionofaself-anesetinR2.Ergod.Th.&Dynam.Sys. 15(1997),77-97.[L1] N. Luzia. Avariational principleforthedimensionforaclassofnon-conformal repellers.Ergod.Th.&Dynam.Sys. 26(2006),821-845.[L2] N. Luzia. Hausdor dimension for an open class of repellers inR2. Nonlinearity19 (2006),2895-2908.12[LW] F. LedrappierandP. Walters. Arelativisedvariational principleforcontinuoustransfor-mations.J.LondonMath.Soc. 16(1977),568-576.[Mu] C. McMullen. TheHausdordimensionof general Sierpiski carpets. NagoyaMath. J.96(1984),1-9.Lecture3: Measureoffulldimensionfornonlinear2-dimensionaldynamics Topologicalpressureandvariationalprincipleforsomenoncompactsets. Existenceofanergodicmeasureoffulldimensionforskew-producttrans-formations: reductionstotheprevioustypesofvariationalprinciples.References[Bo3] R. Bowen. Topological entropy for non-compact sets. Trans.Amer.Math.Soc.184 (1973),125-136.[GP2] D. GatzourasandY. Peres. Thevariational principleforHausdordimension: asurvey.Ergodictheoryof Zdactions(Warwick,19931994), 113125,London Math. Soc. LectureNoteSer.,228,CambridgeUniv.Press,Cambridge,1996.[L3] N. Luzia. Measure of full dimension for some nonconformal repellers. Preprint, 2006.(arXiv:0705.3604v1)[PP] Ya. Pesin and B. Pitskel. Topological pressure and the variational principle for noncompactsets.Funktsional.Anal.iPrilozhen. 18(1984),50-63.[R2] D. Ruelle. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications,5).Addison-Wesley,1978.3Lecture1WebeginbydeningtheHausdordimensionof asetFRn(seethebook[F]formoredetails). Thisisdonebydeningt-Hausdormeasures,wheret0,andthenchoosetheappropriatetformeasuringF. Whent = n,then-Hausdormeasureisjustthen-dimensional exteriorLebesguemeasure. Thediameterof asetU Rnisdenotedby[U[. WesaythatthecountablecollectionofsetsUiisa-coverofFifF

i=1Uiand[Ui[ foreachi. Givent 0and> 0letHt(F) = inf_

i=1[Ui[t: Uiisa-coverofF_.Thenthet-HausdormeasureofFisgivenbyHt(F) =limHt(F)(thislimitexistsbecauseHt(F)isincreasingin). So, givenFRn, whatistheappropriatet-HausdormeasureformeasuringF?Letuslookatthefunctiont Ht(F). Itisnotdiculttoseethatthereisacriticalvaluet0suchthatHt(F) =_ ift < t00 ift > t0.00 0t*t Ht(F)Figure1Denition1. TheHausdordimensionofFisdimHF= t0.Remark1. Inthedenitionof Hausdordimensionwecanusecoversbyballs.More precisely, let Bt(F) be the number obtained using covers by balls of diameters insteadofusingany-coverwhendeningHt(F). Ofcourse,Ht(F) Bt(F)becausewearerestrictingtoaparticularclassof-coversandusinginf overthisclass. Now,givenasetUwith[U[ thereexistsaballB Uwith[B[ 2[U[.ThisimpliesthatBt2(F) 2tHt(F). So,letting 0weobtainHt(F) Bt(F) 2tHt(F),andsothecriticalvaluet0atwhichHt(F)andBt(F)jumpisthesame.Usingthesamearguments,wecanalsousecoversbysquares. Whatwecannotdoistouseonlycoversformedbysetswhicharetoodistorted. ThisiswhythecomputationofHausdordimensionofinvariantsetsfornonconformaldynamicsrevealstobemorecomplicated.Problem: HowtocomputetheHausdordimensionofaset?4 If,forevery> 0,wendsome-coverUiofFsuchthat

i[Ui[ C< ,whereCisaconstantindependentof,thenHt(F) < whichimpliesdimHF t. HowtoobtainanestimationdimHF t?We must use every-cover... A priorithis seems to be an untreatable prob-lem. Is there some non-trivial mathematical object that takes acquaintancewith this?As we shall see now, the answer is yes, and the object is measure.MassDistributionPrincipleSupposethereexistsaBorel probabilitymeasureonRnsuchthat(F)=1,andthereexistaconstantC> 0andt > 0suchthatforeverysetU Rn,(U) C[U[t.Then,ifUiisany-coverofF,

i[Ui[t C1

i(Ui) C1(

iUi) C1(F) = C1.ThisimpliesthatHt(F) C1> 0andsodimHF t.Thereisanon-uniform versionofthisprincipleasweshalldescribenow. ThemainconceptbehindthisistheHausdordimensionof aprobabilitymeasuredenedbyL.-S.YoungasdimH = infdimHF: (F) = 1.Bydenition, if (F)=1thendimHFdimH. Thenon-uniform versionoftheMassDistributionPrincipledealswiththefollowinglimitd(x) =limr0log (B(x, r))log r,whereB(x, r)standsfortheopenball of radiusrcenteredatthepointx, whichiscalledthelowerpointwisedimensionof. Thenwehavethefollowing(seethebook[P]foraproof)Lemma1. (VolumeLemma)(1) If d(x) tfor-a.e. xthendimH t.(2) If d(x) tfor-a.e. xthendimH t.(3) If d(x) = tfor-a.e. xthendimH = t.Corollary1. If(F) = 1andd(x) tfor-a.e. xthendimHF t.Example1. (Self-aneCantorsets)Theseareself-anegeneralizationsofthefamousmiddle-third Cantorset. Theyareconstructedaslimitsetsofn-approximations: the1-approximationconsistsinmdisjointsubintervalsof[0, 1]; the2-approximationconsistsinsubstitutingeachinterval of the 1-approximation by a rescaled self-ane copy of the 1-approximation(from[0,1]tothisinterval),andsoon. InthelimitwegetaCantorsetK.5RR R 12 3R R R1 1 1 2 3 1f011Figure2Until now we have not talked about dynamical systems. These sets are dynam-icallydened(seeFigure2): LetRimi=1beacollectionofdisjointsubintervalsof[0, 1], whichwecallMarkovpartition. Considerthemapf :

mi=1Ri[0, 1]thatsendslinearlyeachintervalRionto[0, 1]. ThenKisthesetofpointsthatalwaysremain in the Markov partition when iterated by f(the n-approximation is the setof pointsthatremainintheMarkovpartitionwheniteratedn 1times). Moreprecisely,K=

n=0_i1,...,inRi1...in(1)wherei1, ..., in 1, ...mandRi1...in= Ri1 f1(Ri2) fn+1(Rin) (2)whichwecallbasicintervalsofordern.NowwearegoingtocomputetheHausdordimensionof Kinterms of thenumbersai= [Ri[, i = 1, .., m,thelengthsoftheelementsoftheMarkovpartition.Estimate. Let>0andnbesuchthat(max ai)n. Sothebasicsetsofordernforma-coverofK. Notethat[Ri1...in[ = ai1ai2...ain.SoHt(K) m

i1=1m

i2=1

m

in=1(ai1ai2...ain)t=_m

i=1ati_n. (3)Whatistheleasttsuchthattheexpressionontherighthandsideof(3)isniteforeveryn?Itistheuniquesolutiont = t0ofthefollowingequationm

i=1ati= 1 (4)whichiscalledMoranformula(notethat

mi=1atiisstrictlydecreasingint, hasvaluem>1fort=0andvalue

mi=1ai 0whichimpliesdimHK t0.6EstimatedimHK t0. LetbetheBernoullimeasureonKwithweigths(Ri) = at0i,i = 1, .., mi.e.(Ri1...in) =n

l=1at0il(notethatf[Kistopologicallyconjugatedtoafull shiftonmsymbols; alsore-memberthat

mi=1at0i= 1). Thenlog (Ri1...in)log [Ri1...in[= t0anditfollowsfromtheVolumeLemmathatdimH = t0andsodimHK dimH = t0.Conclusion.dimHK= t0wheret0isgivenbytheMoranformulam

i=1at0i= 1.Example2. IfKisthemiddle-thirdCantorsetthenitsHausdordimensionisthesolutionof2_13_t= 1i.e.dimHK=log 2log 3.Thisequationhasadynamicalmeaningoftheformdimension =entropyLyapunov exponent. (5)Ingeneral, theLyapunovexponent of a1-dimensional dynamics depends onin-variantmeasures. Forinstance,istheresomeanalogousformulafortheHausdordimension of self-ane Cantor sets?The answer is yes as we shall see now in a moregeneral context. Also, a formula of the type (5) only makes sense for 1-dimensionaldynamics, since for n-dimensional dynamics it is expected to exist several Lyapunovexponents.NonlinearCantorsetsNowweconsidersetswhicharegeneratedusingadynamicsasfortheself-ane Cantor sets, but now the generating dynamics need not be linear. Let Rimi=1beacollectionofdisjointsubintervalsof[0, 1],andf :

mi=1Ri [0, 1]beaC1+transformation (for some > 0) such that f > 1 and f[Ri is a homeomorphismonto[0, 1](seeFigure3).Asbeforeweconsiderthef-invariantsetdenedby(1)and(2). Itfollowsfromtheintermediatevaluetheoremthatthebasicintervalsofordernsatisfy[Ri1...in[ = ((fn)(x))1=_n1

j=0f(fjx)_1forsomex Ri1...in. Usingthefactthatf>1andfC1+weobtainthefollowingresult(seethebook[PT]foraproof)7RR R 12 3f0 11Figure3Lemma2. (BoundedDistortionProperty)ThereexistsC> 0suchthatforeveryn Nandeveryn-tuple(i1, ..., in),(fn)(x)(fn)(y) Cforeveryx, y Ri1...in.If: R,weusethefollowingnotation(Sn)(x) =n1

j=0(fj(x)).Thenwecanwrite[Ri1...in[t supxRi1...inet(Sn log f)(x),where means up to a constant (independent of n), due to the bounded distortionproperty. Asfortheself-anecase,forprovingtheestimateweusethecoverofbybasicintervalsofordern,sowewanttondtheleast tsuchthat

i1,...,in[Ri1...in[t

i1,...,insupxRi1...inet(Sn log f)(x)(6)isniteforalln. Duetotheuniformexpansionoff,(6)growsatanexponentialrateinn: negative, zeroorpositive, dependingonthevalueof t. Sowewanttondthevaluet = t0forwhichlimn1n log

i1,...,insupxRi1...inet0(Sn log f)(x)= 0 (7)(for then,the exponential rate at t = t0 +, > 0,is negative). This motivates thefollowingdenition.Denition2. Let:RbeaHldercontinuousfunction. TheTopologicalPressureof(withrespecttothedynamicsf[)isPf|() =limn1n log

i1,...,insupxRi1...ineSn(x).TheconditionofbeingHldercontinuousistoapplytheboundeddistortionproperty.8Remark2. TheTopological EntropyistheTopological Pressureof theconstantzerofunction:htop(f[) = Pf|(0).Since in our case we are dealing with a dynamics which is topologically congugatedtoafullshiftinmsymbols,wehavethathtop(f[) = log m.So, equation(7)canberestatedas: Thereisauniquesolutiont =t0of thefollowingequationPf|(t log f) = 0, (8)whichisthecelebratedBowensequation. SodimH t0.To prove the other inequality we shall need the well known Variational PrinciplefortheTopological Pressure. Beforestatingthisprinciplewemustintroducethenotionofentropyof aninvariantmeasure. Denoteby/(f[)thesetofall f[-invariantprobabilitymeasures,andby/e(f[)thesubsetofergodicones. Given /(f[),Shannon-McMillan-Breimanstheorem(see[M])saysthatthelimitlimn1n log (Ri1...in)exists for -a.e. x

n=1Ri1...in. Moreover, if is ergodicthenthis limit isconstant-a.e.Denition3. Let /e(f[). TheEntropyof(withrespecttothedynamicsf[)ish(f) =limn1n log (Ri1...in) (9)for-a.e. x

n=1Ri1...in. If/(f[)thenh(f)isjusttheintegral ofthelimitin(9)withrespectto.Note that, if /e(f[) then by Shannon-McMillan-Breimans theorem,Birkhosergodictheoremandtheboundeddistortionproperty,d(x) =limnlog (Ri1...in)log [Ri1...in[=limn1n log (Ri1...in)1n(Sn log f)(x)=h(f)_log fdfor-a.e. x

n=1Ri1...in. So,bytheVolumeLemma,if /e(f[)thendimH =h(f)_log fd. (10)ThenexttheoremisduetoSinai-Ruelle-Bowenandaproofcanbeseenintheexcelentexposition[Bo1].Theorem1. (VariationalPrinciplefortheTopologicalPressure)Let: RbeHldercontinuous. ThenPf|() = supM(f|)_h(f) +_d_.Moreover, thissupremumisattainedat auniqueinvariant measure, whichistheGibbsstateforthepotential (henceergodic)andisdenotedby.9When = 0weobtaintheVariational PrinciplefortheEntropy.Atthistimethereadermightguessthat, forcalculatingHausdordimension,we will use the potential = t0 log fwhere t0is the solution of Bowens equation(8). ItfollowsfromTheorem1and(10)that0 = Pf|(t0 log f) = ht0log f(f) t0_log fdt0 log fandt0=ht0log f(f)_log fdt0 log f= dimHt0 log f .ThusdimH dimHt0 log f = t0aswewish.Conclusion: TheHausdordimensionof isgivenbytheroott0of BowensequationPf|(t log f) = 0(foracompletediscussionsee[Bo2] and[R1]). Moreover, theGibbsstateforthepotentialt0 log fdenotedbyt0 log fis theuniqueergodicinvariant measureonof full dimension. Also, it is veryimportant to keep in mind, in what comes, the validity of the Variational PrinciplefortheDimension i.e.dimH = supMe(f|)dimH, (11)and /e(f[) dimH =h(f)_log fd. (12)1011Lecture2Herewedealwithsetswhicharegeneratedbysmoothplanetransformationsfwhicharenonconformal i.e. possesstwodierentratesofexpansion. Asbeforewealsoconsider sets whicharegeneratedusingaMarkovpartition, andsothebasicsets of order n(see(2)) formanatural cover of bysets withdiameterarbitrarilysmall. Theproblemisthat, duetononconformality, thesesetsarenotapproximately balls, andsocoversformedbybasicsetsarenotsucientlyforcalculatingHausdordimension(seeRemark1).Nevertheless,BowensequationPf|(t log |Df|) = 0still makes sense but, in general, its root does not give us the Hausdor dimension of: itisgreaterthan dimH anddepends onthedynamicsfweusetogenerate.TheaimofthislectureistoconvincethereaderthatthemaintoolforcomputingHausdor dimension in the nonconformal setting is not the Topological Pressure buttheRelativizedTopological Pressureand, especially, thecorrespondingRelativizedVariational Principle.Example3. (GeneralSierpinskiCarpets)LetT2= R2/Z2bethe2-torusandf0:T2T2begivenbyf0(x, y) = (lx, my)wherel >m>1areintegers. Thegridof lines[0, 1]i/m,i =0, ..., m 1,andj/l [0, 1],j= 0, ..., l 1,formasetofrectangleseachofwhichismappedbyf0ontothe entire torus (theserectangles are the domains of invertibilityoff0). Nowchoose some of these rectangles, the Markovpartition, andconsiderthefractal set 0consistingof thosepoints that always remaininthesechosenrectangleswheniteratingf0. Asbefore, 0isthelimit(intheHausdormetric)ofn-approximations: the1-approximationconsistsofthechosenrectangles,the2-approximationconsistsindividingeachrectangleofthe1-aproximationintol msubrectanglesandselectingthosewiththesamepatternasinthebegining,andsoon(seeFigure4).Figure4. l = 4,m = 3;1-approximationand2-approximationWesaythat (f0, 0) is ageneral Sierpinski carpet. TheHausdordimensionof general Sierpinski carpetswascomputedbyBedford[Be] andMcMullen[Mu],independently,obtainingthebeautifulformuladimH 0=log(

mi=1ni )log m12whereniisthenumberofelementsoftheMarkovpartitioninthehorizontalstripi,and =log mlog l.Seehowitgeneralizesthe1-dimensional formula(5)ittwoways: (i)put m=1;(ii)putni= 1wheneverni,= 0,i = 1, ..., m.Self-affinefractalsintheplaneNowweconsidersetswhichareself-anegeneralizationsof general Sierpinskicarpets (andgeneralize the corresponding1-dimensional versions, the self-aneCantorsets).LetS1, S2, ..., SrbecontractionsofR2. Thenthereisauniquenonemptycom-pactsetofR2suchthat =r_i=0Si().Thissetisconstructedlikethegeneral Sierpinski carpets(there, thecontractionsare the inverse branches of f0correspondingtothe chosenrectangles, andtheequationabovesimplymeansthatthesetisf0-invariant). Wewill refertoasthelimitsetofthesemigroupgeneratedbyS1, S2, ..., Sr.Weshall consider theclass of self-anesets that arethelimit sets of thesemigroupgeneratedbythemappingsAijgivenbyAij=_aij00 bi_x +_cijdi_, (i, j) 1.Here1= (i, j) : 1 i mand1 j niisaniteindexset. Weassume0 < aij< bi< 1, (13)foreachpair(i, j),

mi=1bi 1,and

nij=1aij 1foreachi. Also,0 d1< d2