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