CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS

Transcription

1 CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS HRANT A. HAKOBYAN Abstract. We show that middle iterval Cator sets of Hausdorff dimesio are miimal for quasisymmetric maps of a lie. Combiig this with a theorem of Wu we coclude that there are rigid subsets of a lie whose every quasisymmetric image has zero legth ad Hausdorff dimesio.. Itroductio Give M a homeomorphism f : R R is said to be M-quasisymmetric if for every pair of adjacet itervals I ad J of the same legth M fi) fj) M here ad i sequel stads for the -dimesioal Lebesgue measure). A map is quasisymmetric if it is M-quasisymmetric for some M. We deote by QS ad QSM) the set of all quasisymmetric ad M-quasisymmetric homeomorphisms of R respectively. More geerally a homeomorphism f betwee metric spaces X, d X ) ad Y, d Y ) is called η quasisymmetric if there is a self-homeomorphism η : [0, ) [0, ) such that for all x, y, z X ad t > 0 d X x, y) td X y, z) d Y fx), fy)) ηt)d Y fy), fz)). Give a compact set E R we are iterested i the distortio of the Hausdorff dimesio of E uder quasisymmetric maps. If dim H E) = 0 the dim H fe)) = 0 sice quasisymmetric maps are Hölder cotiuous see []). I [2] it is show that if dim H E) > 0 the for ay ε > 0 oe ca fid a quasisymmetric mappig such that dim H fe)) > ε. I the opposite directio Tukia [7] proved that for ay ε > 0 there is a set E R ad f QS such that dim H R\E) < ε ad dim H fe)) < ε. I this article we prove the followig theorem. Theorem.. Middle iterval Cator sets of dimesio are miimal for quasisymmetric maps. 99 Mathematics Subject Classificatio. Primary 30C65; Secodary 28A78. Key words ad phrases. Hausdorff dimesio, quasisymmetric maps.

2 2 HRANT A. HAKOBYAN A set E R is called miimal for quasisymmetric maps if dim H fe)) dim H E), f QS. We say E R is a middle iterval Cator set if it ca be costructed as follows. Fix a sequece c = {c i } of umbers i 0, ). From [0, ], which we will deote by E 0,, remove the middle iterval of legth c cetered at /2. Call the removed iterval J, ad the compoets of the remaiig set E, ad E,2. From the middle of E,i remove the iterval J 2,i of legth c 2 E,i, i =, 2. Cotiue i the same fashio. Let E = 2 =0 j= E,j where E,j = E +,2j J +,j E +,2j, J,j = c E,j ad E +,j = E +,j for ay j ad j. We will deote the set correspodig to a sequece c = {c i } by Ec). E 0, E, J, E,2 E 2, J 2, E 2,2 J, E 2,3 J 2,2 E 2,4 Figure. A middle iterval Cator set The followig defiitios are by ow stadard. The coformal dimesio of a metric space X is the ifimal Hausdorff dimesio of quasisymmetric images of X C dim X) = if{ dim H Y ) f : X Y quasisymmetric} whereas the quasicoformal or global coformal dimesio of a set E R is defied as the ifimum over a smaller collectio of maps QC dim E) = if{ dim H fe)) f : R R quasisymmetric}. Usig this termiology Tukia s theorem says that there are subsets of R of dimesio ad quasi)-coformal dimesio < while our defiitio of miimality becomes a particular case of the followig. E R is miimal for quasisymmetric maps) if QC dim E) = dim H E). I [3] it has bee show that for every α there are miimal Cator sets i R, 2, of Hausdorff dimesio α. All the previously kow examples of miimal subsets of R had positive Lesbegue measure, these are quasisymmetrically thick sets. Recall from [8] that E R is called a quasisymmetrically thick set if fe) > 0, f QS. I [4] it was show that a middle iterval Cator set Ec) with c i < /2 is quasisymmetrically thick if ad oly if cp i <, p > 0. Our theorem gives a partial egative aswer to the followig questio from [3] see page 370): If E is ot a quasisymmetrically thick set, the is there a quasisymmetric image of E of Hausdorff dimesio <? I other words: If a set is ot quasisymmetrically thick the does it ecessarily have coformal dimesio <? Theorem. shows that every middle iterval Cator set of zero legth ad dimesio oe is a example of a set which is ot quasisymmetrically thick ad has

3 CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS 3 quasicoformal dimesio. Is it true i this case that C dim E) =? More geerally: is there a set E R such that QC dime) = but C dime) <? Oe could also ask: Is there a rigid set whose every QS-image has dimesio ad legth 0? I [8] a set is called quasisymmetrically ull if all its QS-images have zero legth. I [9] Wu had show that if c = c i / l p, p the E = Ec) is ull. She oticed that i particular ull sets ca have dimesio. Therefore combiig theorem. with Wu s theorem we get a affirmative aswer to the questio above. Corollary.2. There are quasisymmetrically ull sets which are miimal. It s eough to take the sequece { /i) i c i = /i) i if i = 2 m if i 2 m ad costruct the correspodig Cator set Ec). We leave it to the reader to verify that the set has dimesio ad that Wu s coditio is also satisfied. Akoledgemets. I would like to thak my advisor Christopher Bishop for his advice ad ecouragemet as well as Saeed Zakeri for umerous commets o a earlier versio of this paper which made the expositio much more clear ad cocise. 2. Backgroud Material Recall that the Hausdorff t-measure of a metric space E is defied by { } H t E) = lim if diamu i ) t : E U i, diamu i < ε, ε 0 where {U i } is a ope cover of E. The Hausdorff dimesio of E is dim H E) = if{ t : H t E) = 0}. Oe usually gives a upper boud o the Hausdorff dimesio of a set by fidig explicit covers for it. Lower bouds ca be obtaied from the mass distributio priciple: If E R supports a positive Borel measure µ satisfyig µe I) C I d, for some fixed costat C > 0 ad every iterval I R the dim H E) d. Let NE, ε) be the miimal umber of ε balls eeded to cover E. Upper ad lower Mikowski dimesios of E are defied as dim M E) = lim sup ε 0 log NE, ε), dim log /ε M E) = lim if ε 0 log NE, ε) log /ε respectively. Whe this two umbers are the same the commo value is called the Mikowski dimesio of E. Geerally for a subset of a lie oe has dim H E) dim M E) dim M E) see [6]). Ad therefore whe dim H E) = the the Mikowski dimesio of E exists ad is equal.

4 4 HRANT A. HAKOBYAN Lemma 2.. If E = Ec) ad dim M E) = the 2.) c i ), 2.2) c p i 0, 0 < p <. Proof. From the defiitio of Mikowski dimesio we get dim M E) = lim log log 2 2 Q c i) = lim log 2 log c i) =. Therefore 2.) holds. Now, from the usual iequality betwee geometric ad arithmetic meas c i) c i) see [5]) we get that c i) or, equivaletly, c i 0. Combied with Jese s iequality cp i ) p c i for p < this gives 2.2). Our mai tool for provig theorem. will be the followig lemma from [9]. Here we reformulate it i a rescaled maer. Lemma 2.2. If f is a M-quasisymmetric fuctio the for ay two itervals J I ) J q + M) 2 fj) ) J p 2.3) I fi) 4 I where p = pm) = log 2 + /M), q = qm) = log 2 + M). 3. Proof of theorem. Proof. Fix d <, M ad take ay f QSM). We will costruct a measure µ o fe) satisfyig µi) C I d for some costat C > 0 ad therefore will coclude by the mass distributio priciple that dim H fe)) d. Sice d is arbitrary it will follow that dim H fe)) =. First ote that E has a tree structure where each paret iterval has exactly two childre ad the same is true for fe). Deote I,j = fe,j ) ad defie the measure µ iductively as follows: µi 0, ) = µi,j ) = I,j d I,j d + I,j d µi,k), where I,k is the paret of I,j ad I,j is the other iterval with the same paret as I,j. Now pick a iterval I,j ad compute µi,j )/ I,j d. There is a uique sequece of ested itervals I,j I,j... I 2,j2 I,j I 0 = [0, ] ad to simplify the otatio we deote I k,jk by I k. Also

5 CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS 5 lettig G = I \I I ) for =, 2,... from Wu s iequalities it follows that c q i + M 2 G i I i 4cp i, ad hece by iductio we get 3.) µi ) I d = I d + I d I d I d + I... I d d I d + I I 0 d = I + G + I ) d I d + I d I + G + I )d I d + I... d... I + G + I ) )d I i d + I i I d + I = d d I i + G i + I i )d = p i ). The secod equality above uses the fact that I k = I k + G k + I k, k. We estimate from below the product i the paretheses usig 2.3). The idea is to use the left ad right iequalities of 2.3) for small ad large c i -s respectively. First ote that 4x x) 5 for x < /0 ad let { )} p S = i N : c i < mi 0,, 3 S = S {i } s = cards ). From 2.) it follows that s / as. For i S the secod iequality i 2.3) gives us the followig estimate 3.2) p i = I i d + I i d I i + I i )d I i d + I i d I i + I i )d 4cp i )d = I i + I i )d I i + I i )d I i + G i + I i )d = I i d + I i d + ) I i d I i + I i I i ) d 4c p i )d G i I i Now sice c i < /3, I i ad I i are images of two itervals E i ad E i of the same legth which are at most E i away from each other. Therefore, twice applyig the defiitio of quasisymmetry we immediately get that M + M ) I i / I i = fe i ) / fe i) MM + ). Cosiderig the fuctio x +xd for d < oe ca easily see that o a iterval [M + +x) d )/M 2, MM + )] its smallest value is attaied at MM + ) ad is equal to +MM+))d +MM+)) d >. We will deote this value by C M, d). The importat ) d

6 6 HRANT A. HAKOBYAN thig is that it is larger tha. Therefore 3.2) fially gives us the followig estimate 3.3) p i C M, d) c p i )5d for i S. For i / S we use the first iequality of 2.3) to get I i d + I i p i = d I i + G i + I i = I i d + I ) i d Ii d I )d I i d = + i I i I i ) 2 Ei dq ) 2 ci ) dq 3.4) + M) 2d = E i + M) 2d 2 = C 2 M, d) c i) dq Combiig 3.),3.2) ad 3.4) we get p i C c p i )5d C2 c i) dq i S {i }\S 3.5) Cs C s c p i )5d c i ) dq 2 i S where C, C 2 >. Now 3.5) together with s / imply that there is a costat CM, d) > such that 3.6) p i C c i ) dq. i S c p i )5d C The theorem will be proved if we show that the right had side above teds to ifiity as. Ideed, from 3.6) we will get that p as ad therefore 3.) would imply that there is a costat C such that µi ) C I d for every iterval I. For the secod term i 3.6) oe has ) log C c i ) dq log C + dq log c i ). By 2.) log c i) 0 ad therefore C c i) dq. O the other had the first term i 3.6) ca be bouded from below i the followig maer: C 3.7) c p i )5d C c p i ) 5d i S ) d

7 CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS 7 where c i = mic i, p 0.) p 0. <. To show that the right had side i 3.7) teds to ifiity it s eough to show that log c p i ) = log c p i ) 0. To see the later first ote that log x) > 2x for 0 < x 0.. Sice c p i 0. we have ) 0 > log c p i ) > 2 c p i > 2 c p i + s 0 0 by 2.) ad by the fact that s /. So we get that C i S c p i )a, ad hece have proved that the growth coditio of the mass distributio priciple holds for all itervals of the form fe i,j ). To complete the proof we eed to show that it holds for ay iterval I R. So fix a iterval I. Let l i = E i,j, i, j. Clearly l i 0 so there is a i such that l i+ f I) < l i. It follows the that there are at most 2 itervals of geeratio i which itersect f I) ad therefore at most 4 such itervals of geeratio i +. Deotig the latter oes by E,..., E 4 some of these may be empty) we get µi) µfe )) µfe 4 )) C fe ) d fe 4 ) d ). Now sice E i f I) it follows that E... E 4 3f I) 3f I) is just the dilatio of f I)) ad hece fe ) d fe 4 ) d 4 f3f I)) d. From the defiitio of quasisymmetry it follows that f3f I)) d + 2M) d ff I)) d = CM, d) I d ad hece combiig the last three iequalities we coclude the desired growth µi) C I d for some costat C ad ay iterval I. As we oted i the begiig if follows that dim H fe)) = sice d could be chose as close to as oe would like. Refereces. L.V. Ahlfors, Lectures o Quasicofomal Mappigs, Va Nostrad, C.J.Bishop, Quasicoformal mappigs which icrease dimesio, A. Acad. Sci. Fe ), C.J.Bishop, J.Tyso, Locally miimal sets for coformal dimesio, A. Acad. Sci. Fe ),

8 8 HRANT A. HAKOBYAN 4. S.Buckley, B. Haso, P. MacMaus, Doublig for geeral sets, Math. Scad ), G. Pólya, G. Szegö, Problems ad Theorems i Aalysis I, Spriger, P. Mattila, Geometry of sets ad measures i Euclidea spaces, Cambridge Uiversity Press, P. Tukia, Hausdorff dimesio ad quasisymmetric mappigs, Math. Scad ), S. Staples, L. Ward, Quasisymmetrically thick sets, A. Acad. Sci. Fe. Math ), Jag-Mei Wu, Null sets for doublig ad dyadic doublig measures, A. Acad. Sci. Fe. Math ), Departmet of Mathematics, Stoy Brook Uiversity, Stoy Brook, NY address: hhakob@math.suysb.edu