Fractal dimensions for the fibres of certain type of carpets

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Fractal dimensions for the ﬁbres of certain type of carpets

Tae Hee Kim and Hung Hwan Lee

(Received May 27, 2002) (Revised March 13, 2003)

Abstract. We deﬁne the familyFof certain type of carpets, and calculate the fractal dimensions of ﬁbres Fx for all FAF and for almost all x.

1. Introduction and preliminaries

Let Fbe the family of all carpets obtained by partitioning the unit square into four subsquares, discarding one of them and repeating this on each of the remaining squares, with no constraints on the positions of the discarded subsquares. For FAF and xA½0;1, set vertical ﬁbres Fx¼ fyA½0;1 j ðx;yÞAFg.

H. Furstenberg conjectured that almost all vertical ﬁbres of all FAFhave positive Hausdor¤ dimension [5].

In [2], I. Benjamini and Y. Peres showed for allF AF, the lower bound of Hausdor¤ dimension ofFx is¹₂ for almost allxA½0;1with respect to Lebesgue measure.

In this note, we consider the family Fða;bÞ of certain type of carpets which are constructed as follows:

Partition the unit square into four rectangles, with the ratio of side length, a:b, where a;bAR^þ (the set of positive real numbers), aþb¼1, aab and discard one of them. Again partition each of the three remaining rectangles into four subrectangles, with the same ratio of side length, a:b, and discard one of them. Apply the same operation to each of the remaining rectangles, with no constraints on the positions of the discarded subrectangles, and repeat this operations to obtain a limit set, a type of carpet F in Fða;bÞ.

For above FAFða;bÞ, we ﬁnd a lower bound and an upper bound of fractal dimensions for the vertical ﬁbres Fx, which gives us the result of I.

Benjamini and Y. Peres [2] as corollary.

We now review some deﬁnitions and the known-results.

2000 Mathematics Subject Classiﬁcation. 28A78, 28A80.

Key words and phrases. Hausdor¤ measure, Hausdor¤ dimension, upper box dimension, ﬁbre of carpet.

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Definition 1.1. Suppose that F is a bounded subset of Rⁿ and s is a nonnegative real number. The Hausdor¤ measure of F is deﬁned as

H^sðFÞ ¼lim inf

d!0

X^y

i¼1

jUij^s:FH 6^y

i¼1U_i;jUij<d

( )

where jUj denotes the diameter of a set U.

The Hausdor¤ dimension of the set F is deﬁned as

dim_H F¼inffs:H^sðFÞ ¼0g ¼supfs:H^sðFÞ ¼yg:

Definition 1.2. The upper box dimension of bounded FHRⁿ are deﬁned as

dim_BF ¼lim sup

d!0

logN_dðFÞ logd ;

where N_dðFÞ denotes the number of d-mesh cubes that intersect F. Here we note that

dimH FadimBF: 2. Main results

We are now in a position to prove our main results.

Theorem 2.1. For all F AFða;bÞ, with a;bAR^þ, aþb¼1 and aab dimHFxb2abfbðlogblogaÞ þlogag

2ab²ðlogblogaÞ þloga for almost all xA½0;1 with respect to Lebesgue measure.

Proof. For all xA½0;1Þ, we can deﬁne xx^ as follows; Bisect the interval

½0;1Þ with ratio a:b and we write

x₁ ¼ 0; if xlies in½0;aÞ; 1; otherwise

and repeat to bisect ½0;aÞ and ½a;1Þ with the same ratio a:b, and we write x2¼ 0; if xlies in the resulting interval ½0;a²Þor½a;aþabÞ;

1; otherwise.

Repeating this method, we have a sequence xx^¼ ðx1;x2;. . .Þ for xA½0;1Þ.

Let F be the family of functions

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j:6^y

n¼0ðf0;1gⁿ f0;1gⁿÞ ! f0;1g² where f0;1g⁰ f0;1g⁰¼ fqg.

Then for each jAF, we construct FðjÞHf0;1g^y f0;1g^y by FðjÞ ¼ fðxx;^ ^yyÞ ¼ ððx1;x₂;. . .Þ;ðy1;y₂;. . .ÞÞ jEnb1;x_n;y_nAf0;1g

and jððx1;x2;. . .;xn1Þ;ðy1;y2;. . .;yn1ÞÞ0ðxn;ynÞg and deﬁne a map f on FðjÞ by

fð^xx;yyÞ ¼^ 7^y

n¼0ðCnð^xxÞ CnðyyÞÞ^

where Cnð^xxÞ and CnðyyÞ^ are the n-th step intervals containing xx^ and yy,^ respectively. If we write FðjÞ ¼ fðFðjÞÞ, then we easily see that Fða;bÞ ¼ fFðjÞ jjAFg.

Now consider Foða;bÞ ¼ fFðjÞ:jAF and jðx1;. . .;xn;y1;. . .;ynÞ0 ð;1Þ;Eng, that is, Foða;bÞ is the subfamily of Fða;bÞ whose elements are constructed by discarding one of the two-top subrectangles for all steps. Note that for any F AFða;bÞ, there exists FF~AFoða;bÞsuch that dim_HFF~_xadim_HF_x for all xA½0;1 (c.f. [3], [5]). Hence we will compute a lower bound of dim_HFF~_x instead of a the lower bound of dim_HF_x.

Let F ¼FðjÞAFo and x;zA½0;1Þ, and write xx^¼ ðx1;x2;. . .;Þ, ^zz¼ ðz1;z2;. . .;Þ corresponding to x and z, respectively. We deﬁne

P~

P_x:f0;1g^y! f0;1g^y by PP~_xð^zzÞ ¼ yy;^

where yy^¼ ðy1;y2;. . .;yn;. . .Þ are deﬁned inductively; s1 is deﬁned by jðqÞ ¼ ðs1;1Þ and then

y₁¼ 0; s1¼x1; z1; s10x1

If y1;y2;. . .;yn are deﬁned, thensn ¼snðx1;x2;. . .;xn1;z1;z2;. . .;zn1Þis deﬁned by

jðx1;. . .;xn1;y₁;. . .;yn1Þ ¼ ðsn;1Þ and

yn¼ 0; sn¼xn; zn; sn0xn:

Therefore for given x;zA½0;1Þ, we can deﬁne PxðzÞ ¼ y; where y¼ 7

y

n¼1

Cnð^yyÞ

and

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Fx¼ 6

zA½0;1Þ

PxðzÞ

where CnðyyÞ^ is the n-th step interval containing yy^ with length jCnðyyÞj ¼^ a^ðnT^k¼1ⁿ ^y^k^Þb^T^k¼1ⁿ ^y^k.

And if m denotes Lebesgue measure on ½0;1, then mP¹_x is a measure on the ﬁbre Fx.

Therefore choose ðx;zÞA½0;1Þ² randomly according to Lebesgue measure mm and then we have

mP¹_x ðCnðyyÞÞ ¼^ b^T^k¼1ⁿ ^y^ka^ðT^k¼1ⁿ ^ðs^k^l^x^k^ÞT^k¼1ⁿ ^y^k^Þ where l denotes the sum mod 2.

Since sk is a function of ðx1;. . .;xk1;z1;. . .;zk1Þ and yk is a function of ðx1;. . .;x_k;z₁;. . .;z_k;s₁;. . .;s_kÞ,fsklx_kg^y_k¼1 andfy_kg^y_k¼1 are independent, thus for a. a. ðx;zÞA½0;1²,

1 n

Xⁿ

k¼1

ðsklxkÞ !2ab and

1 n

Xⁿ

k¼1

y_k !2ab²; as n!y:

Therefore for a. a. ðx;zÞA½0;1², logmP¹_x ðCnðyyÞÞ^

logjCnðyyÞj^ ¼ðPn

k¼1y_kÞlogbþ ðPn

k¼1ðsklx_kÞ Pn

k¼1y_kÞloga ðn ðPn

k¼1ykÞÞlogaþ ðPn

k¼1ykÞlogb

!2abfbðlogblogaÞ þlogag

2ab²ðlogblogaÞ þloga 1y; as n!y: If we write

LðxÞ ¼ yAFxj lim

n!y

logmP¹_x ðCnð^yyÞÞ logjCnðyyÞj^ ¼y

;

then we obtain that for a. a. x with respect tom,mP¹_x ðLðxÞÞ ¼1 by Fubini’s theorem. Hence

dim_HF_xby

for a. a. x with respect to m (see [3]). r

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Corollary2.2 [1]. LetF be as in our introduction. Then for all F AF, dim_HF_xb¹₂, for a. a. x with respect to m.

Proof. Put a¼b¼1 and M¼2 in Theorem 2.1. r On the other hand, an upper bound of Hausdor¤ dimension for ﬁbres F_x of FAFða;bÞ goes as follows.

Theorem 2.3. For all F AFða;bÞ with a;bAQ^þ, a¼_M^c, b¼_M^d dimBFxalogðM²d²Þ

logM 1

for a. a. xA½0;1 with respect to Lebesgue measure m.

Proof. We easily see that F is covered at most ðM²d²Þⁿ squares of side length Mⁿ, for every nb1.

Put

b¼logðM²d²Þ logM : Suppose that there exists F AF and e>0 such that

mfxjdim_BðFxÞbb1þ2eg1d>0:

Then by Egorov’s theorem there exists noAN such that m xjsup

kbn

logN_1=MkðFxÞ

nlogM bb1þe

b 1

Md; Enbno:

Therefore we have at least _M¹ dMⁿ intervals of length Mⁿ on the x- axis above which F intersects more than M^nð^b1þeÞ squares for inﬁnitely many nbn_o.

Then it leads to contradiction since 1

MdM^nðbþeÞ>ðM²d²Þⁿ; for some n>logMlogd

elogM : r Remark. We don’t know yet whether Theorem 2.3 is true or not for the case of irrational number.

Corollary2.4 [1]. Let Fbe as in the introduction. Then for all F AF, dim_BF_xa^{log 3}_{log 2}1, for a. a. x with respect to m.

Proof. Put a¼b¼1 and M¼2, in Theorem 2.3. r

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References

[ 1 ] R. B. Ash, Real Analysis and probability, United Publishing and Promotion Co., LTD, 1972.

[ 2 ] I. Benjamini and Y. Peres, On the Hausdor¤ dimension of ﬁbres, Israel J. Math., 74 (1991), 267–279.

[ 3 ] C. D. Culter, A note on equivalent interval covering systems for Hausdor¤ dimension onR, Internat. J. Math. and Math Soc., 11 (1990), 643–650.

[ 4 ] K. Falconer, Fractal Geometry, John Wiley and Sons, 1990.

[ 5 ] H. Furstenberg, Intersections of Cantor sets and transversality of semigroups in problems in Analysis, A Symposium in Honor of S. Bocher (R. C. Gunning. Ed.), Princeton University Press, 1970, 41–59.

Tae Hee Kim Department of Mathematics Kyungpook National University

Taegu 702-701, Korea E-mail: [email protected]

Hung Hwan Lee Department of Mathematics Kyungpook National University

Taegu 702-701, Korea E-mail: [email protected]