The Hausdorff dimension of deformed self-similar sets

The Hausdor¤ dimension of deformed self-similar sets

Tae Hee Kim, Sung Pyo Hong and Hung Hwan Lee

(Received October 12, 1999) (Revised June 12, 2001)

Abstract. We define deformed self-similar sets which are generated by a sequence of similar contraction mappingsff_s:sASgonR^d,f_shaving its contraction ratiors, and calculate thier Hausdor¤ dimension.

1. Introduction

Hutchinson [4] proved that there exists a unique compact set FHR^d such that F¼6ⁿ

i¼1f_iðFÞ for any given finite set ff_ig_i¼1ⁿ of similarities in R^d with ratio ri, 1aian. He also showed that dimHF ¼dimBF ¼dimpF¼s andPn

i¼1r_i^s¼1 if ff_ig_i¼1ⁿ satisfies the open set condition, that is, there exists a bounded non-empty open setOsuch that6ⁿ

i¼1f_iðOÞHOandf_iðOÞVf_jðOÞ ¼j if i0j.

Recently, S. Ikeda [5] defined the loosely self-similar set F which is gen- erated by a sequence of mappingsff_i1i2...ikg ðijAf1;2;. . .;ngÞ,f_i1i2...ik having its contraction ratio r_ik, and showed that dim_HF ¼s andPn

i¼1r_i^s¼1 if ff_i1i2...ikg satisfies the disjoint condition.

In this paper, we will generalize loosely self-similar sets [5] and perturbed Cantor sets [1]. The construction is as follows.

Fix mb2, write Sk ¼ f1;2;. . .;mg^k and S¼6^y

k¼1Sk. Consider a se- quence of similar contraction mappingsff_s:sASgonR^d. Suppose that each f_shas a contraction ratiors, that is,jf_sðxÞ f_sðyÞj ¼rsjxyjfor anyx;yAR^d, where j j is the Euclidean norm. We further assume there exists 0<a;b<1 such thata<r_s<b for anysASand there exists a bounded open set VHR^d such that

(1) f_sðVÞHV for any sAS

(2) f_i1i2...ik1ikðVÞVf_{i1i2...ik1ik0}ðVÞ ¼j, ik0i⁰_k.

It is obvious that there exists a non-empty compact set XHV such that the properties (1) and (2) are satisfied when V is replaced by X.

2000 Mathematics Subject Classification. 28A78, 28A80.

Key words and phrases. Deformed self-similar set, Hausdor¤ measure, Hausdor¤ dimension.

The first author was supported by KOSEF and the second author was supported by Com2MaC- KOSEF. The third author was supported by TGRC and Korean Research Foundation 1998-015- D00019.

For brevity, we write

F_s1f_i1f_i1i2 f_i1i2...ik

Rs1ri1ri1i2. . .ri1i2...ik

for anys¼i₁i₂. . .i_k AS_k. Then using a compact setXgiven above we obtain a unique compact set K,

K¼ 7

y

k¼1

6

sASk

F_sðXÞ:

We call this K a deformed self-similar set. If we take r_s¼r_ik for any s¼

i₁i₂. . .i_k, the obtained set K becomes a loosely self-similar set in S. Ikeda’s

sense. Moreover, if we takef_i1i2...ik¼f_ik for all k, the obtained setKis a self- similar set in Hutchinson’s sense.

2. Preliminaries and main theorem

We begin to recall the well-known s-dimensional Hausdor¤ measure and dimension: Let E be a bounded subset of R^d and sb0. For every d>0 we define

H_d^sðEÞ ¼inf X^y

i¼1

jUij^s:EH 6

iUi;jUijad

( )

;

where jAj denotes the diameter of A. Then we obtain the s-dimensional Hausdor¤ measure of E by

H^sðEÞ ¼lim

d!0 H_d^sðEÞ:

The Hausdor¤ dimension of E is defined by

dim_H E¼supfsb0:H^sðEÞ ¼yg:

Then we see that

dim_HE¼inffsb0:H^sðEÞ ¼0g:

To calculate the Hausdor¤ dimension ofK we recall a measure on subsets of K due to Rogers [7]. For any sb0, nAN and EHK, let

M_n^sðEÞ ¼inf X

jFsðXÞj^s:EH 6

sFsðXÞ;sA 6

kbn

Sk

( )

:

Then we get a measure of Hausdor¤ type on subsets of K by letting M^sðEÞ ¼ lim

n!yM_n^sðEÞ:

We note that if M^tðEÞ>0 then M^sðEÞ ¼y for all s<t. Therefore we see that

supfsb0:M^sðEÞ ¼yg ¼inffsb0:M^sðEÞ ¼0g:

Now we are ready to show a method to get the Hausdor¤ dimension of a deformed self-similar set.

Theorem 2.1. Let K be a deformed self-similar set as defined in the intro- duction. Puta¼supfsb0jM^sðKÞ ¼yg. Then we have0<H^aðKÞ<yand

dim_H K¼a:

Proof. By the definition ofa;M^sðKÞ<y for anys>a. Thus, for such an s, H_d^sðKÞaM_n^sðKÞaM^sðKÞ if dbbⁿ, where b is an upper bound of con- traction ratios rs. Letting d!0, we have

H^sðKÞaM^sðKÞ<y fors>a;

which implies dim_HKaa.

To prove that dimHKba, we claim thatsadimH K, for alls<a. For 0<s<a,M^sðKÞ ¼yby the definition ofa. Then there exists a compact subset FHK and a constant b>0 such that 0<M^sðFÞ<y and M^sðFVFsðXÞÞa bR_s^s for all sAS. (See Proposition 3.1 [3].) Define a Borel measure m by

mðAÞ ¼M^sðFVAÞ forAHK:

Then 0<mðKÞ<y and mðFsðXÞÞabR^s_s for all sAS.

Let Vbe the open set given in the introduction and letUHR^d satisfy 0<

jUjajVj. Set

Q¼ fsAS:jFsðVÞj<jUjandjFsj ðjsj1ÞðVÞjbjUjg;

where sjl¼i₁i₂. . .i_l fors¼i₁i₂. . .i_k andlak. We see that ajUj<jFsðVÞj<

jUj for sAQ. Then Qo¼ fsAQ:UVFsðVÞ0 jg has at most some finite m_o elements which is independent of U sincefFsðVÞ jsAQ_og is disjoint. (See Lemma 9.2 [2].)

Hence

mðUÞa X

sAQo

mðFsðXÞÞ

ab X

sAQo

R_s^s

abmojVj^sjUj^s

which, by Mass distribution principle 4.2 [2], implies thatH^sðKÞ>0. That is, dimHKbs for any s<a. Therefore dimH Kba. r

Remark(cf. [3]). LetKbe a non-empty compact set generated by relaxing equality (3) to inclusion, that is,

KH 7

y

k¼1

6FsðKÞ:

Then it can be proved that

dim_HK¼supfsb0:M^sðKÞ ¼yg by the same proof as in Theorem 2.1.

Corollary2.2 (I. S. Baek [1]). Put X ¼ ½0;1. Define a sequence of con- traction maps ff_i1i2...ikg on X for i₁i₂. . .i_k Af1;2g^k, k¼1;2;. . .; such that

f_i1i2...ikðxÞ ¼ r^ðkÞ₁ x; if ik ¼1 r^ðkÞ₂ xþ ð1r^ðkÞ₂ Þ; if ik ¼2 8<

:

and there exist 0<a;b<1 such that

a<r^ðkÞ_i <b; and a<1X²

i¼1

r^ðkÞ_i <b for all k. Put

K¼ 7

y

k¼1

6

sASk

FsðXÞ:

Then K is a perturbed Cantor set and dimH K¼supfsb0:M^sðKÞ ¼yg

¼sup sb0:lim inf

n!y

X

i1i2...inAf1;2gⁿ

ðr^ð1Þ_i1 r^ð2Þ_i2 . . .r^ðnÞ_in Þ^s¼y 8<

:

9=

;: Proof. Let

b¼sup sb0:lim inf

n!y

X

i1i2...inAf1;2gⁿ

ðr^ð1Þ_i1 r^ð2Þ_i

2 . . .r^ðnÞ_in Þ^s¼y 8<

:

9=

;

¼sup sb0:lim inf

n!y

X

sAf1;2gⁿ

jFsðXÞj^s¼y 8<

:

9=

;:

To show that dimH Kab, suppose that 0<s<dimHK. Then lim

n!yM_n^sðKÞ ¼ y, that is, for givenL, there existsnosuch thatM_n^sðKÞbLfornbno. Hence

P

sAf1;2gⁿ

jFsðXÞj^s>L for nbno. Therefore lim inf

n!y

P

sAf1;2gⁿjFsðXÞj^s¼y, so sab. Now supposes<b. Then lim inf

n!y

P

sAf1;2gⁿ

jFsðXÞj^s¼y. Define a finite Borel measure m on X such that

mðFsðXÞÞ ¼ jFsðXÞj^s P

sAf1;2gⁿ

jFsðXÞj^s:

Then mðFsðXÞÞ ¼P²

i¼1

mðFsiðXÞÞ and this measure m is supported on K, thus, mðKÞ ¼1. Since s<b, it is easy to see that from the definition of b that

lim inf

n!y

jFi1i2...inðXÞj^s

mðFi1i2...inðXÞÞ¼lim inf

n!y

X

sAf1;2gⁿ

jFsðXÞj^s¼y

for all xAK, where i1i2. . .in. . . is the uniquely obtained sequence such that x¼ lim

n!yFi1i2...inðXÞ. Hence there exists c>0 such that H^sðKÞbc mðKÞinf

xAK lim inf

n!y

jFi1i2...inðXÞj^s mðFi1i2...inðXÞÞ¼y: (See Theorem 2.1 [8] and Lemma 2.2 [6].)

Therefore M^sðKÞbH^sðKÞ ¼y. So, M^sðKÞ ¼y, which implies ba

dimHK. r

Corollary 2.3 (S. Ikeda [5]). Let K be a deformed self-similar set such that ri1;...;ik ¼rik for alls¼i1. . .ikAS. Then K is a loosely self-similar set and

dim_HK¼supfsb0:M^sðKÞ ¼yg

¼d with X^m

i¼1

r_i^d ¼1:

Proof. The proof is similar to that of Corollary 2.2. r Open question: Let K be a deformed self-similar set. We wonder if the packing dimension [8] of K is equal to

sup sb0: lim

n!y M_n^sðKÞ ¼y

n o

where M_n^sðKÞ ¼sup PjFsðXÞj^s:FsðXÞVFs⁰ðXÞ ¼j;s0s⁰ands;s⁰A 6

kbn

Sk

( )

.

Acknowledgement. We wish to thank the referee for his helpful comments about the proof of Corollary 2.2 and Corollary 2.3.

References

[ 1 ] I. S. Baek, Dimensions of the perturbed Cantor set, Real Analysis Exchange 19 (1993/

1994), 269–273.

[ 2 ] K. J. Falconer, Fractal geometry-Mathematical foundations and applications, Wiley, 1990.

[ 3 ] K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc. 347 (1995), 3121–3129.

[ 4 ] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J.30(1981), 713–747.

[ 5 ] S. Ikeda, On loosely self-similar sets, Hiroshima Math. J.25 (1995), 527–540.

[ 6 ] S. Ikeda and M. Nakamura, Dimensions of Measures on perturbed Cantor set, preprint.

[ 7 ] C. A. Rogers, Hausdor¤ measures, Cambridge Univ. Press, 1970.

[ 8 ] S. J. Taylor and C. Tricot, The packing measure and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699.

Tae Hee Kim and Hung Hwan Lee Department of Mathematics Kyungpook National University

Taegu 702-701, Korea

E-mail: T. H. Kim: thkim64@knu.ac.kr E-mail: H. H. Lee: hhlee@knu.ac.kr

Sung Pyo Hong Department of Mathematics

Pohang University of Science and Technology Pohang 790-784, Korea

E-mail: sungpyo@vision.postech.ac.kr