Asymptotic behavior of lebesgue measures of cantor sets arising in the dynamics of tangent family $t_alpha(theta)=alpha tan(theta/2)$

ASYMPTOTIC BEHAVIOR OF LEBESGUE MEASURES OF CANTOR SETS ARISING IN THE DYNAMICS OF TANGENT FAMILY T_a(9) = o:tan(9/2)

HONG OH KIM, JUN Kyo KIM AND JONG WAN KIM

1. Introduction

Let 0 < 0: < 2 and let Ta(⁹⁾ = 0: tan( 9/2). Ta has an attractive fixed point at 9=o. We denote by C( 0:) the set of points in I = [-7r, 7r]

which are not attracted to 9= 0 by the succesive iterations ofTa. That is, C( 0:) is the set of points in I where the dynamics ofTa is chaotic.

It is also related to the Julia set of the family of point-mass singular inner functions exp(0:_~+~) and is shown to be a Cantor set, a closed, totally disconnected, perfect subset ofI in [KKl,KK2]. Since Ta has a slope 0:/2 at 9 = 0 which is less than one, it has two more fixed points in I other than the attractive fixed point at 9 = o. They are symmetrically located at -90(0:) and 90(0:). See Figure 1. We denote the interval ( -90 (0: ), 9(0:»by Eo (0: ). From the graphical analysis, the points on the interval Eo(0:) are shown to be attracted to the attrac- tive fixed point at 9= 0 under Ta. Therefore, C(o:) C I\Eo(o:). Since the Lebesgue measure IEo(o:)1 of Eo(o:) tends to 27r as 0: -+ 0, the Lebesgue measure IC(o:)1 of C(o:) tends to 0 as 0: -+

o.

more precisely on the asymptotic behavior of IC(o:)1 as 0: ^-+ 0 in the following theorem.

THEOREM.

(a) IC(o:)1 =0(0:⁵) as 0: -+ 0, analytically.

(b) IC(o:)1 =0(0:⁷⁾ as 0:-+ 0, by use of MATHEMATICA.

Itis a reasonable conjecture that IC(0:)1is of infinite order as0: ^-+

o.

Received September 17, 1994.

1991 AMS Subject Classification: 26A18.

Key words: Dynamics, Julia set, Cantor set.

The author was partly supported by TGRC(KOSEF).

Figure 1 2. Analytical proof

Let E1(a)be the set of points 8 inI\Eo(a) such thatTa(8) E Eo(a) (mod 271"). Then E1(a) =U{E?d: k₁ =±1,±2,···}, where

E~kl) = {B El: Ta(B) E Eo(a)+2kl71"}.

Let Ez(a) be the set of points 8 inI\(Eo(a)UE1(a» such thatT~(8) E Eo(a) (mod 271"). ThenEz(a) is given as

where

E~kl,k2) = {B El: T_a(8) E E~kl)(a)+2k_z71"}.

Inductively, we let E_{n (}a) be the set of points () in1\_Uj~;Ej(a) such that T:;(B) E Eo(a) (mod 271"). Then

En(a)

=

=

kz ..., , n-,k - 0 ±1 ±2 ... }, , ,

where

E~kl,k2,···,kn)= {8 El: T_o(8) EE~~c,kn-d +2k_n

1r}.

Then clearly C(a) is given by

00

C(a) = 1\

U

n=O

For an n-tuple (k), kz,'" ^{,kn )} of nonzero integer k₁ and intege rs kz, ... ,k_{n ,} we set

(0) ( (0) ( )

81 = - 80 a) ,8₂ = 80 a ,

( 0») (0»)

8~1) = 2tan 2k¹

1r:

1r:

and, inductively,

O\n)

ekn?r: 0~n-1») ,O\n)

(2kn?r:0\n-1}

~ 1 2 T h E(k1 ," ,kn ) (ll(n) ll(n») Of ll(n) , J.or n = , , . . . . en n = 171 ,l7z. course,17i s depend on the choice of k}, kz, ... ^,kn but we suppressed its dependence for a notational simplicity.

Now, we estimate IEn(a)1 up to order 4 in its Taylor expansion at

a = O. The following identities are easy to obtain.

2.1 LEMMA.

(i)

L

k#O(2k)2 - 1 -

(iii) '"" ( 1 )3 _ 1 _ 311"2

~ (2k)2 - 1 - 32

k#O

k² 11"2

(v)

L

k#O

00 1 11"4

(vii) '""_~ - - (2k+ 1)4 - 48

- 0 0

To begin with, the leading terms ofIEn(a)1 is generally given by

IEn(a)1 =

~ (~)

Proof. For an integer k1 =1=0

IEj',)1⁼ 2tan-¹

ekl~: 8\0)) _

ekl?r:8\0»)

_ 2tan-1 ( 2a9o(a) ) - (2k11r)2 - 95(a)+^a2

2a 2

= 21r· 1r2((2k

1)2 _ 1) +O(^{a ).}

2a n+l)

1r2(2kn+1)2 +O(a .

2.3. LEM.MA.

(i) IEo(a)1 =211"-~a-~a2+(~-~)a3+(~-~)a4+0(a5), (ii) lE₁(a)1 = .i_11'a - ^{(1. _}_{11' 1r1l"}⁸ )a2 _ ⁽¹⁶_31r1l" ^_ 32)a3_11'5 + (-l. -_611' ~_311'5 +

~)a4+Oe(^5),

(iii) IE2^(a)1 = ~a2 - (-!r - -!a)a3 - _(6~+~ - ~)a4 +O(a5), (iv) IE3(a)1 = -!ra3 - _(2~ - _~)a4+0(a5),

(v) _IE4(a)1 = _2~a4 +0(a5).

Proof. From the relation Oo(a) = atan(00(a)/2) and lima_oOo(a)

=11", we have the following Taylor expansion ofOo(a) near0: =o.

( 2 4

(2 16)

(16 80)

00 a)=1I"--a--a + - - - a + - - - a +O(a).

11" 11"3 311"3 11"5 311"5 11"⁷ Therefore,

I ()IEo a = 211"--a--a^{4 8 2}+^{( 4}- - -³²⁾ a³+⁽³²- - -

160)

For a nonzero integer k1 , we have the following expansion by a routine calculation:

where the sum runs over all nonzero integers k1 .

Next, for an integer k2 , we have

IE~kl,k2)1

0(2) 0(2) -1 (2k211"+0~1)) ^-1 (2k211"+OP))

=

=

(

0'(0(1) - 0(1» )

_ 2tan-1 2 1

- .. (2k 211"+oP)(2k₂1r+8~1) ^+0'2 0'(0(1) _ 0(1)

=^{2 2 1} +0(0'5)

(2k211"+0~1»(2k211"+0~1) +0'2

80'2 3 { 64kl

=

r«

16 3 2 }

- 1I"5«2kt} - 1)(2k2+1)2 - 1I"5«2kt}2 - 1)2(2k2+1)2

4{ 1 2 8 k 1 256k 1

+a - 1I"7«2kt}2 _ 1)2(2k2+1)3 - 1I"7«2k1)2 - 1)3(2k2+1)3

( 16 4 ) 8 8

+ 11"4 - 311"2 1I"3«2kt}2 - 1)3(2k2+1)2 + 1I"3«2k1)2 - 1)2(2k2+1)2

( 4 1)

1

11"4 - 11"2 - 11"7 «2k1)2 - 1)(2k2+¹⁾² +1I"7«2k1)2 -1)3(2k2+¹⁾⁴

8 32

1I"5«2kt}2 - 1)(2k2+¹⁾⁴ 1I"7«2kt}2 - 1)2(2k2+¹⁾⁴

32 3 2 }

+ 1I"7«2kt}2 - 1)(2k1+1)2(2k2+1)3 - 1I"7«2kt}2 - 1)(2k1 - 1)2(2k2+1)3 +0(0'5).

Therefore, we haveby use of Lemma 2.1

where the sum runs overall nonzero integers k1 and all integers k_{2 •} Next, for integers k2 and k3 , we compute

IE~kl,k2,k3)1⁼ 9~3)^_ 9~3)^{= 2tan-1 .(} 0'(9~2)^- 9~2» ⁾

(2k271"+8~2»(2k271"+9~2» +0'2 (9(2) 9(2»

= 2 ~2)^{2 - 1 (1)} +0(0'5)

(2k271"+9₁ )(2k271"+_{92 )}+0'2

16 1 ( 1 ) 2 ( 1 ) 2

= 71"5 (2kl)2 - 1 2k2+1 2k3+1 0'3

+ :;

{«2kl~~1_1)2

lr

1 ( 1 ) 2 ( 1 )2

(2kt}2 - 1 2k2+1 2k3+1

2 ( 1 ) 2 ( 1 )2

«2kt}2 - 1)2 2k2+1 2k3+1

+ (2k

1:2 _ 1

(2k2~ J

+

lr}

Therefore, we have by the use of Lemma 2.1

IE₃^{(a)1 =}

L

= 2:.._7ra^{3 _}

(--!- _

where the sum runs over all nonzero integers k1 and all integers k_{2 ,}k3 •

Finally by Lemma 2.2, we have

IE4(a)1 = _a1 ⁴ +O(a^5).

27r

2.4 Proof of Theorem (aJ. By Lemma 2.3, we have

4 4

I

U

^L

n=O n=O

Since IEn(a)1 = ~ (~r+O(aⁿ+¹⁾ by Lemma 2.2, we have

00 00

I

U

L

n=5 n=5

ex> 4 ex>

I

U

L

L

n=O n=O n=5

and hence we have our main result,

3. Proofby mathematica

We used the MATHEMATICA to obtain the Taylor expansion of IEn(a)1 up to order 6. They are given as follows:

(i) IEo(a)1 =^{27l" _} i a _ '!'a,.2⁺

(-±- _

7l" ~ 37l"3 7l"5 37l"5 7l"7

4 ... stt·^c·S96- 5 184···l11n--5376 .. 6 7 . - ( - - - + - ) 0 ' - ( - - - + - ) 0 ' +0(0'),

57l"5 7l"7 7l"9 157l"7 7l"9 7l"11 (ii) IEl(a)l=ia-(~_.!.)a2-(~-^32)0'3

7l" 7l" ~ 37l"3 7l"5

( 1 SO 160) 4 ( 1 32 160 896) 5

+ 67l" - 37l"5 + -;7 a + 3~ + 157l"5 - -;7+ 7 ^a

_ (_1 1_ _

-±- _

607l" 67l"3 37l"5 157l"7 37l"9 7l"11 (Hi) IE2(a)1= ~a2^_ (.!- __~)0'3 _ (2-+.!. _16)0'4

11" 11" ~ 611" 11"3 11"5

+(_1 1_ _ 28 + 80)0'5 +(_1_ _...!... _^{56 +} 448)0'6 +0(0'7),

127l" 37l"3 37l"5 7l"7 607l" 37l"5 7l"7 7l"9

(iv) lE ()I 1 3 (1 2 ) 4 (1 + 1 8) 5

3 a =;0' - 27l" - 7l"3 a - 127l" 7l"3 - 7l"5 a +(_1 1_ _ ~⁺ ~)a6 +0(0'7),

247l" 67l"3 37l"5 7l"7

(v) IE4(a)1 = 2-0'4 _ (..!... _..!...)0'5 _ (_1_+^_1__ ~^)0'6+0«(1'7),

27l" 411" ~ 247l" 2~ 7l"5

(vi) IE5(a)1= 41

7l" 0'5 - (81

7l" - 2:3)0'6 + 0(0'7), (vii) IE6(a)1 = -0'6 +0(0'7),

87l"

Since IEn(a)1 = !(~)n +O(an+¹⁾ by Lemma 2.2, we have

ex> ex>

I

U

L

n=7 n=7

Therefore, we have

00 6 00

I

U

L

L

n=O n=O n=7

and hence,

4. Concluding remarks

The Cantor set C(a) is atypical as compared with the classical Can- tor set. When we encountered with the Cantor set C(a), we wondered whether the Lebesgue measure of C(a) is positive for all 0<a <2 or not. We could only prove that IC(a)1 =O(a⁵) and conjecture that it is of infinite order as a --+ O. Also it would be an interesting question to know the Hausdorff dimension of the set C(a).

References

[D] R. L. Devaney,An introduction to Chaotic Dynamical Systems, Addison-We sley, Menlo Park, 1989.

[KK1] H. O. Kim and H. S. Kim, Iteration,Denjoy- Wolff point andergodic prop- erties of point-mass singular inner functions, J. of Math. Anal. and Appl.

174 (1993), 105-117.

[KK2] H. O. Kim and H. S. Kim, Dynamics of two parameter of point-mass singular inner junctions,J. Korean Math. Soc. 33 (1996), 39-46.

Department of Mathematics KAIST

Taejon 305-701, Korea