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

Hong Oh Kim; Jun Kyo Kim; Jong Wan Kim

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 1996; 33(1): 47-55

Printed March 1, 1996

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

Hong Oh Kim, Jun Kyo Kim, and Jong Wan Kim

KAIST, KAIST, and KAIST

Abstract

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Abstract

The set of points on the interval $[-\pi, \pi]$ which are not attracted to zero under the succesive iterations of the function $T_\alpha(\theta) = \alpha \tan(\theta/2)$ is a Cantor set $C(\alpha)$, a closed , totally disconnected, perfect subset of $[-\pi, \pi]$ if $0 < \alpha < 2$. The Lebesgue measure of $C(\alpha)$ is shown to be of order 5 analytically and of order 7 by use of MATHEMATICA as $\alpha \rightarrow 0$. It is a reasonable conjecture that it is of infinite order.

Keywords: Dynamics, Julia set, Cantor set

MSC numbers: 26A18