March  2007, 19(1): 121-138. doi: 10.3934/dcds.2007.19.121

Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }

1. 

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India, India

Received  March 2006 Revised  March 2007 Published  June 2007

In this paper, the dynamics of transcendental meromorphic functions in the one-parameter family

$\mathcal{M} = { f_{\lambda}(z) = \lambda f(z) : f(z) = \tanh(e^{z}) \mbox{for} z \in \mathbb{C} \mbox{and} \lambda \in \mathbb{R} \setminus \{ 0 \} }$

is studied. We prove that there exists a parameter value $\lambda^$* $\approx -3.2946$ such that the Fatou set of $f_{\lambda}(z)$ is a basin of attraction of a real fixed point for $\lambda > \lambda^$* and, is a parabolic basin corresponding to a real fixed point for $\lambda = \lambda^$*. It is a basin of attraction or a parabolic basin corresponding to a real periodic point of prime period $2$ for $\lambda < \lambda^$*. If $\lambda >\lambda^$*, it is proved that the Fatou set of $f_{\lambda}$ is connected and, is infinitely connected. Consequently, the singleton components are dense in the Julia set of $f_{\lambda}$ for $\lambda >\lambda^$*. If $\lambda \leq \lambda^$*, it is proved that the Fatou set of $f_{\lambda}$ contains infinitely many pre-periodic components and each component of the Fatou set of $f_{\lambda}$ is simply connected. Finally, it is proved that the Lebesgue measure of the Julia set of $f_{\lambda}$ for $\lambda \in \mathbb{R} \setminus \{ 0 \}$ is zero.

Citation: M. Guru Prem Prasad, Tarakanta Nayak. Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 121-138. doi: 10.3934/dcds.2007.19.121
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