DYNAMICS OF {λ tanh(ez) : λ ∈ R \ {0}}

Abstract

Abstract. In this paper, the dynamics of transcendental meromorphic func- tions in the one-parameter family M= {f(z) = f(z) : f(z) = tanh(ez) for z 2 C and 2 R \ {0} }is studied. We prove that there exists a parameter value −3.2946 such that the Fatou set of f(z) is a basin of attraction of a real fixed point for > and, is a parabolic basin corresponding to a real fixed point for = . It is a basin of attraction or a parabolic basin corresponding to a real periodicpoint of prime period 2 for < . If > , it is proved that the Fatou set of f is connected and, is infinitely connected. Consequently, the singleton components are dense in the Julia set of f for > . If , it is proved that the Fatou set of f contains infinitely many pre-periodic components and each component of the Fatou set of f is simply connected. Finally, it is proved that the Lebesgue measure of the Julia set of f for 2 R \ {0} is zero.