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

Abstract

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