Dynamics of certain class of critically bounded entire transcendental functions

Let E denote the class of all transcendental entire functions f (z) = ∞n=0 anzn for z ∈ C and an 0 for all n 0 such that f (x)>0 for x <0 and the set of all (finite) singular values of f forms a bounded subset of R. For each f ∈ E, one parameter family S = {fλ ≡ λf : λ > 0} is considered. In this paper, we mainly study the dynamics of functions in the one parameter family S. If f (0) = 0, we show that there exists a positive real number λ∗ (depending on f ) such that the bifurcation and the chaotic burst occur in the dynamics of functions in the one parameter family S at the parameter value λ = λ∗. If f (0) = 0, it is proved that the Julia set of fλ is equal to the complement of the basin of attraction of the super attracting fixed point 0 for all λ > 0. It is also shown that the Fatou set F(fλ) of fλ is connected whenever it is an attracting basin and the immediate basin contains all the finite singular values of fλ. Finally, a number of interesting examples of entire transcendental functions from the class E are discussed. © 2006 Elsevier Inc. All rights reserved.