Dynamics of certain class of critically bounded entire transcendental functions

Abstract

Let E denote the class of all transcendental entire functions f ( z ) = ∑ n = 0 ∞ a n z n for z ∈ C and a n ⩾ 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 = { f λ ≡ λ f : 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 f λ 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 ( f λ ) of f λ is connected whenever it is an attracting basin and the immediate basin contains all the finite singular values of f λ . Finally, a number of interesting examples of entire transcendental functions from the class E are discussed.