Dynamics of certain class of critically bounded entire transcendental functions

Abstract

Let E denote the class of all transcendental entire functions Click to view the MathML source for Click to view the MathML source and angreater-or-equal, slanted0 for all ngreater-or-equal, slanted0 such that f(x)>0 for x<0 and the set of all (finite) singular values of f forms a bounded subset of Click to view the MathML source. For each fset membership, variantE, one parameter family Click to view the MathML source is considered. In this paper, we mainly study the dynamics of functions in the one parameter family Click to view the MathML source. 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 Click to view the MathML source 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 Click to view the MathML source 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.