Abstract:

One possible analogy of cubic polynomials is entire transcendental functions with two free singular values, where one of them is an asymptotic value with one finite preimage and the other is a critical value. Our main goal is to understand the topology of the main hyperbolic component of this family, i.e., the set of parameter values for which there is an attracting fixed point with an immediate basin of attraction containing the two singular values. We concentrate on the central slice of the main hyperbolic component in the parameter space, assuming that the fixed point is superattracting. This slice can be parametrized by the family of functions $f$_a(z)=a(exp(z)(z-1)+1) which has two singular values: one superattracting fixed point at $z=0$ and one free asymptotic value at $z=a$. We study the capture components in parameter plane, that is, the set of parameter values $a$, for which the asymptotic value is in the basin of attraction of $z=0$, also observe the structure of dynamical plane, basically the number and boundedness of Fatou components in the basin of attraction of 0 also estimates for the size of the immediate basin of attraction, in terms of location of asymptotic value as parameter in the parameter plane.