Julia set under q deformation on a Quadratic Map

The long-term investigation of dynamical systems has served as inspiration for research on the dynamics of families of mappings. The investigation of the behaviour of the mappings on intervals and Cantor sets was made possible by many of these discoveries. In order to comprehend the nature of families of mappings produced by initialising a complex number, Julia sets are essential. Any function subject to q-deformation effectively undergoes alteration, and in the limit where q → 1, the initial function is restored. In this instance, we use a quadratic map in its complex form. We also employ an entirely imaginary deformation parameter ε. In this study, we apply the Julia set's q-deformation to a quadratic map and generate Julia sets that correspond to various deformation parameter values. We next analyse for what values of the parameter ε the Julia sets cease to exist or fill the entire space. We also plot the heat maps for these ε values.