Multicorn Sets of <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mover accent="true"><mi>z</mi><mo>¯</mo></mover></mrow><mi>k</mi></msup><mo>+</mo><msup><mi>c</mi><mi>m</mi></msup></mrow></semantics></math></inline-formula> via S-Iteration with <i>h</i>-Convexity

Abstract

Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets and their dynamic nature. In this paper, we study the dynamics of multicorn sets of z¯k+cm, where k≥2, c≠0∈C, and m∈R, by using S-iteration with h-convexity instead of standard S-iteration. We develop escape criterion z¯k+cm for S-iteration with h-convexity. We analyse the dynamical behaviour of the proposed conjugate complex function and discuss the variation of iteration parameters along with function parameter m. Moreover, we discuss the effects of input parameters of the proposed iteration and conjugate complex functions of the behaviour of multicorn sets with numerical simulations.

Keywords

• multicorns
• S-iteration
• <i>h</i>-convexity
• escape criterion