Bicomplex Mandelbar Dynamics Associated with Three Conjugations

In this paper, we investigate the antiholomorphic counterpart of bicomplex quadratic dynamics by introducing bicomplex Mandelbar sets associated with the three natural involutive conjugations of the bicomplex algebra. For each conjugation †m, m∈{1,2,3}, we study the iteration (Fm)C(η)=(η†m)2+C on BC and define the associated parameter set via the boundedness of the orbit of the origin. Using the idempotent decomposition, we obtain a conjugation-dependent classification of the dynamics: †3 yields the idempotent product of two classical Mandelbar sets, whereas †2 and †1 generate cross-coupled quadratic systems, holomorphic and antiholomorphic in one step, respectively. We further prove that the †1- and †2-dynamics are equivalent up to complex conjugation of an idempotent parameter; consequently, the principal three-dimensional slices of the corresponding Mandelbar sets are congruent, mirror-symmetric copies of one another, although neither possesses this diagonal symmetry individually. We visualize these slices and rigorously establish their reflection symmetries. The results clarify the role of bicomplex conjugations in antiholomorphic dynamics and reveal structural phenomena absent from the holomorphic bicomplex Mandelbrot setting.