On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map

A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009, 19:2123–2129) and (Nonlinear. Dyn. 2013, 73:1155–1163), the authors presented the two kinds of fractal sets of a class of alternated complex map and left some visually observations to be proved about the boundedness and symmetry properties of these fractal sets. In this paper, we improve the previous results by giving the strictly mathematical proofs of the two properties. Some simulations that verify the theoretical proofs are also included