In this paper, we consider the family of rational mapswhere n ≥ 2, d ≥ 1, and λ ∈ C. We consider the case where λ lies in the main cardioid of one of the n − 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps F λ and F µ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy µ = ν j(d+1) λ or µ = ν j(d+1) λ where j ∈ Z and ν is an (n − 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.
For the family of rational maps z n + λ/z n where n ≥ 3, it is known that there are infinitely many small copies of the Mandelbrot set that are buried in the parameter plane, i.e., they do not extend to the outer boundary of this set. For parameters lying in the main cardioids of these Mandelbrot sets, the corresponding Julia sets are always Sierpiński curves, and so they are all homeomorphic to one another. However, it is known that only those cardioids that are symmetrically located in the parameter plane have conjugate dynamics. We produce a dynamical invariant that explains why these maps have different dynamics.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.