Search

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ2×2 and Julia set of one dimensional complex dynamic system.

Let f be a transcendental entire function with order ρ < ½ and let σ be a sufficiently large constant. We prove that if there exists r 0 > 1 such that, for all r > r 0 and any small ɛ > 0, $$M(r^\sigma ,f) \geqslant M(r,f)^{\sigma + \varepsilon } ,$$ then every component of the Fatou set F(f) is bounded.

We study the dynamics of commuting rational maps with coefficients in ℂ p . By lifting the dynamics from ℙ1(ℂ p ) to Berkovich projective space ℙ Berk 1 , we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in ℂ p have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.

In this paper, we consider the dynamics of the map z ↦ exp(z)/z on the punctured plane ℂ* = ℂ\{0}. We show that for almost every point z ∈ ℂ*, the ω-limit set of z is equal to {0,∞}. In particular, the map is not recurrent.

In this paper, we discuss the rational maps $F_\lambda (z) = z^n + \lambda /z^n ,n \geqslant 2$ with the positive real parameter λ. It is shown that the immediately attracting basin B λ of ∞ for F λ is always a Jordan domain if the Julia set of F λ is not a Cantor set. Furthermore, B λ is a quasidisk if there is no parabolic fixed point on the boundary of B λ . It is also shown that if the Julia set of F λ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpinski curve is given.

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions.

Suppose f(z) is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim (J(f) < 2.

In this paper, we study the dynamics of the family of rational maps fλ(z) = zn + λ/zm, n ≥ 2, m ≥ 1, λ ∈ ℂ. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.

We shall show that for certain holomorphic maps, all Fatou components are simply connected. We also discuss the relation between wandering domains and singularities for certain meromorphic maps.