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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 ℂ and Julia set of one dimensional complex dynamic system. [ABSTRACT FROM AUTHOR]

We study the dynamics of commuting rational maps with coefficients in ℂ. By lifting the dynamics from ℙ(ℂ) to Berkovich projective space ℙ, 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 have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the rational maps with the positive real parameter λ. It is shown that the immediately attracting basin B of ∞ for F is always a Jordan domain if the Julia set of F is not a Cantor set. Furthermore, B is a quasidisk if there is no parabolic fixed point on the boundary of B. It is also shown that if the Julia set of F 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 Sierpiński curve is given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of the family of rational maps f λ( z) = z n + λ/ z m, 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 f λ. [ABSTRACT FROM AUTHOR]