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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]

Holomorphic dynamics has been a key area of study for 100 years. In this survey, we focus on the dynamics of rational functions. [ABSTRACT FROM AUTHOR]

This article studies the iterative behaviour of a quasiregular mappingthat is an analogue of a sine function. We prove that the periodic points ofSform a dense subset of. We also show that the Julia set of this map isin the sense that the forward orbit underSof any non-empty open set is the whole space. The mapSwas constructed by Bergweiler and Eremenko (Ann. Acad. Sci. Fenn. Math. 36 (2011), pp. 165–175) who proved that the escaping setis also dense in. [ABSTRACT FROM PUBLISHER]

Abstract: We study the dynamics on the Julia set for holomorphic endomorphisms of . The Julia set is the support of the so-called Green current T, so it admits a natural filtration , where for we put . We show that for a generic point of there are at least “Fatou directions” in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions. [Copyright &y& Elsevier]

We consider one-parameter family of transcendental meromorphic functions of the form , , . We study the geometry and dynamics of hyperbolic functions and define the Mandelbrot-like set ℳ for this family. We also characterize the hyperbolic components and give combinatorial description of their deployment in the parameter plane. [ABSTRACT FROM PUBLISHER]

We show that, for many parameters aϵC, the set I(fa) of points that converge to infinity under iteration of the exponential map fa(z) = ez + a is connected. This includes all parameters for which the singular value a escapes to infinity under iteration of fa. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice j = 0. [ABSTRACT FROM AUTHOR]

For a nonconstant function F and a real number h∈]0, 1] the relaxed Newton's method NF,h of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z)=P(z)eQ(z), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that NF,h is a rational map with a parabolic fixed point at infinity. [ABSTRACT FROM AUTHOR]