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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 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]

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]