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One possible reference is MR2304299 Gelfert, Katrin; Wolf, Christian Topological pressure for one-dimensional holomorphic dynamical systems. Bull. Pol. Acad. Sci. Math. 55 (2007), no. 1, 53–62 Some …

There has been indeed much development in the dynamics of several complex variables in the last 20 years or so. The directions which the subject took focus on different aspects. E.g., is one interest …

An answer before the numerics start: First, note that for $w=u+iv \in \mathbb{C}$ and a fixed $a \in \mathbb{R}$ we have $|w+a|=\sqrt{u^2+v^2+2au+a^2}=\sqrt{|w|^2 + 2au+a^2}$ . Next, consider the im …

An incomplete answer, too long for a comment: Such questions deal with ``purity of the branch locus''. Let $f: X \to Y$ be a finite surjective morphism between irreducible (complex) projective variet …

Only a partial and rather distant answer: There is ``this sort of expression" in dynamical Nevanlinna theory. The mean proximity of $f$ with respect to $a \in \hat{\mathbb{C}}$ is defined as $m(a,f)= …

Edited: The previously found sufficient condition is indeed necessary, but even better, it is satisfied by all polynomials of degree at least two. Thus the conjecture is true: Theorem: Let $p$ be a c …

The unique measure of maximal entropy $\mu_f$ supported on the Julia set of a rational map $f$ of degree $d \geq 2$ is indeed the unique balanced measure for $f$, i.e., the only probability measure $\ …

An attempt to extend the Fatou-Julia theory to continuous self-maps of a connected Riemannian manifold $M$ with a $\mathcal{C}^\infty$ smooth metric was made in the following book: MR1784605 (2001i:37 …

With your background, the book by Milnor or by Beardon looks like a good choice. In addition to the references already suggested, there also are: MR1224235 Steinmetz, Norbert: Rational iteration. Co …