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Yes there are such examples, the simplest one is $\sin(z^3)$. It has a stronger property that the area of non-escaping set is finite. MR2213937 Hemke, Jan-Martin, Recurrence of entire transcendental …

To complement the answer of Lasse Rempe-Gillen. If there is one component, then it is completely invariant. It is either an immediate domain of an attracting fixed point or of a petal. If it is the do …

If $f$ is rational, then $g$ is also rational. It is a very rare, exceptional situation when such two functional equations are satisfied. All such cases have been explicitly described by J. Ritt in Pe …

This is true for every subsequence. Indeed, if $z$ is on the set of normality (where $f^n$ is normal), then evidently every subset of $f^n$ is normal. If $z$ is on the Julia set, then there is a repel …

The mistake is in the statement that $\partial B\subset F(f)$. There can be points on $\partial B$ and $\partial V_m$ which are in $J(f)$, namely preimages of $z_0$ :-) An example is $f(z)=z+1-1/z$. …

The correct statement is this. If the Fatou set $F(f)$ of an ENTIRE function has a multiply connected component, then all components of $F(f)$ are bounded. The idea of the proof is the following. Let …

Yes, there is an explanation, but not in all cases. Rotational symmetries of polynomial Julia sets are explained in the paper MR0951972. For the case of rational functions, see also MR1092156. And al …

The answer is given in @Misha's comment. This is an extended comment. You do not need the theory of quasiconformal (quasisymmetric) maps on arbitrary sets $X$ here, because there is a stronger form of …

Yes. This was proved by Fatou, Sur les equations fonctionnelles, Bull SMF, 48 (1920) on p. 81. See also: MR1295160 F. Przytycki, A. Zdunik, Density of periodic sources in the boundary of a basin of …

Yes. All periodic components of the set of normality of any transcendental entire function are simply connected. This is a theorem of Baker, The domains of normality of an entire function. Ann. Acad. …

Your statement that iterates of the Newton method converge to a cycle almost everywhere is equivalent to the statement that for every polynomial $f$ the Julia set of the rational function $z-f(z)/f'(z …

It is not quite clear what you mean by calculate. This limit is a familiar, classical object: it is called the Green function of the complement of the Julia set, or the equilibrium potential of the Ju …

All fixed fixed points of the Newton map $N(z)=z-P(z)/P'(z)$ are superattracting. So there are no invariant Siegel disks. But of course there are periodic Siegel disks of periods greater than $1$. Fo …

The answer to a) is yes, and this was proved by Fatou in 1919. Sur les équations fonctionnelles Bulletin de la S. M. F., tome 48 (1920), p. 208-314. There are many generalizations of this fact. For o …

There is a book by Erik Fornaess and Nessim Sibony, MR1363948, survey papers of the same authors, MR1810536, MR1748606, MR1285389 and on various specific questions I also recommend papers of Misha Lyu …