Tsuji functions with segments of Julia

Lines of Julia of Integral Functions.

Dimensions of Julia Sets of Meromorphic Functions

It is shown that for any meromorphic function f the Julia set J(f) has constant local upper and lower box dimensions, d(J(f)) and d(J(f)) respectively, near all points of J(f) with at most two exceptions. Further, the packing dimension of the Julia set is equal to d(J(f)). Using this result it is shown that, for any transcendental entire function f in the class B (that is, the class of function...

Entire Functions with Julia Sets of Positive Measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log logM(r, f) ≥ 1 2 N log r − O(1), where M(r, f) denotes the maximum modulus of f . We show that if f doe...

Meromorphic functions with linearly distributed values and Julia sets of rational functions

If the preimage of a four-point set under a meromorphic function belongs to the real line then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle. If the full preimage of a two-point set under a rational...

Connectivity properties of Julia sets of Weierstrass elliptic functions

We discuss the connectivity properties of Julia sets of Weierstrass elliptic ℘ functions, accompanied by examples. We give sufficient conditions under which the Julia set is connected and show that triangular lattices satisfy this condition. We also give conditions under which the Fatou set of ℘ contains a toral band and provide an example of an order two elliptic function on a square lattice w...