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Permutable rational functions have identical Julia set. However, is the converse right? We have found a class of rational functions with degree two, they have identical Julia set but are not permutable. Moreover, they have hyperbolic periodic points only.

We prove that for any finite collection of quadratic Julia sets, a polynomial and a transcendental entire function exist whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

Let f be an entire transcendental map, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B(z0) of an attracting fixed point z0. We study the structure of the Julia set of f, which is equal to the boundary of B(z0), and the behaviour of the Riemann mapping onto B(z0) using the technique of geometric coding trees of preimages of points from B(z0). We show that for a given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow no faster than the iterates of an exponential function, then there exists a point in the Julia set with this itinerary. Moreover, we determine cluster sets for and show that has an unrestricted limit equal to ∞ at points of a dense uncountable set in the unit circle.

For the Potts models on the diamond-like hierarchical lattice, the domains of the complex phases are indeed the Fatou components of a family of rational maps. In this paper, we deal with the relationships between this family of Fatou components and the Jordan domains and describe the topological structures of this family of Fatou components completely.

We discuss the existence of a rational map whose Julia set is homeomorphic to a given topological self-similar set. We prove that a self-similar set K is homeomorphic to the Julia set of a postcritically finite rational map if and only if K is embedded in S 2 such that the dynamics of K can be extended to a postcritically finite branched covering on S 2 .

Let fn and f be entire functions and suppose that fn converges to f locally uniformly on C. Then if the Fatou set of f consists only of basins of attracting cycles or is empty, the Julia set of fn converges to that of f in the Hausdorff metric. We also show that expanding f implies the above assumption. Next we show that for each singular value c of f there exists a singular value c(n) of fn (for each sufficiently large n) converging to c. As an application, we propose a criterion which determines by the approximating sequence (fn)n=0infinity whether the Julia set of S is the whole sphere C for a certain class of entire functions.

When -2

For any rational map T:C to C the topological pressure P(t) of the function -t log mod T' mod is well defined, although its Julia set may contain critical points. The authors extend the class of Sullivan's (1982, 1983) conformal measures on Julia sets to the class of measures which are conformal except on some finite set and show that the minimal exponent for which such measures exist, the supremum of Hausdorff dimensions of ergodic T-invariant measures with positive entropy and the minimal zero of the function P(t) are equal. This result may be regarded as a version of the Bowen-McCluskey-Manning formula. A large subclass of rational maps is found for which the same statement is true for conformal measures in the sense of Sullivan.

We consider a sequence of entire functions (g(m)) converging to a limit function g locally uniformly on C. It is claimed by Kisaka that, if the Fatou set F(g) of the limit function is the union of the basins of attracting periodic orbits, then the Julia sets J(g(m)) converge to the Julia set J(g) in the Hausdorff metric. We show that this is not true in general.

It is argued that since any real space renormalisation R is a transformation under which the partition function is invariant, those R which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space. In this extension the Julia set of R (now extended to include R which are algebraic) is an approximation to the locus of the Yang-Lee zeros. Finite-size scaling using Onsager's solution of the two-dimensional Ising model is used to construct R which are highly accurate in real temperatures in the critical region; these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases.

We calculate the fractal dimension D (t) of the Julia set associated with the partition function zeros of the ferromagnetic Ising model on the Cayley tree. In the low-temperature phase the Julia set is the unit circle (D (t) = 1) in the complex fugacity plane. In the paramagnetic phase the Julia set is a Fatou dust (Cantor set) on the unit circle whose fractal dimension is a decreasing function of the temperature. Like an order parameter D (t) displays a singular behaviour with a critical exponent βD = 1/2.

The self-replicating properties of the Julia sets Jc(p) for the iterative processes z implies zp+c are examined for integers p)1. It is shown that the corresponding Mandelbrot sets M(p) contain (p-1)-fold symmetries, which results in Jc(p) having p-fold symmetries.