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By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992.

In this paper, we prove that two rational maps with the Cantor Julia sets are quasiconformally conjugate if they are topologically conjugate.

The local connectivity of Julia sets for the family of biquadratic polynomials fc(z) = (z2 − 2c2)z2 with a parameter c is discussed. It is proved that for any parameter c, the boundary of the immediately attracting domain of fc is a Jordan curve.

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.

Let ƒ(z = e2πiθz(1 + z/d)d, θ ∈ ℝ \ ℚ be a polynomial. If θ is an irrational number of bounded type, it is easy to see that ƒ(z has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of ƒ(z satisfies Dim(J(ƒ)) < 2.

For a sequence ( c n ) of complex numbers, the quadratic polynomials and the sequence ( F n ) of iterates are considered. The Fatou set is defined as the set of all such that ( F n ) is normal in some neighbourhood of z, while the complement of (in ) is called the Julia set. The aim of this paper is to study the stability of the Julia set in the case where ( c n ) is bounded. A problem put forward by Bruck is solved.

A sufficient and necessary condition is given for the continuity of Julia sets in the space of all rational maps with degreek>1.

The topological structures of the Julia sets of rational and entire functions have been investigated and the complexity of the Julia sets of rational functions has been described. For entire functions, it is proved that the dynamics on the Fatou sets will influence the topological complexity of Julia sets.