Biquadratic Polynomial Dynamics

The present Ph.D dissertation is concerned with the dynamics of biquadratic polynomials.At the end of the last century, C.J.Yoccoz is made significant contributions to the theory of complex dynamics, one of which is the study of the local connectivity of the Julia sets of quadratic polynomials pc(z) = z2 + c and the Mandelbrot set M. In his work. Yoccoz introduced a powerful puzzle technique and obtained the result that if the quadratic polynomial pc(z) with , which has no indifferent cycle, is not infinitely renormalizable, then the Julia set J(pc) of pc is locally connected while dM is also locally connected at c.Almost at the same time, B.Branner and J.H.Hubbard discussed the Julia sets of cubic polynomials. With the similar technique, they gave a complete description of the connectivity and the totally disconnectivity of the Julia sets of cubic polynomials.Now the technique developed by Bianner, Hubbard and Yoccoz is called Branner-Hubbard-Yoccoz puzzle theory. A further application of the puzzle theory, due to Faught and Hubbard, is the study of the Julia sets and the connectedness locus of a cubic family with one critical fixed point.In all the above cases, the polynomials have only one critical point to be considered. For the dynamics of polynomials with higher degree, for example, the quar-tic polynomials, the Branner-Hubbard-Yoccoz puzzle theory should be extended. A.Douady had suggested to study dynamics of biquadratic polynomials. By definition, a biquadratic polynomial is the composition of two quadratic polynomials, so it is an even quartic polynomials which can be written asf(z) = z4 + az2 + 6,where a, b are parameters.It is easy to see that / has a critical point at 0 and two symmetric criticalpoints at . In this family, two critical points should be considered. However, because of the symmetry of the critical points it allows us to extend Branner-Hubbard-Yoccoz puzzle theory to study our case.The first result of our work is on the connectivity of Julia sets of biquadratic polynomials. By an extension of Branner-Hubbard-Yoccoz puzzle theory, a result analogous to Branner and Hubbard's in the cubic case is obtained. Theorem 1. Let f be a biquadratic polynomial, then(1) the Julia set of f is connected if and only if the orbits of all critical pionts are bounded;(2) the Julia set of f is totally disconnected if and only if every connected component of the filled Julia set of f containing critical points is not periodic under iterations:(3) if f does not satisfy the conditions (1) and (2), then the Julia set of f has infinitely many non-trivial connected components, where a component is trivial if it contains only one point.In the case (3). we also have that each critical component and its inverse images under iterations are homeomorphic to the filled Julia set of a quadratic polynomial with connected Julia set. Each of the other components of A'(/) is a single point.Next, we discuss a family of biquadratic polynomials with one parameter so that the critical point 0 is a fixed point. We write this family in the formfc(z)= z4-2c2z2where c is a parameter. It has one critical fixed point 0 and two symmetric critical points 眂 under this parameterization.The second result is on the local connectivity of the Julia set of fc in its dynamical plane. We obtainTheorem 2. Let the Julia set J(fc) of fc is connected. If fc has no parabolic periodic points and is not renormalizable, then J(fc] is locally connected.Here fc is renormalizable if there are simply connected domains U and U' and a positive integer n such that contains the critical point c or -c1 and is a quadratic-like mapping with connected Julia set.Furthermore, for the immediately attracting domain at 0, we can remove the non-renormalizable condition.Theorem 3. // the Julia set J(fc) of fc is connected, then the boundary of the immediately attracting domain of fc at 0 is a Jordan curve. Hence it is locally connected.Finally, we turn to the parameter plane of the family f...