Connectivity Of Filled Julia Sets For A Kind Of Quartic Polynomials

In this paper, based on the theory of connectivity of filled Julia Setsfor even quartic polynomials, and local connectivity of Julia sets, connectivity offilled Julia sets for a class of quartic polynomials are concerned. Firstly, by theextend Puzzle technique of Branner-Hubbard and Yoccoz , we study the deepof the puzzle piece of 0 of a kind of quartic polynomials f's filled Julia sets.Secondly, even quartic polynomial has three finit critical points. In the case ofthe first critical point is supperattracting and fixed, the second is unboundedand the last one is bounded. It is obtained that a connected component of thefilled Julia sets is non-trivial (that is, it has two point s at least) if and only ifit is a periodic critical component, or an inverse image of some periodic criticalcomponent under the iteration of f. Finally, we demonstrate a example ofpolynomial with the above property.u...