Totally Disconnectedness Of Fractal Squares

This thesis studies the connectedness of square fractal sets. Let()iixaS xn+=,where2{0,1,..., 1}ia ?n - and i =1,2,...,q. The self-similar set1(qiiFS F==è is called asquare fractal set.Fix n 32. Our aim is to determine a critical value of q such that, if q is not greaterthan the critical value, then there exists a totally disconnected square fractal set; if q isgreater than the critical value, then all the square fractal sets contain a non-trivial connectedcomponent. We conjecture that the critical value is21[ ]2nq n n-= - -.This thesis obtains some results on this problem. In Chapter 3, we construct two kinds oftotally disconnected square fractal sets for21[ ]2nq n n-= - -. Thus the critical value is atleast21[ ]2nn n-- -. In Chapter 4, we prove that all the square fractal sets contain anon-trivial connected component when2 q =n -n. Thus the critical value is less than2 n -n.In Chapter 5, we further study the properties of totally disconnected square fractal sets, obtaina necessary and sufficient condition for totally disconnected. Based on this condition, we alsopresent two necessary conditions which are easy to verify.