Abstract:

Complex trees are introduced as a unified way of generating and studying a broad class of fractals. The tip sets of these infinitely ramified n-ary trees are attractors for linear IFS $lbrace mathbb {C},lambda_0 z + 1,lambda_1 z + 1, ...,lambda_n z + 1 brace$ which include classical Cantor sets, Koch curves, Sierpinski gaskets and much more. We characterize all the branch tips by power series encoded by infinite words of the alphabet $A = lbrace 0, 1, ..., n brace$ and we provide a general condition for ensuring that the tip set is connected.

Finally we go through several examples of parameterized families of tip set connected complex trees, and we study the Mandelbrot sets associated to them.