Recurrent Sets, Super Self-similar Sets And Cantor Sets

Anomaly" geometrical objects that appear in the nature such as the bank of clouds, the outline of mountain, snowflake, the boundary of coast are all difficult to describe with the straight line, smooth curve, smooth surface of classic geometry . But gathering of these anomalies can better describe many natural phenomena .In nineteen eighties fractal geometry founded by B.B.Mandelbrot provided thoughts, methods and techniques to study this anomaly geometry. Specially in recent years, this newly arisen subject reach large achievement in mathematics, physics, chemistry, biology , medical science, geology, etc. At the same time, the large quantity of problems questioned by different subjects stimulated the fractal geometry to develop.This paper gives a superficial discussion of three types of classical fractals 梤ecurrent sets, super self-similar sets and Cantor sets.Recurrent sets were first introduced by Dekking in paper [l],[2], [3] in 1982, who deeply discussed their properties of fractal, algebra and combination. Bedford [4] got Hausdorff dimension of self-similar recurrent sets under open set condition. Zhiying Wen . Liming Wu and Hongliu Zhong determined Hausdorff dimension and Box dimension of recurrent sets under common situation . Wenxia Li[8] extended the concept of recurrent sets.In this paper we consider the condition under which hypothesis (1)holds, L#'s being integral in the sense of similitude; we give an example for which Kn := Lσ-nK[σn(w)] converges even if Lσ isn't expanding and an sufficient condition for Kn to fail to converge when Lσ isn't expanding.Self-similar sets are a kind of the simplest fractal sets, whose dimension proper is well known. There are many kinds of extension of self-similar sets, such as super self-similar sets, low self-similar sets, sub-self-similar sets, random self-similar sets, etc.. The definition of super self-similar set is given in [6]. Superself-similar sets defined by this definition are self-similar sets themselves, whose Hausdorff dimension and Box dimension are equal.In this thesis we first make a modification for the definition of super self-similar sets , then construct the set A, which is a super self-similar, but whose Hausdorff dimension and Box dimension are not equal, which tell us that Hausdorff dimension and Box dimension of super self-similar sets defined by this thesis aren't equal .The Cantor set is the most classical set in fractal, whose fractal dimension and Hausdorff measure are well known. The character and fractal dimension of intersection of Translations the Cantor set are given in [5].At last , we get the extended form of the dimension formula of the Cantor set on 1-dimension n-section Cantor sets.