| This dissertation consists of 5 chapters of which the second to the fourth chapters are contributed to the studies of several kinds of fractal sets related to overlaps and the fifth chapter is devoted to generating of colorful images with symmetries from dynamics.1. A class of Sierpinski carpets with overlaps:The self-similar sets with overlaps structure have been subject of several studies. Ngai and Wang [9] introduced the notion of finite type and described a scheme for computation of the Hausdorff, Box dimension when the finite type occurs, which, however, forces the ratios to be exponentially commensurable. The result, recently, was extended by Lau and Ngai [10] to generalized finite type without restriction on the ratios. In Chapter 2 of this dissertation, we make use of Lau-Ngai's scheme to analyze a class of Sierpinski carpets with parameters. Based on this scheme, a sufficient and necessary condition for an IFS of contractive similitudes to satisfy the open set condition is given, that is the IFS needs to be generalized finite type and has no complete overlap. Since the IFS {φi}i5=1 of the Sierpinski carpets is generalized finite type when (λ1, λ2) ∈Q × Q by the consequent in [10,9], we then give a characterization of the rational pairs (λ1, λ2) for which the corresponding IFS {φi}i5=1 are of complete overlap. In addition, we also give an explicit criterion for a class of rational pairs for which the IFS {φi}i5=1 are of no complete overlap2. On the intersection of translation of middie-α Cantor setsThe middle-α Cantor set Γα in the interval [0,1] is a straightforward generalization of the classical middle third Cantor set. There has been a fast growth in general interest in the structure of intersection of two Cantor sets. In chapter 3 we mainly explore the Hasdorff, Box, Packing dimension of Γα ∩ (Γα + t). The main difficulties in the study of the dimension of the intersection lie in that the number of similitudes used in the construction may vary from step to step. Under some conditions, however, Γα ∩ (Γα +t) can be provedto be a generalized Moran set. But in many cases, the structure of intersection is more complicated, a suitable equivalence relation on the component sets of Γα ∩ (Γα + t) is then defined, a graph-directed set with OSC coinciding with Γα ∩ (Γα +t) is constructed, whose Hausdorff dimension (and thus the Hausdorff dimension of the Γα ∩ (Γα+t)) can be computed using the theory developed by Mauldin and Williams [34]. Our ideas of the construction of the graph-directed set come from [35, 37,36,9].3. On the intersection of translation of Cantor sets with different contractionsIn chapter 4, we further investigate the Hausdorff, Box, Packing dimension of the intersection of Cantor sets with different contractions. We show that the equivalence relation defined in chapter 3 no longer works, moreover, the set Γβ — Γβ generally is not a self-similar set, which implies that the method developed by Li and Xiao [25] also fails. But by modifying the definition of equivalence relation and generalize the level of iteration, a graph-directed set with OSC coinciding with Γβ ∩(Γβ + t) can also be constructed. Also, Our ideas of the construction of the graph-directed set come from [35,37,36,9].4. Orbit trap rendering method for generating colorful images with cyclic or dihedral symmetryOrbit trap rendering method for generating pseudo 3D-effect colorful images of M-J sets has proved successful. Traditional orbit trap rendering method, however, has nothing to do with the symmetry, but symmetry is everywhere in the nature world and then is crucial not only from the point of theory but also from view of application. Chapter 5 explores combining chaotic functions having cyclic or dihedral symmetry with orbit trap rendering method. By setting symmetric orbit traps (in particular, by using more complex bounded sets, such as chaotic attractors, as orbit traps), visually fascinating images with cyclic or dihedral symmetry can be created in this way.
|
PDF Full Text Request