Measures And Dimensions Of Fractals, Lipschitz Equivalence And The Open Set Condition

In this paper, we study four problems of fractal geometry: the measures and di-mensions of fractals, Lipschitz equivalence, on single-matrix graph directed self-affneIFS and dual system of algebraic graph directed IFS.It is diffcult to determine the exact Hausdorff measures of fractal sets. We canonly deal with the Hausdorff measures of some fractal sets with Hausdorff dimension< 1 or integers. As far as we know, there is no example of fractal set such that itsHausdorff dimension is no-integer and greater than 1, but its Hausdorff measure can begiven explicitly. The diffculty lies on that we lack a general tool and hence a very ac-curate estimate is needed. We construct a class of so called circular Moran sets whoseHausdorff dimension is strictly between 1 and 2. Thanks to the iso-perimetric inequal-ity, we gain an accurate estimate of these circular Moran sets on the basis of improvingformer technologies and get its exact Hausdorff measure by an explicit formula.Lipschitz equivalence is a kernel problem on geometric measure theory, whichhas very strong challenge, see David and Semmes[14] as well as Falconer[20,21] etc.People are very concerned about the problem of seeking effective Lipschitz invariance.We introduce a notion ofδ-connected of sets and then define the gap sequences ofcompact sets of Rd. We post the relations among the gap sequence, fractal dimensionand Lipschitz equivalence. Finally, a new Lipschitz invariance characterized by gapsequence is got, which is a stronger Lipschitz invariance than the classic upper-boxdimension invariance.The study of self-affne sets is one of the most diffcult problems in dynamicalsystem and fractal geometry. Since little is known about self-affne sets, every progresson it appeals people's attention. He and Lau discussed the open set condition of aclass of single-matrix self-affne IFS by using the Hausdorff measure and Hausdorffdimension w.r.t. a weak norm of Euclidean space. We generalize the study of He andLau to single-matrix graph iterated self-affne IFS according to a finer estimate. Hence we can apply the results to the study of fractal structures of atomic surfaces.The dynamical system induced by a directed graph IFS with algebra numbers asits parameters is one of the most active studies. These studies focus on two aspects,one isβ-numeration system tiling and the other is atomic surface of substitution byprojection. We introduce a notion of dual IFS by the algebraic dual of parameters.The dual IFS defined in this way has many important properties as the initial system,including the open set condition. This new viewpoint and method unify those twostudies in one frame.