| This paper is a report on recent results related to analysis on graph-directed fractals.It related to the work of Mauldin and Williams on calculation of fractal dimensions of graph-directed self-similar sets[22],the work of Rao,Wen,Ngai,Lau,Wang on the study of fractal dimensions of certain self-similar sets with overlaps[21,25,26],the work of Hambly and Nyberg on the analysis of finitely ramified graph-directed sets[13,14],and the work of Cao and Qiu etc.on the harmonic structures and Laplacians on a more general class of finitely ramified self-similar sets[7,8].In this paper,firstly,we discuss on two conditions of self-similar sets named finite neighboring type condition and finite chain length condition,and show that they lead to the existence of a finitely ramified graph directed(f.r.g.d.)construction of a self-similar set K.Secondly,by means of the finite ramified of finite type(f.r.f.t.)cell structure,which is equivalent to the f.r.g.d.construction of K,we consider the relationship between harmonic structures of different f.r.g.d.constructions on K.Finally,we introduce the spectral decimation property of a special pair of graph-directed fractals T and S,and make a full description of the Neumann and Dirichlet spectra of the Laplacians on both the fractals.Based on these former works,we add some examples of f.r.f.t.self-similar sets,and obtain the harmonic structures and Laplacians on these fractals.In addtion,at the end of this paper,we make a tentative study on the existence of any non-trivial spectral decimation property of the Laplacians on the Sierpinski gasket in a graph-directed manner. |
