| The thesis mainly deals with the singularity of certain self-similar measure , the structure and the correlation dimension of the multiplicative sequence, and the thesis consists of three independent chapters.The self-similar measure has been studied since 1930's, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems and Hausdorff dimension estimation. But the most fundamental question is for its -absolute continuity and singularity. In Chapter 2, we provide the Hausdorff dimensions of the sections of finite type for self-similar sets. And a sufficient condition for certain self-similar measures being singular is obtained , using this criterion we also prove the projection of the self-similar measure with respect to the Sierpinski carpet onto a line with rational slope is singular.The substitutive sequences have been studied for many years, many non-periodic phenomena such as the geometry structure and the properties of spectrum for the quasicrystal can be described by it. With the development of fractal geometry, finite automata and dynamical system, the substitutive sequences play a more and more important role. In Chapter 3, we mainly study the symmetric substitutive sequence—multiplicative sequence, the recurrence relations are obtained for the correlation function, using the relation between the decay rate of correlation function and the characteristics of singular Fourier spectra, we give the exact value for the correlation dimension of the spectral measure for the infinite multiplicative sequence, which extends the results in [50]. This sequence also plays a role in studying of the one-dimension discrete Schrodinger operator, its singular continuous spectrum had been recovered in connection with some physical phenomena.In Chapter 4, we prove the m—tuplings Morse sequenc and the other two sequences generated by it are admissible sequences, which has many applications in the β—expansion.
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