The Development Of Fractal Projection

This thesis describes the development of projection theory in fractal geometry, This article centers on the paper which refer to the fractal projections published in year 1954.The main body can be divided into three parts.Part One: It mainly introduces two famous projection theorems in Marstrand's paper and its corollary and discusses other important content in his paper related to fractal geometry.Part Two: It mainly introduces a projection theorem proof by Kaufman which proves to be more enlightening and concise compared with Marstrand's. Besides, it introduces the theoretical method involved in the proof and the concept of exception set led by the certification process.Part Three: It mainly introduces a series of theories which is expanded out during the development process of projection, including integer dimension sets, self-similar sets,imitation sets, random sets as well as the projection theory in the restricted direction. In addition, as a counterpart packing dimension of Hausdorff Dimension appeared, the correlation theories have been transformed to the packing dimension from Hausdorff Dimension.The general train of thought of this article is that first introduced the origin of the fractal projection,the two classical projection theories on s-set in John Marstrand's paper.On this basis, Roy Davies presented the change of dimension and measure when Borel-set of2 R projects onto the line at angle ? to the x-axis,according to the conclusion that every Borel-set in the infinite Hausdorff measure of s-dimension contains one s-set. While Marstrand's projection theorem is useful, it is difficult to summarize and extend the theory in that the proving is too complex. Therefore, Kaufman presented a brief proving by potential theory and Fourier transform method. Enlightened by Kaufman, Mattila created some theories from high-dimensional space to the subspace projection. In addition, Mattila found that some exceptional sets do not meet the projection theorem and got a series of conclusions, among which the integral nature of the parameter ? can be applied to many other parameterized mapping. Thus, the concept of generalized projection is drawn forth.Jarvenpaa got some conclusions under the packing dimension projection by transforming the related conclusions of projection theorem from Hausdorff dimension to the packing dimension. Afterwards, a series of conclusions appeared in succession on the projection in the restricted direction of integer dimension sets, self-similar sets, imitation sets and random sets. In the future, John Marstrand's theory will continue to promote and guide the research in this field.