Affine Embedding Of Self-similar Set

This paper describes the four parts: iterated function system and self-similar set, symbol space, the parameterization of Koch curve and the affine embedding of Cantor set.In the first part, we mainly give a introduction of some content of fractal geometry, firstly we give a brief introduction of iterated function system and self-similar set, then we know that self-similar set is a product of special iterated function system. Secondly we discuss the iterated function system and self-similar set by four examples:the middle third Cantor set, Koch set, Cλ set and Julia set. Thirdly, we describe the a important property of fractal geometry——dimension, which includes the overview of Hausdorff dimension and box-counting dimension.In the second part, we mainly introduce the important theory of fractal geometry——symbolic space, which play an important role in the next two parts. we give a measure of the symbolic space first and thereby induce a topology to elicit that the proposition (Σ∞,d) is a measure space. Secondly, a discussion of its discrete nature is given. Thirdly, we conclude the compactness of symbolic space from that Σ∞is Homeomorphism to the middle third Cantor set.In the third part, we mainly account for the parameterization of the fractal Koch set. we first get mapping: φ[0,1]→K, φ(t)=π(?)τ-1(t) by defining two mappings: π: and Then, the surjective and continuous nature proves that Koch set is a simple curve. Secondly, we discuss the—Holder continuity, preservation of measure and self-similarity of Koch curve.In the last part, we discuss the affine embedding of self-similar set, actually Cλ set. we first discuss that when the λ of the Cλ set is less than1/4and its logarithmic ratio is a positive integer, it can affine embedding. Then we take into account that when its logarithmic ratio is not a positive integer, whether there is possible that it can affine embedding and we give a special case.