Lipschitz Equivalence Of Self-affine Lalley Sets And Assouad Dimensions Of Moran Sets

Fractal geometry provides the ideas, tools and skills for studying non-regular geometry objects. The various forms of dimensions, such as Hausdorff dimensions,(upper and lower) Boxing dimensions, Packing dimensions, give a portrait of its complexity. However, these dimensions often change un-der homeomorphic maps. People found that the dimension of a set couldn’t change under bi-Lipschitz map. Namely, the bi-Lipschitz map doesn’t influ-ence the complexity of set. This is because the bi-Lipschitz map is "close" to self-similar map, it keeps the geometric structure of the set to some de-gree. Since the geometric structure of two sets with the same dimension has great difference, we may not set up bi-Lipschitz map between them. So the bi-Lipschitz map which is used as further analysis of sets with the same fractal dimension from the geometric structure is a power tool. The two sets are Lipschitz equivalent means that we can set up a bi-Lipschitz map be-tween them. In recent decades, people thoroughly studied on the Lipschitz equivalence of sets and obtained lots of interesting conclusions. But for self-affine sets, since the difference of local geometric property between self-affine sets and self-conformal sets, so far, there is a little research on the Lipschitz equivalence of them. In this thesis, we will study the Lipschitz equivalence of a special self-affine sets-self-affine Lalley sets, and get some results on the Lipschitz equivalence of self-affine Lalley sets.The Assouad dimensions of sets is important indicator to describe their geometric structure, it derives from the "doubling properties" of sets. An important property of Assouad dimension is that it is not less than the upper Boxing dimension. For some classical fractal sets, such as the self-similar sets which satisfy open set condition, people showed that the Assouad dimension just equal to the upper Boxing dimension. In this thesis, we will study the Moran sets which are broader sets than self-similar sets and determine their Assouad dimension. the results show that the Assouad dimension can be strictly greater than upper Boxing dimension.The thesis is mainly divided into the following several parts:(I) In the first chapter, we introduce some usual fractal dimensions and the latest results on related-issues.(II) In the second chapter, we show that two self-affine Lalley sets which satisfy the dust-like condition (namely strong separation condition) are Lip-schitz equivalent.(III) The third chapter consider two self-affine Lalley sets, which don’t request the dust-like condition, replace with HBSC condition (namely, hor-izontal block separation condition) and the contraction ratios of them are equal in the same horizontal line, show the Lipschitz equivalence of them. The missing of dust-like condition brings huge difficulty to construct the bi-Lipschitz map.(IV) In the fourth chapter, we study the graph directed self-affine Lalley sets and obtain that two graph directed self-affine Lalley sets satisfying the dust-like condition are Lipschtiz equivalent.(V) In the last chapter, we give the Assouad dimensions of Moran sets.