| In this dissertation, we study two kinds of questions.The first question is quasisymmetric minimality on packing dimension for Moransets.Hakobyan [33] proved that the middle interval Cantor sets of Hausdorf dimension1are all quasisymmetrically Hausdorf-minimal; Hu and Wen [48] generalized this resultunder the uniform Cantor sets of Hausdorf dimension1with supnk<∞; Dai, Wen,Xi and Xiong [13] generalized this result under a large class of Moran sets on line withHausdorf dimension1. The known conclusions are all about quasisymmetric packingminimality. In our paper, we study the quasisymmetric minimality on packing dimen-sion for two kinds of Moran sets on line. We construct a probability measure whichis supported on the image set, construct a subset which satisfies the mass-distributiontheory. We get two results:(1)homogenous Moran sets on line of packing dimension1with supnk<∞are all quasisymmetrically packing-minimal;(2) Moran sets on line ofpacking dimension1with c=infk,jck,j>0are all quasisymmetrically packing-minimal.The second question is about the equivalent conditions of quasisymmetric mappings.It is usually very difcult to check whether a given mapping is quasisymmetric. Tukiaand Va¨is¨al¨a [90] introduced the notion of weakly H-quasisymmetric. If a mapping is qua-sisymmetric, then it is weakly H-quasisymmetric, but not all weakly H-quasisymmetricmappings are quasisymmetric. V¨ais¨al¨a [104] and Heinonen [40] show thatμif X is a con-nected and doubling space, Y is a doubling space, then the mapping f: X→Y is weaklyH-quasisymmetric if and only if f is quasisymmetric.There is a natural question: can we weaken the definition of connectedness? By theexample of chapter4, the result of [104] and [40] is not holds even X is a self-similarset satisfying the strong separation condition. Therefore, we introduce the definition ofweakly (λ, H)-quasisymmetric§and obtain the following result: the mapping which mapsa compact λ uniformly perfect set on line with λ>1into a doubling metric space is aquasisymmetric mapping if and only if it is a weakly (cλ, H)-quasisymmetric mapping forsome H>0§where c≥1is a constant satisfies λ2cλ1≤0. Then we apply result (1)to the Cantor-like set on line, and obtain an equivalent condition. Furthermore, notice thespecial structure of Cantor-like set, we use the skill of the proof of (1), and obtain anotherequivalent condition, the result is:(2)let E Rbe a Cantor-like set on line satisfying theλ-small gap condition with λ>1§Y is a doubling metric space, f is a mapping from Xto Y§then the following conditions are equivalent:(i) f is a quasisymmetric mapping; (ii)f is a weakly (c(1+λ), H)-quasisymmetric mapping for some H>0§where c≥1isa constant satisfies (1+λ)2c(1+λ)1≤0;(iii)f is a weakly (λ, H)-quasisymmetricmapping for some H>0.(3)Furthermore, we give some equivalent conditions to thequasisymmetric mappings between two quasisymmetric equivalent Cantor-like sets. |
