Some Studies About Doubling Measures And Quasisymmetric Maps

According to the Lebesgue measure of sets, we divide the subsets of Euclidean Space into two main classes, fat sets and thin sets. Let E (?) [0, 1]n. We say that E is fat if Ln(E)> 0 and that E is thin if Ln(E)= 0, where Ln(E) denotes the n-dimensional Lebesgue measure of E.we further divide fat sets into three subclasses, which are called very fat, fairly fat, and minimally fat. Similarly, divide thin sets into three subclasses, which are called very thin, fairly thin, and minimally thin.Let E (?) [0,1]n be a fat set. We divide all fat sets into three subclasses as follows.(1) E is very fat if?(E)> 0 for all doubling measures?on [0, 1]n.(2) E is fairly fat if there are constants 1< K1< K2<?such that?(E)> 0 for any K1-doubling measure?on [0, 1]n and?(E)= 0 for some K2-doubling measure v on [0,1]n.(3) E is minimally fat if for every K> 1 there is a K-doubling measure?on [0, 1]n such that?(E)=0.Similarly, all thin sets can be divided into three classes as follows. Let E (?) [0, 1]n be a thin set.(1) E is very thin if?(E)= 0 for all doubling measures?on [0, 1]n.(2) E is fairly thin if there are constants 1< K1< K2<?such that?(E)=0 for any K1-doubling measure?on [0, 1]n and?(E)> 0 for some K2-doubling measure v on [0,1]n.(3) E is minimally thin if for every K> 1 there is a K-doubling measure?on [0, 1]n such that?(E)> 0.We study the fatness and thinness of uniform Cantor sets and Sierpinski carpets.This thesis contains two parts.In part one, i.e. Chapter 3, we study the fatness and thinness of uniform Cantor sets in real line.We divide uniform Cantor sets into six subclasses as above. Han-Wang-Wen [37] obtained the necessary and sufficient condition of a uniform Cantor set to be very fat or very thin. Motivated by Han-Wang-Wen [37], we obtain the necessary and sufficient condition for a uniform Cantor set to be fairly fat, minimally fat, fairly thin or minimally thin, respectively. We prove that, if E= E({nk},{ck}) is a fat uniform Cantor set, then E is a minimally fat set if and only if?(nkck)p=?for all 0< p< 1 and that E is a fairly fat set if and only if there are constants 0< p< q< 1 such that?(nkck)q<? and?(nkck)p=?.We also get the similar result about thin uniform Cantor set.We get,if E=E({nk},{ck})is a thin uniform Cantor set,then E is a minimally thin set if and only if?(nkck)p<?for all p>1 and that E is a fairly thin set if and only if there are constants 1<p<q<?shch that?(nkck)q<?.and?(nkck)p=?In part two,i.e.Chapter 4,we study the fatness and thinnness of Sierpinski carpets in plane.The construction of Sierpinski carpet,in plane is similar to middle interval Cantor set. We also divide the Sierpinski carpets into six subclasses:which are called very fat,fairly fat,minimally fat,very thin,fairly thin,and minimally thin,respectively.We characterize these subclasses in terms of doubling measures on[0,1]2. We prove that,if S({mk})is a fat Sierpinski carpet,then S({mk})is very fat if and only if?k=1?(mk-1)p<?for all p?(0,2)?S({mk})is fairly fat if and only if there are constants 0<p<q<2 such that?k=1?(mk-1)q<?and?k=1?(mk-1)p=?;S({mk})is minimally fat if and only if?k=1?(mk-1)p=?for all p?(0,2). We also get the similar result about thin Sierpinski carpet.If S({mk})is a thin Sierpinski carpet,then S({mk})is very thin if and only if?k=1?(mk-1)p=?for all p>2?S({mk})is fairly thin if and only if there are constants 2<p<q<?such that?k=1?(mk-1)q<?and?k=1?(mk-1)p=?;S({mk}) is minimally fat if and only if?k=1?(mk-1)q<?for all p>2.