Holder Embedding And Egoroff’s Theorem

In this paper,we prove that the symbolic space{0,1)N can be Holder embedded into any complete Ahlfors l-regular space. As an application of this result,we prove that the exceptional set in Egoroff’s theorem is full dimension,namely,we prove the following theorem:There exists a sequence of measurable,real-valued functions{∫n(x)}n=1∞ defined in each complete Ahlfors l-regular space X such that lim ∫n(x)=0((?)x∈X),and for any decreasing sequence{δn)n=1∞ of positive numbers with lim δ0=0,the set△一{x∈ X:lim fn(x)/δn=∞} has Hausdorff dimension t.This thesis is crganized as follows:In Chapter 1,brief introduction of fractal geometry and Egoroff’s theorem,research background are presented.In Chapter 2,we introduce the definitions properties of Holder embedding and Hausdorff dimensionï¼›meanwhile,some results related to Egoroff’s theorem are givern.Chapter 3 and Chapter 4,the main part of this thesis,include main results of my research.In Chapter 3,firstly,we prove that for any closed ball B(x,r)(?)X and 0<s<t, where X is a Ahlfors t—regular space,there exist 2q disjoint closed balls{B(xi,2 q/s rr)}i=12q such that each ball is included in B(x,r) and the distance between any two distinct balls is not less than 2-q/sr. Then based on this,we construct a s-1-Holder embedding Ï€:{0,1)Nâ†'X,where X is a complete Ahlfors t-regular space.In Chapter 4,first of all,we obtain the dimemsipm of some special Moran sets. Then,by the theorem in Chapter 3,for any sequence of closed balls{Bk)k=1∞ in complete Ahlfors l-regular space X,and any increasing sequence{sk)k=1∞ of positive number with lim sk=t,we can sk 1-Holder embedding Ï€k:{0,1}Nâ†'Bk.Further,by the sk1-Holder embedding Ï€k,we construct a sequence of measurable,real-valued functions {fn}n=1∞ defined in X.Finally,combining the results on the dimension of Moran sets,we prove that the setâ–³has Hausdorff dimension t.