On Hausdorff Dimension Of Self-affine Sets

In this paper,we discuss the Hausdorff and box dimensions of self-affine sets in Euclidean spaces under various separation conditions.We prove that Falconer's lower bound estimate of Hausdorff dimension is valid with the strong separation condition replaced by a certain open set condition.The paper is divided into three parts.In the first part,we present the background of the problem and its significance.Then we give some basic concepts,notations and known lemmas needed for the study.In particular,we recall the definition and some basic properties of the singular value function of a nonsingular linear contracting mapping.In the second part,we review Falconer' s results on the Hausdorff and box dimen-sions of self-affine sets.In 1988,Falconer showed that,for self-affine sets with the open set condition,their Hausdorff and box dimensions coincide for almost all trans-lations.In 1992,using the singular value function,he defined a measure of Hausdorff type on the symbolic space,by which he obtained a lower bound of Hausdorff dimen-sion for self-affine sets with the strong separation condition.Regretfully,this lower bound is usually not equal to the previous dimensional value.Thus,the computation of Hausdorff dimension of self-affine sets remains a difficult question,even if the strong separation condition is satisfied.In addition,for self-affine sets with the open set con-dition and a certain projection property he determined the exact values of their box dimensions.In the third part,we introduce,for self-affine sets F,a condition as follows:F satisfies the open set condition for an open set U with the property that every x ? F has an open half ball S(x,a)lying in U.Replacing the strong separation condition by this condition,we prove that Falconer's lower bound for Hausdorff dimension of self-affine sets remains to be true.