On The Study Of The Pointwise Densities Of The Cantor Measure

This dissertation consists of four parts:Chapter One Introduction:Chapter Two Preliminaries; Chapter Three The pointwise densities of Cantor measure (?) and Chap-ter Four The pointwise densities of Cantor measure (?).In Chapter Three, for a given symmetrical Cantor set. under certain conditions, we obtain the explicit formulae of pointwise densities of the Cantor measure on it. Moreover, we discuss a key quantity in these formulas. Our results generalize the known results.Let K(a) be the symmetrical Cantor set generated by?0(x)= ax and?1(x)= ax+(1-a), where 0< a< 1/2. Let s be the Hausdorff dimension of K(a) and?the Cantor measure.Let (?)*s(?,x) and (?)*s(?,x) denote the lower and upper s-densities at x?K(a) respectively. And let (?)s(?,x) be s-density at x?K(a).For the above Cantor measure?, the following results are found in [8,24,25,34]: (1) For every (2) There exist constants 0< d< d?1, such thatFeng, Hua and Wen [12] obtained the explicit formulae of (?)*s(?,x) and (?)*s(?,x) for every point x?K(a), where a?(0,1/3]. Moreover, they proved that?(x) = 0 for?-a.e. x?K(a), where?(x) is a key quantity in these formulae. Their results make the above conclusions (1) and (2) precise.In this thesis, under the hypothesis that we obtain the explicit formulas of the upper and lower s-densities (?)*s(?,x) and (?)*s(?,x) for every point x?K(a). Moreover, we characterize a key quantity (?)(x) in these formulae.In Chapter Four, we investigate a general case for the symmetrical Cantor set K(a). We propose a conjecture about the explicit formulae of s-upper densities (?)*s(?,x) for every point x?K(a), where 0< a< 1/2. Moreover, we give the proof under the additional condition thatOur results generalize and enrich the results in [12,22,26].