关键词: 位序-规模法则, Zipf定律, 空间结构, 双分形, 多分形, 对称性, 美国城市

Abstract: Urban hierarchy as a result of some fractal recursion is generalized to the binomia l multiplicative process and accordingly city-size distributions can be describ ed by multifractal measures. Based on the generalized Beckmann-Davis model advan ced by the authors P_m=P₁r_p^1-m，f_m=f_1f^m-1，where r_f=f_m+1/f_m denotes number ratio of cities in the mth and (m+1)th level of central-place hierarchies, the spectrum of fractal dimensions D_q as a function of the moment order q of city class-size relationships is expressed asD_q=p^q+(1-p)^q］/［(1-q)lnr_f］,where p=P(2)/［P(2)+P(3)］,(P(2) is the population of rank 2,etc.), thus the mass exponent can be given by the formula, τ(q)=(1-q)D_q. By means of the Legendre transformation, the Lipschitz-Holder exponent α(q) for the mass can be derived as α(q)=［p^qlnp+(1-p)^qln(1-p)］/{［p^q+(1-p)^q］ln(1/r^f)}, correspondingly, the fractal dimension of the set supporting this exponent will yield through the equation f(α)=qα(q)+τ(q). An empirical analysis is made with the population data of the US cities to verify the theory and models developed in this paper, which will contribute to reconcile the apparent difference between the hierarchical step-like frequency distribution of city sizes suggested by central place theory and the smooth curve reflected by the work on the rank-size rule.

Key words: rank-size rule, Zipf's law, city-size distributions, bi-fractals, multifracta ls, symmetry, the US cities