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

摘要： 基于城市等级体系的分形递归模型P_m=P₁r_p^1-m，f_m=f_1f^m-1及其等价形式三参数Zipf定律P(r)=C(r-α)^-d_z 提出关于城市位序规模分布的多分形模型，得到多分维谱的二标度表达形式D_q=ln[p^q+(1-p)^q/(1-q)lnr_f]，这里p=P(2)/[P(2)+P(3)]为概率尺度(括号中的数字表示城市的位序，P为对应城市的人口)；然后借助Legendre变换给出相应的参量表达，包括质量指数 τ(q)、关于质量的Lipschitz-Holder指数α(q)以及指数支集的分维函数f(α)。导出关于城市体系人口分布的空间维数谱模型D_p(q)=D_qD_f以及有关的参量表达式，实现了区域城市人口多分维的可计算性。多重Zipf维数模型不仅可以有效地统一中心地的等级阶梯与位序-规模法则反映的连续分布，而且可以揭示城市体系演化的更多信息和隐含法则。以美国城市体系(1998年的数据)为实证对象，给出了城市规模分布的多分维D_p以及f(α)曲线等部分数值和图谱。

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