Representations for real numbers and their ergodic properties
Representations for real numbers and their ergodic properties
复制标题
DOI：
10.1007/bf02020331
复制
发表时间：
1957-09
期刊：
影响因子：
--
通讯作者：
A. Rényi
中科院分区：
文献类型：
--
作者：
A. Rényi
我们将考虑实数л‘由正函数у的无限迭代表示-f(X)以“/-展开”(1)x-s0+/(/L+/(*S+/(**H)···)的形式表示，其中“数字”í“=”(X)(n 0，1，…))而“余数”(2)r“(X)=/(é”+il-/(S“+2+/(S K+si-)···)(n 0，1，...)(n 0，1，...)由下列递归关系定义：*o(X)=[x]，Гп(х)(X)，*n+i(X)=L/)(r”(X))]，г11+1(х)=(г/>；(г“(х)(n 0，1，...)其中[г]表示实数z的整数部分和(г)，xy(Y)是y=f(X)的反函数。在§1中，我们将研究施加在函数f(X)上的条件是充分的，以确保每个实数x应该有一个/-展开式(1)形式的表示。1
We shall consider representations of a real number л'by infinite iteration of a positive function у-----f (x) in the form of the “/-expansion”(1) x—S0+/(/l+/(* s+/(** H)•••) where the “digits” í „= é „(x)(n 0, 1,...) and the “remainders”(2) r „(x)=/(é „+ il-/(s „+ 2+/(s K+ si----)•••)(n 0, 1,...} are defined by the following recursive relations:* o (x)=[x], Гп (х)(x),* n+ i (x)= L/)(r „(x))], г11+ 1 (х)=(г/>(г „(х)))(n 0, 1,...) where [г] denotes the integral part and (г) the fractional part of the real number z and xy (y) is the inverse function of y= f (x). In § 1 we shall investigate what conditions imposed on the function f (x) are sufficient to ensure that every real number x should have a representation in the form of the/-expansion (1). 1