First digit phenomenon and ergodic theory
For any k belongs to I/sup +/ identical with the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = )n: k = (n/10/sup r/) for some r belongs to I/sup +/), where (x) denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = (lim/n ..-->.. infinity) (S intersection (1, N))/N, where S contains or equals I/sup +/ and (S) denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers K, and if the mapping T that maps S into (2S) union (2S + 1) preserves the density d, then the result of Cohen is that d(p) = log/sub 10/ (1 + (1/p)) for p belongs to D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a belongs to R/sup +/ be a positive real number. Let theta (x) = ax, where x belongs R/sup +/, and consider the sequence p = (p/sub 1/, p/sub 2/,...) of first digits of the orbit of theta: )x, theta(x), theta/sup 2/ (x), ...). That is p/sub k/ = (theta /sup k/ (x)/b/sup r/) belongs to D where r = (log/sub 10/ theta/sup k/ (x)). The purpose of this paper is to investigate the dynamical properties of the sequence p. 7 references.
- Research Organization:
- Univ. of California, Santa Barbara, CA (United States)
- DOE Contract Number:
- W-7405-ENG-26
- OSTI ID:
- 5936327
- Journal Information:
- J. Math. Anal. Appl.; (United States), Vol. 95:2
- Country of Publication:
- United States
- Language:
- English
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