An Inversion Algorithm for One-Dimensional F-Expansions

• Published in: The Annals of mathematical statistics Vol. 41; no. 5; pp. 1472 - 1490
• Main Author: Guthery, Scott Bates
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics 01.10.1970; The Institute of Mathematical Statistics
• Subjects: Continued fractions; Ergodic theory; Integers; Lebesgue measures; Markov processes; Mathematics; Perceptron convergence procedure; Real numbers; Stochastic processes; Terminology
• ISSN: 0003-4851; 2168-8990; 2168-8990
• DOI: 10.1214/aoms/1177696793

If f is a monotone function subject to certain restrictions and φ its inverse, then one can associate with any x, a real number between zero and one, a sequence { an} of integers such that x = f(a1+ f(a2+ f(a3+ f(a4+ ⋯. If T is the transformation$\langle\varphi(x)\rangle$where$\langle\rangle$stands for the fractional part, it has been shown that there is a unique measure μ invariant under T which is absolutely continuous with respect to Lebesgue measure. Examples are f(x) = x/10 which gives rise to the decimal expansion with invariant measure Lebesgue measure, or f(x) = 1/x which gives rise to the continued fraction, with measure dx/ln 2(1 + x). This induces a measure P on the sequences { an} which is stationary ergodic and has other interesting properties. However, a large class of pairs {f, μ} gives rise to the pair {{ an}, P}. The paper is concerned with the problem of how, given a measure μ to find, when possible, and f, which corresponds to a pair {{an}, P}, or given an { f, μ} pair, to reduce it to a canonical form. Interesting observations about the "memory" of the process arise from the "canonical form".