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October, 1970 An Inversion Algorithm for One-Dimensional $F$-Expansions
Scott Bates Guthery
Ann. Math. Statist. 41(5): 1472-1490 (October, 1970). DOI: 10.1214/aoms/1177696793


If $f$ is a monotone function subject to certain restrictions and $\varphi$ its inverse, then one can associate with any $x$, a real number between zero and one, a sequence $\{ a_n\}$ of integers such that $x = f(a_1 + f(a_2 + f(a_3 + f(a_4 + \cdots.$ 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 $\mu$ 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 $\{ a_n\}$ which is stationary ergodic and has other interesting properties. However, a large class of pairs $\{f, \mu\}$ gives rise to the pair $\{\{ a_n\}, P\}$. The paper is concerned with the problem of how, given a measure $\mu$ to find, when possible, and $f$, which corresponds to a pair $\{\{a_n\}, P\}$, or given an $\{ f, \mu\}$ pair, to reduce it to a canonical form. Interesting observations about the "memory" of the process arise from the "canonical form".


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Scott Bates Guthery. "An Inversion Algorithm for One-Dimensional $F$-Expansions." Ann. Math. Statist. 41 (5) 1472 - 1490, October, 1970.


Published: October, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0214.43703
MathSciNet: MR276170
Digital Object Identifier: 10.1214/aoms/1177696793

Rights: Copyright © 1970 Institute of Mathematical Statistics


Vol.41 • No. 5 • October, 1970
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