An Inversion Algorithm for One-Dimensional $F$-Expansions

Abstract

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".