## An inversion algorithm for one-dimensional $$f$$-expansions.(English)Zbl 0214.43703

Summary: 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 $$\langle\cdot\rangle$$ 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 $$dx/\ln2(1+x)$$. 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”.

### MSC:

 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A99 Classical measure theory 60B99 Probability theory on algebraic and topological structures
Full Text: