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

Guthery, Scott Bates

Ann. Math. Stat. 41, 1472-1490 (1970).

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