Division point measures resulting from triangle subdivisions
If the unit interval \([0,1]\) is divided into \([0,\alpha]\) and \([\alpha,1]\) and then subdivided iteratively in the same proportions, then it is well-known that the weak*-limit measure of the uniform measure on the division points is mutually singular to Lebesgue if \(\alpha\neq\frac12\). \textit{S. Kakutani} [Lect. Notes Math. 541, 369--375 (1976; Zbl 0363.60023)] showed that dividing at each stage only those intervals of greatest length in the same proportions gives convergence to Lebesgue measure for any \(\alpha\in(0,1)\). Here natural analogues of these results are found for certain division schemes of a right triangle.
scientific article; zbMATH DE number 6050463
scientific article