A quadrature mirror filter (QMF) function can be considered as the transition function for a Markov process on the unit interval. The QMF functions that generate scaling functions for multiresolution analyses are then distinguished by properties of their invariant sets. By characterizing these sets, we answer in the affirmative a question raised by Gundy.
"Invariant sets for QMF functions." Rocky Mountain J. Math. 48 (8) 2559 - 2571, 2018. https://doi.org/10.1216/RMJ-2018-48-8-2559