Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations

An open problem in polarization theory is to determine the bi nary operations that always lead to polarization when they are used in Arıkan-like constructions. This paper , which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binar y operation to be polarizing. This (first) part of the paper introduces the mathematical framework that we will us e in the second part [1] to characterize the polarizing operations: uniformity-preserving, irreducible, ergodi c and strongly ergodic operations are defined. The concepts of a stable partition and the residue of a stable partition ar e introduced. We show that an ergodic operation is strongly ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly ergodic if and only if all the operations in the se qu nce are strongly ergodic. In the second part of the paper, we provide a foundation of polarization theory based on the ergodic theory of binary operations that we develop in this part.