Ergodic Theory Meets Polarization. II: A Foundation of Polarization Theory

Understand

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Ar{\i}kan style constructions.

• This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing.
• This (second) part provides a foundation of polarization theory based on the ergodic theory of binary operations which we developed in the first part.
• We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly ergodic.