Limit theorems and wrapping transforms in bi-free probability theory

We characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus.The bi-free analogous Lévy triplet of an infinitely divisible distribution on the bi-torus without nontrivial idempotent factors is obtained.This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one.Also, the relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions, are analyzed.The analysis relies on the convergence criteria for limit theorems and the use of push-forward measures induced by the wrapping map from the plane to the bi-torus.