Doubly stochastic operators with zero entropy

TL;DR

This paper investigates doubly stochastic operators with zero entropy, extending key theorems to better understand their spectral properties and genericity within dynamical systems.

Contribution

It generalizes three fundamental theorems—Rokhlin's, Kushnirenko's, and Halmos-von Neumann's—to the context of doubly stochastic operators with zero entropy.

Findings

Generalization of Rokhlin's theorem on zero entropy genericity

Extension of Kushnirenko's theorem on discrete spectrum equivalence

Representation of zero entropy operators as group rotations

Abstract

We study doubly stochastic operators with zero entropy. We generalize three famous theorems: the Rokhlin's theorem on genericity of zero entropy, the Kushnirenko's theorem on equivalence of discrete spectrum and nullity and the Halmos-von Neumann's theorem on representation of maps with discrete spectrum as group rotations.