Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem

TL;DR

This paper proves the ergodic action of positive rationals on finite adeles for certain parameters and establishes the uniqueness of KMS states in the Bost-Connes system, linking ergodic theory and number theory.

Contribution

It demonstrates ergodicity of rational actions on finite adeles for eta in (0,1] and derives the uniqueness of KMS states in the Bost-Connes phase transition.

Findings

Positive rationals act ergodically on finite adeles for eta in (0,1].

Uniqueness of KMS_eta-states for the Bost-Connes system is established.

Connects ergodic theory with phase transition phenomena in number theory.

Abstract

For each \beta\in(0,+\infty) there exists a canonical measure \mu_\beta on the ring A_f of finite adeles. We show that the positive rationals act ergodically on (A_f,\mu_\beta) for \beta\in(0,1], and then deduce from this the uniqueness of KMS_\beta-states for the Bost-Connes system.