2022/06/27 by Yunping Jiang, Jiang, Yunping
Mathematics · #28D05 #37E05 #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Primary 37A05 #Secondary 28D20
paper · pdf · doi:10.48550/arxiv.2206.13569
openalex publication_date 2022/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore the geometric aspects of the Furstenberg conjecture, proving that a non-atomic probability measure on the unit circle, invariant under both p- and q-actions for coprime integers p,q>1, must be the Lebesgue measure if it exhibits balanced geometry for one of these actions. Within rigidity theory, we show that balanced geometry is equivalent to the Lipschitz property. A consequence is that an orientation-preserving homeomorphism of the circle conjugating both p- and q-actions and preserving the Lebesgue measure must be the identity if one of these conjugations satisfies the Lipschitz property. Our approach does not rely solely on ergodicity, and we conclude by proposing conjectures and open problems that frame the Furstenberg conjecture through geometric and quasisymmetric perspectives.