Escape of mass in homogeneous dynamics in positive characteristic

TL;DR

The paper reveals a striking contrast between positive and zero characteristic dynamics for $A_\infty$-periodic measures on the space of $2$-lattices as one moves along Hecke rays. By embedding the problem in Bruhat–Tits trees and Hecke trees, the authors prove linear growth of $A_\infty$-orbits, establish uniform escape of mass along all rational rays, and show that uncountably many rays induce escape or accumulation on measures not absolutely continuous to the homogeneous one. A key innovation is an effective equidistribution for sector-spheres, derived from exponential decay of matrix coefficients, which underpins the exotic mass-accumulation phenomena along Hecke rays. Overall, the results highlight deep arithmetic- dynamical differences in function fields and open questions about genericity and the structure of escaping sets in positive characteristic.

Abstract

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of 2-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue; that is, that for a large set of rays the measures fail to converge to the uniform probability measure on the space of 2-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of 2-lattices.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 5
• Conjecture 6
• Lemma 7
• Proposition 8
• Remark 9
• Lemma 10