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Escape of mass in homogeneous dynamics in positive characteristic

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  • 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: 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.

    Mathematics Subject Classification: Primary: 20G25, 37A17, 20E08, 22F30; Secondary: 20H20, 20G30, 20C08, 37D40.


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  • Figure 1.  The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$

    Figure 2.  Back and forth paths in cuspidal rays

    Figure 3.  A partition of the Hecke sphere $S_\nu(n)$

    Figure 4.  Rational Bruhat-Tits rays

    Figure 5.  Sector-spheres in Hecke trees

    Figure 6.  Iterated construction of nested sectors

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