# American Institute of Mathematical Sciences

2017, 11: 369-407. doi: 10.3934/jmd.2017015

## Escape of mass in homogeneous dynamics in positive characteristic

 1 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 2 Laboratoire de mathématiques déOrsay, UMR 8628 Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France 3 Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000 Israel

Received  December 20, 2015 Revised  December 05, 2016 Published  May 2017

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.

Citation: Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
##### References:

show all references

##### References:
The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$
Back and forth paths in cuspidal rays
A partition of the Hecke sphere $S_\nu(n)$
Rational Bruhat-Tits rays
Sector-spheres in Hecke trees
Iterated construction of nested sectors
 [1] Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008 [2] Wenxiu Gong, Zuoliang Xu. An alternative tree method for calibration of the local volatility. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020146 [3] Kathryn Dabbs, Michael Kelly, Han Li. Effective equidistribution of translates of maximal horospherical measures in the space of lattices. Journal of Modern Dynamics, 2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229 [4] Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443 [5] Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511 [6] Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237 [7] Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209 [8] Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1 [9] Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83 [10] Frédéric Bernicot, Bertrand Maury, Delphine Salort. A 2-adic approach of the human respiratory tree. Networks & Heterogeneous Media, 2010, 5 (3) : 405-422. doi: 10.3934/nhm.2010.5.405 [11] Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021183 [12] Jean-Baptiste Bardet, Bastien Fernandez. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 669-684. doi: 10.3934/dcds.2011.31.669 [13] Sergei Avdonin, Yuanyuan Zhao. Leaf Peeling method for the wave equation on metric tree graphs. Inverse Problems & Imaging, 2021, 15 (2) : 185-199. doi: 10.3934/ipi.2020060 [14] Sergei Avdonin, Jonathan Bell. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems & Imaging, 2015, 9 (3) : 645-659. doi: 10.3934/ipi.2015.9.645 [15] Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017 [16] Alberto Bressan, Michele Palladino. Well-posedness of a model for the growth of tree stems and vines. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2047-2064. doi: 10.3934/dcds.2018083 [17] Reuven Cohen, Mira Gonen, Avishai Wool. Bounding the bias of tree-like sampling in IP topologies. Networks & Heterogeneous Media, 2008, 3 (2) : 323-332. doi: 10.3934/nhm.2008.3.323 [18] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2021, 16 (1) : 1-29. doi: 10.3934/nhm.2020031 [19] Wael Bahsoun, Christopher Bose. Quasi-invariant measures, escape rates and the effect of the hole. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1107-1121. doi: 10.3934/dcds.2010.27.1107 [20] Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227

2020 Impact Factor: 0.848