Équidistribution non archimédienne et actions de groupes sur les arbres

Broise-Alamichel, Anne; Parkkonen, Jouni; Paulin, Frédéric

doi:10.1016/j.crma.2016.08.001

Cited by 2 publications

References 6 publications

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Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Broise-Alamichel¹,

Parkkonen²,

Paulin³

2019

Progress in Mathematics

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In this survey based on the book by the authors [BPP], we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphazising the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) growth type. 1 1 A Patterson-Sullivan construction of equilibrium states We refer to [PPS, Chap. 3, 6, 7] and [BPP, Chap. 2, 3, 4] for details and complements on this section. Let X be (see [BPP] for a more general framework) ‚ either a complete, simply connected Riemannian manifold Ă M with dimension m at least 2 and pinched sectional curvature at most´1, ‚ or (the geometric realisation of) a simplicial tree X whose vertex degrees are uniformly bounded and at least 3. In this case, we respectively denote by EX and V X the sets of vertices and edges of X. For every edge e, we denote by opeq, tpeq, e its original vertex, terminal vertex and opposite edge. Let us fix an indifferent basepoint x˚in Ă M or in V X. Recall (see for instance [BH]) that a geodesic ray or line in X is an isometric map from r0,`8r or R respectively into X, that two geodesic rays are asymptotic if they stay at bounded distance one from the other, and that the boundary at infinity of X is the space B 8 X of asymptotic classes of geodesic rays in X endowed with the quotient topology of the compact-open topology. When X " Ă M , up to a translation factor, two asymptotic geodesic rays converge exponentially fast one to the other, and B 8 Ă M is homeomorphic to the sphere S m´1 of dimension m´1. When X is a tree, up to a translation factor, two asymptotic geodesic rays coincide after a certain time, and B 8 Ă M is homeomorphic to a Cantor set.

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Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Broise-Alamichel¹,

Parkkonen²,

Paulin³

2019

Progress in Mathematics

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Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

Ciobotaru

Finkelshtein

Sert

2019

International Mathematics Research Notices

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We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

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Équidistribution non archimédienne et actions de groupes sur les arbres

Cited by 2 publications

References 6 publications

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

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