0000000000345526
AUTHOR
Anne Broise-alamichel
Bruhat–Tits Trees and Modular Groups
In this chapter, we give background information and preliminary results on the main link between the geometry and the algebra used for our arithmetic applications: the (discrete-time) geodesic ow on quotients of Bruhat{Tits trees by arithmetic lattices.
Equidistribution of Common Perpendicular Arcs
In this chapter, we prove the equidistribution of the initial and terminal vectors of common perpendiculars of convex subsets, at the universal covering space level, for Riemannian manifolds and for metric and simplicial trees.
Équidistribution non archimédienne et actions de groupes sur les arbres = Non-Archimedean equidistribution and group actions on trees
We give equidistribution results of elements of function fields over finite fields, and of quadratic irrationals over these fields, in their completed local fields. We deduce these results from equidistribution theorems of common perpendiculars in quotients of trees by lattices in their automorphism groups, proved by using ergodic properties of the discrete geodesic flow. Nous donnons des résultats d'équidistribution d'éléments de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux complétés. Nous déduisons ces résultats de théorèmes d'équidistribution de perpendiculaires communes dans des quotients d'arbres par des réseaux de leur g…
Equidistribution of Equidistant Level Sets to Gibbs Measures
Before stating this equidistribution result, we begin with a technical construction that will also be useful in the following chapter.
Negatively Curved Geometry
Let X be a geodesically complete proper CAT(–1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X.
Rate of Mixing for the Geodesic Flow
The main part of the chapter then consists in proving analogous bounds for the discrete-time and continuous-time geodesic ow for quotient spaces of simplicial and metric trees respectively.
Equidistribution and Counting of Quadratic Irrational Points in Non-Archimedean Local Fields
We use these results to deduce equidistribution and counting results of quadratic irrational elements in non-Archimedean local fields.
Symbolic Dynamics of Geodesic Flows on Trees
In this chapter, we give a coding of the discrete-time geodesic ow on the nonwandering sets of quotients of locally finite simplicial trees X without terminal vertices by nonelementary discrete subgroups of Aut(X) by a subshift of finite type on a countable alphabet.
Equidistribution and Counting of Cross-ratios
The following properties of relative heights are easy to check using the definitions, the invariance properties of the cross-ratio, and Equation (17.1).
Potentials, Critical Exponents,and Gibbs Cocycles
Let X be a geodesically complete proper CAT(–1) space, let x0 ∈ X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X.
Explicit Measure Computations for Simplicial Trees and Graphs of Groups
In this chapter, we compute skinning measures and Bowen{Margulis measures for some highly symmetric simplicial trees X endowed with a nonelementary discrete subgroup Г of Aut(X).
Rate of Mixing for Equilibrium States in Negative Curvature and Trees
In this survey based on the recent book by the three authors, 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, emphasizing 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) grow…
Équidistribution non archimédienne et actions de groupes sur les arbres
Resume Nous donnons des resultats d'equidistribution d'elements de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux completes. Nous deduisons ces resultats de theoremes d'equidistribution de perpendiculaires communes dans des quotients d'arbres par des reseaux de leur groupe d'automorphismes, demontres a l'aide de proprietes ergodiques du flot geodesique discret.
Equidistribution and Counting of Integral Representations by Quadratic Norm Forms
In the final chapter of this book, we give another arithmetic equidistribution and counting result of rational elements in non-Archimedean local fields of positive characteristic, again using our geometric equidistribution and counting results of common perpendiculars in trees summarised in Section 15.4.
Equidistribution and Counting of Rational Points in Completed Function Fields
Let K be a (global) function field over Fq of genus g, let v be a (normalised discrete) valuation of K, let Kv be the associated completion of K, and let Rv be the affine function ring associated with v.
Patterson–Sullivan and Bowen–Margulis Measures with Potential on CAT(–1) Spaces
In this chapter, we discuss geometrically and dynamically relevant measures on the boundary at infinity of X and on the space of geodesic lines gX.
Skinning Measures with Potential on CAT(–1) Spaces
In this chapter, we introduce skinning measures as weighted pushforwards of the Patterson{Sullivan densities associated with a potential to the unit normal bundles of convex subsets of a CAT(–1) space.
Equidistribution and Counting of Common Perpendiculars in Quotient Spaces
In this chapter, we use the results of Chapter 11 to prove equidistribution and counting results in Riemannian manifolds (or good orbifolds) and in metric and simplicial graphs (of groups).
Random Walks on Weighted Graphs of Groups
Let X be a locally finite simplicial tree without terminal vertices, and let X = ∣X∣1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut(X).
Fields with Discrete Valuations
In the present chapter, before embarking on our arithmetic applications, we recall basic facts on local fields for the convenience of the geometer reader.