Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Abstract

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.

On the nonarchimedean quadratic Lagrange spectra

“…In this section, we recall the basic notations and properties of function fields K over F q and their valuations v, the associated Bruhat-Tits trees T v and modular groups Γ v acting on T v . We refer to [Gos, Ros, Ser] for definitions, proofs and further information, see also [BPP,Ch. 14 and 15].…”

“…the union of the orbits of α and α σ under the projective action of Γ, with Θ α = Θ α, Γv . Note that α σ = α, since an irreducible quadratic polynomial over K which is inseparable does not split over K v (see for instance [BPP,Lem. 17.2]), and that there exists a loxodromic element γ α ∈ Γ v such that ]α, α σ [ = Ax γα (see for instance [BPP,Prop.…”

“…where t → y t , t → z t are the geodesic lines starting from ∞, through ∂H at time t = 0, ending at the points at infinity y, z respectively. By for instance [BPP,Eq. (15.2)], we have |y − z| v = d H∞ (y, z) ln qv .…”

“…Now let us fix α ∈ K (2) . We denote by α σ ∈ K (2) the Galois conjugate of α over K. The complexity h(α) = 1 |α−α σ | of α was introduced in [HP] and developped in [BPP,§17.2]. It plays the role of the (naive) height of a rational number in Diophantine approximation by rationals, and is an appropriate complexity when studying the approximation by elements in the orbit under the modular group of a given quadratic irrational.…”

“…3 We refer to the above references for motivations and results, in particular to [HP,Thm. 1.6] for a Khintchine type result and to [BPP,§17.2] for an equidistribution result of the orbit of α under PGL 2 (R).…”