TENT PROPERTY AND DIRECTIONAL LIMIT SETS FOR SELF-JOININGS OF HYPERBOLIC MANIFOLDS

(1) Let Γ = (ρ1 × ρ2)(∆) where ρ1, ρ2 : ∆ → SO (n, 1) are convex cocompact representations of a finitely generated group ∆. We provide a sharp pointwise bound on the growth indicator function ψΓ by a tent function: for any v = (v1, v2) ∈ R , ψΓ(v) ≤ min(v1 dimH Λρ1 , v2 dimH Λρ2). (0.1) We obtain several interesting consequences including the gap and rigidity property on the critical exponent. (2) For each v in the interior of the limit cone of Γ, we prove the following on the v-directional conical limit set Λv ⊂ S n−1 × S : ψΓ(v) max (v1,v2) ≤ dimH Λv ≤ ψΓ(v) min (v1,v2) . We also study the local behavior of the higher rank PattersonSullivan measures on each Λv. (3) Our approach gives an analogous result as (0.1) for any Anosov subgroup of a semisimple Lie group of rank at most 3. For instance, our result implies that ψΓ(diag(t1, · · · , td)) ≤ min1≤i≤d−1(ti − ti+1) for any Hitchin representation Γ < PSL(d,R) with d ≤ 4.