Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Abstract

Let Γ be some discrete subgroup of SO o (n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO o (n + 1, R) = KAN . We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.Definition. Let µ be some (Borel) probability measure on R n (n ≥ 2). Assume that µ has exact dimension δ. We say that µ is regular if for any m-plane V in R n (1 ≤ m ≤ n − 1) the orthogonal projection of µ onto V has dimension inf{δ, m} almost everywhere.Obviously, if this is the case, then the orthogonal projection of µ onto V is in fact exact dimensional.Theorem B (Theorem 5.2). Let Γ be a discrete non-elementary subgroup of G = SO o (1, n + 1). Assume that Γ is Zariski-dense and has finite BMS measure. Let µ be the Patterson-Sullivan measure (of exponent δΓ) associated with Γ. For µ-almost every ξ ∈ ∂H n+1 , the push-forward of µ through the inverse stereographic mapping ∂H n+1 \ {ξ} → R n is a regular measure.The proof of Theorem B relies on results of Hillel Furstenberg, Pablo Shmerkin and Michael Hochman.

The case of equality in the dichotomy of Mohammadi–Oh

“…converges weakly to P (recall equation (2)) as T → +∞. This set has full BMS measure ( [4], Lemma 5.4). Now fix some x0 ∈ Ω such that for σ(x0)-almost every h ∈ N , x0h ∈ Ω (see Lemma 2).…”

“…Recall that we assume that Γ is Zariski-dense and has finite BMS measure, so that P is an Ergodic Fractal Distribution (EFD) in the sense of Hochman (see [5], Definition 1.2, and [4], Lemma 5.3 for a proof that P is indeed an EFD).…”

“…on the space of Radon measures on N/U . It is straight-forward to check that P T is an Ergodic Fractal Distribution (see [4], Lemma 5.3). Since P T has dimension n − m (see [4], Theorem 4.1) this is possible only if P T = Dirac(Haar N/U )…”

“…dissipative [4] recurrent and ergodic [9] BR totally dissipative if Γ convexcocompact [9] recurrent if Γ convex-cocompact recurrent and ergodic [9] Note that it follows immediately from the definitions that if the BMS measure is recurrent, so is the BR measure. The other implications are not so obvious.…”

Radial projections along chains

“…As in [5], we introduce the distribution on M * P = dm(x)Dirac(σ(x) * ) and this is again a fractal distribution, ergodic with respect to St for any t = 0. In [5] we deal with the setting of the real hyperbolic space; the proof is identical in the complex hyperbolic case. Let us make the distribution P more explicit.…”