Abstract
Let $d \geq 2$, and let $\Gamma \subset PSL(2, \mathbb{R})^d$ be an irreducible, cocompact lattice. We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $\Gamma \backslash PSL(2, \mathbb{R})^d$.
Highlights
Hyperbolicity has been studied since the 1960’s and shortly thereafter, the need to relax its definition became apparent, so partial hyperbolicity was introduced, [2], [3]
A one-parameter family of diffeomorphismst∈R on a manifold M is a partially hyperbolic flow if for every x ∈ M, there is an invariant splitting of the tangent bundle TxM = Es(x) ⊕ Ec(x) ⊕ Eu(x) and there are constants C > 0 and 0 < λ1 ≤ μ1 < λ2 ≤ μ2 < λ3 ≤ μ3 with μ1 < 1 < λ3 such that for every t ∈ R+, C−1λt1 v ≤ Dftv ≤ Cμt1 v for v ∈ Es(x), C−1λt2 v ≤ Dftv ≤ Cμt2 v for v ∈ Ec(x), C−1λt3 v ≤ Dftv ≤ Cμt3 v, for v ∈ Eu(x)
Dolgopyat proved various limit theorems for partially hyperbolic flows under some effective mixing properties of the flow in [7]
Summary
Hyperbolicity has been studied since the 1960’s and shortly thereafter, the need to relax its definition became apparent, so partial hyperbolicity was introduced, [2], [3]. The main point in the argument is that the invariant distributions for coordinate horocycle flows are defined in unitary Sobolev representations of PSL(2, R) on L2(Γ\ PSL(2, R)d), so they are already well-understood by the work of Flaminio and Forni on the equidistribution of horocycle flows (see Theorem 1.1 and Theorem 1.4 of [9]). Using a recent result of Kelmer-Sarnak on the strong spectral gap property of irreducible, cocompact lattices in PSL(2, R)d, we estimate the contribution of these invariant distributions to the ergodic integral of a given coordinate horocycle flow by iteratively applying the relevant coordinate geodesic map, which is a method of Flaminio and Forni in [9]. By (5), the classification of Ui-invariant distributions in I(M ) into irreducible, unitary representation spaces is given by the corresponding classification of horocycle flow-invariant distributions from Theorem 1.1 of [9].
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