Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

Wave Computation on the Hyperbolic Double Doughnut

“…We know, see, e.g., [2,3,6], that K is a two dimensional C ∞ compact manifold, without boundary, its sectional curvature is constant, equal to −1, and its genus, that is the number of "holes", is 2. The geodesic flow is very chaotic : it is ergodic, mixing (theorems by G.Hedlung, E. Hopf), Anosov and Bernouillian (theorems by D. Ornstein, B. Weiss).…”

“…The geodesic flow is very chaotic : it is ergodic, mixing (theorems by G.Hedlung, E. Hopf), Anosov and Bernouillian (theorems by D. Ornstein, B. Weiss). As regards as these properties of the geodesics flow, we can refer to [2], [3], [6].…”

Bounding hyperbolic and spherical coefficients of Maass forms

“…A straightforward computation shows that this is O (1). Similarly, the upper bounds in the remaining parts of Lemma 4 are unchanged.…”

“…Fix the standard identification of the unit tangent bundle T 1 (H) of H with PSL 2 (R) by associating the point (i, ↑) ∈ T 1 (H) with the identity element in PSL 2 (R). Then the geodesic flow on T 1 (H) is given by the action of A, the group consisting of (classes of) diagonal matrices (see [1,II. §3]).…”

Multi-dimensional metric approximation by primitive points

“…It follows from Moore's ergodicity theorem (see [15], Theorem 5 and Proposition 6; see also [1]) that the action of the lattice SL(ν j , Z) ⊂ SL(ν j , R) on Ω n,ν j is ergodic, since the latter can be realised as an homogeneous space Ω n,ν j = L j \SL(ν j , R) where L j is the stabiliser of a point in Ω n,ν j , and L j is a non-compact closed subgroup. Thus the action of Γ π on the product Ω is as well ergodic.…”