Geodesic and Horocyclic Trajectories (Universitext)
Language: English
Pages: 176
ISBN: 085729072X
Format: PDF / Kindle (mobi) / ePub
Geodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature -1, namely the geodesic and horocycle flows. Written primarily with the idea of highlighting, in a relatively elementary framework, the existence of gateways between some mathematical fields, and the advantages of using them, historical aspects of this field are not addressed and most of the references are reserved until the end of each chapter in the Comments section. Topics within the text cover geometry, and examples, of Fuchsian groups; topological dynamics of the geodesic flow; Schottky groups; the Lorentzian point of view and Trajectories and Diophantine approximations.
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disjoint. Moreover one can assume that the horodisks (γ(H + (x)))γ∈Γ x∈Lp (Γ )∩Dz (Γ )(∞) are pairwise either disjoint or identical. To prove this, let x and y in Lp (Γ ) ∩ Dz (Γ )(∞), with y not in Γ (x). Consider the set A of γ ∈ Γ such that γH + (x) ∩ H + (y) = ∅. Let B denote the set of two-sided cosets Γy \A/Γx (i.e., equivalence classes via two relations). If B is finite, it suffices to replace H + (x) with a smaller horodisk than H + (x), for which γH + (x) ∩ H + (y) = ∅ for all γ ∈ Γ .
It follows that nk nk+1 −1 γ = gk T(−1) are both odd or even, then (k+1) · · · T(−1)k gk . If k and k −n −1 + the sequence F (gk−1 (γ + )) is periodic. Otherwise, F (T(−1)k+1 (k+1) gk (γ )) is periodic. In both cases, the sequence F (γ + ) is almost periodic. (ii) Let γ be a hyperbolic isometry in PSL(2, Z). After conjugating γ by a translation, one may suppose that γ + > 0. According to the end of the nk n if k and k are both proof of part (i), γ is conjugate to T1 k+1 · · · T(−1) k 4
that the group G acts by isometries on H, if we replace the global Euclidean scalar product , by the scalar product gz depending on each z ∈ H and defined on each tangent plane Tz H by: → → u,− v)= g z (− 1 → − → u,− v . Im z 2 The family of (gz )z∈H defines a Riemannian metric on H, called the hyperbolic metric. Throughout this text, we consider H equipped with the metric (gz )z∈H and call it the Poincar´e half-plane. By construction the angles defined by this metric are the same as the Euclidean
3 of the sequence associated to An+1 contain the sequence associated to An , which proves the property for p = n + 1. Take some p n + 1. Assume now that the property is true for this p and let us show that it is true up to p + 1. Consider Ap+1 = mp g2 Ap g2 Ap g2 Ap g2 mp . Choose a block B of length 6 n + 3 of the sequence associated to Ap+1 . If B is a block of Ap or of mp , the induction hypothesis applies. Otherwise B is a block of the sequence associated to one of the following words wi for
leads us to define for each irrational number x the quantity ν(x) = inf c > 0 | ∃ (pn /qn )n |x − pn /qn | 1 ∈ Q, c/qn2 and lim |qn | = +∞ . n→+∞ By Theorem 3.1, this quantity is less than 1/2 for all x. The following theorem is more precise. It can be proved, for example by associating a sequence of circles to the sequence of rational numbers given by the continued fraction expansion, and by studying their relative positions [52, Chap. 6, Theorem 6.25] (see also [30]). Theorem 3.3. For
