Geodesic and Horocyclic Trajectories offers an creation to the topological dynamics of 2 classical flows linked to surfaces of curvature -1, particularly the geodesic and horocycle flows. Written essentially with the assumption of highlighting, in a comparatively trouble-free framework, the life of gateways among a few mathematical fields, and some great benefits of utilizing them, historic elements of this box usually are not addressed and lots of the references are reserved till the top of every bankruptcy within the reviews part. themes in the textual content hide geometry, and examples, of Fuchsian teams; topological dynamics of the geodesic move; Schottky teams; the Lorentzian viewpoint and Trajectories and Diophantine approximations.
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We elect S(g1 , g2 ) suﬃciently “small” in order that: L(Γ ) = L(S(g1 , g2 )). It follows from estate three. three, that the set P (Ωg (T 1 S0 )) is a formal compact subset of Ωg (T 1 S) that is invariant with appreciate to the geodesic ﬂow. Take the picture through P of a geodesic trajectory on Ωg (T 1 S0 ), that's neither periodic nor dense (Proposition three. 1). This snapshot is a geodesic trajectory which now not dense in Ωg (T 1 S) and never periodic (Lemma three. 5). We concentration now at the lifestyles of minimum compact units that are invariant with appreciate to the geodesic ﬂow on Ωg (T 1 S).
Allow f be in O0 (2, 1) such that y = f (y0 ) is a course of D. One has γn (y) = γn f (y0 ) , as a result limn→+∞ γn (y) = zero if and provided that limn→+∞ BD0 (x0 , f −1 γn−1 (x0 )) = +∞. The equivalence partially (i) can then be deduced from the relation (∗) BD0 (x0 , f −1 γn−1 (x0 )) = BD (f (x0 ), x0 ) + BD (x0 , γn−1 (x0 )). (ii) back allow us to start with the case the place D = D0 . via workout 1. 5(iii), a metamorphosis f in O0 (2, 1) may be decomposed into kt at kt with kt , kt ∈ ok and at ∈ A. One has dL (x0 , f (x0 )) = dL (x0 , at (x0 )) and dL (x0 , at (x0 )) = |t |.
U )) in T 1 S be a periodic aspect for the geodesic ﬂow. ApplyLet π 1 ((z, − ing Proposition three. 1, we receive a hyperbolic isometry γ ∈ Γ ﬁxing the issues u(+∞) and u(−∞) such that t = d(z, γ(z)). on the grounds that Γ is discrete, the subgroup of hyperbolic isometries ﬁxing u(+∞) and u(−∞) is generated by means of one primitive point γ0 = identification (i. e. , there isn't any isometry h in Γ enjoyable hn = γ0 for n > 1). It follows that the set of genuine numbers t such that → → u ))) = π 1 ((z, − u )), that's a closed subgroup of (R, +), is the set gt (π 1 ((z, − → → u ))) = π 1 ((z, − u )), Z (γ0 ).
Because the staff Γ is geometrically ﬁnite, Corollary 2. eleven means that a few u→ compact subset of T 1 S is intersected by means of all trajectories gR (π 1 ((zn , − n ))). − → It follows that once conjugating γn , exchanging (zn , un ) with a component of u→ gR ((zn , − n )), and passing to a subsequence, one might suppose that the se→ − u→ quence ((zn , − n ))n 1 converges to (z , u ). One has d(γn (zn ), zn ) = (γn ). in addition, passing to a different subsequence, ( (γn ))n 1 converges. as a result there exists ε > zero and M > zero such that d(γn (z ), z ) ε for all n M .
Five. allow x ∈ R and allow (r(s))s∈R be an arclength parametrization of the orientated geodesic (∞x). the subsequent are similar: (i) there exists a chain (sn )n 1 of confident genuine numbers such that limn→+∞ sn = +∞ and π(r(sn )) belongs to the horocycle π(Ht ); (ii) there exists a series (γn )n 1 in Γ − Γ∞ such that |x − γn (∞)| 1/(2tc2 (γn )) and lim c(γn ) = +∞. n→+∞ evidence. (i) ⇒ (ii). we are going to basically recycle the arguments utilized in the facts of Proposition I. three. 20. the truth that π(r(sn )) belongs to π(Ht ) implies that there exists γn ∈ Γ such that r(sn ) ∈ γn (Ht ).