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 belief of highlighting, in a comparatively easy framework, the life of gateways among a few mathematical fields, and the benefits of utilizing them, ancient features of this box should not addressed and many of the references are reserved until eventually the tip of every bankruptcy within the reviews part. subject matters in the textual content hide geometry, and examples, of Fuchsian teams; topological dynamics of the geodesic move; Schottky teams; the Lorentzian perspective and Trajectories and Diophantine approximations.
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Additional resources for Geodesic and Horocyclic Trajectories (Universitext)
I. three. 2). → estate 1. nine. enable (z, − u ) be in T 1 H and (gn )n 1 be a chain in G. (Bu(+∞) (i, gn−1 (i)))n 1 (i) The series has a tendency to +∞ if and provided that → u ))) )n 1 converges to zero. ( Mgn (vect((z, − (ii) the purpose u(+∞) is fixed by way of a parabolic isometry g ∈ G − {Id} if and in simple terms → → u ))) = vect((z, − u )). if Mg (vect((z, − workout 1. 10. turn out estate 1. nine. 2 The horocycle flow on a quotient allow us to think of a Fuchsian team Γ . we are going to hold the notation brought in Sect. III. 1. 2. The commutativity of hR and G proved in workout 1. 2, permits one to define the horocycle flow hR at the quotient T 1 S = Γ \T 1 H (Fig. V. 5): → for all (z, − u ) in T 1 H, we set → → ht (π 1 ((z, − u ))) = π 1 (ht ((z, − u ))). by means of definition of the topology of T 1 S, a series (π 1 ((zn , − u→ n )))n 1 con− → 1 verges to π ((z, u )) if and provided that there exists a chain (γn )n 1 in Γ such that limn→+∞ γn (zn ) = z and limn→+∞ γn (un (+∞)) = u(+∞). → → u ))))n 1 converges to π 1 ((z , − u )) if and provided that there hence (htn (π 1 ((z, − exists (γn )n 1 in Γ such that lim γn zn = z n→+∞ and lim γn (u(+∞)) = u (+∞), n→+∞ → the place zn is the projection on H of htn ((z, − u )). 114 V Topological dynamics of the horocycle flow Fig. V. five. Γ = PSL(2, Z) 2. 1 A vectorial viewpoint on hR allow MΓ denote the subgroup of {± Id}\ SL(2, R) which includes all Mγ with γ in Γ . This crew of linear differences is isomorphic to Γ . the next proposition relates its dynamics on E to these of hR on T 1 S. → → u ) be in T 1 H. There exists a Proposition 2. 1. enable (z, − u ) and (z , − → u ))))n 1 converges to series (tn )n 1 in R such that (htn (π 1 ((z, − → − 1 π ((z , u )) if and provided that there exists a series (Mγn )n 1 in MΓ such → → u ))))n 1 converges to vect((z , − u )). that (Mγn (vect((z, − → facts. imagine that there exists (γn )n 1 in Γ for which (γn htn ((z, − u )))n 1 → − → converges to (z , u ). Its photograph through the map vect converges to vect((z , − u )), when you consider that vect is continuing. we've → → u ))) = Mγn (vect((z, − u ))), vect(γn htn ((z, − → → u ))))n 1 converges to vect((z , − u )). therefore (Mγn (vect((z, − − → Conversely, think that (Mγn (vect((z, u ))))n 1 converges to → u )). through definition of the map vect, the sequences (γn (u(+∞)))n 1 vect((z , − and (Bγn (u(+∞)) (i, γn (z)))n 1 converge to u (+∞) and Bu (+∞) (i, z ) re→ spectively. examine the true quantity tn such that htn (γn ((z, − u ))) is tangent to the geodesic passing via z having γn (u(+∞)) as an endpoint. → u ))) = (zn , − u→ Set htn (γn ((z, − n ). considering that un (+∞) = γn (u(+∞)), the series (un (+∞))n 1 converges to u (+∞). in addition Bγn (u(+∞)) (i, γn (z)) = Bun (+∞) (i, zn ), accordingly (zn )n 1 converges to z . One therefore obtains → − lim htn (π 1 ((z, − u ))) = π 1 ((z , → u )). n→+∞ → → Corollary 2. 2. permit (z, − u ) be in T 1 H. The trajectory hR (π 1 ((z, − u ))) is closed − → in T 1 S if and provided that the orbit of the vector vect((z, u )) with admire to the crowd MΓ is closed in E. workout 2. three. end up Corollary 2. 2. 2 The horocycle flow on a quotient a hundred and fifteen 2.