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Geodesic flow on surfaces without conjugate points - part 2

By Khadim Mbacke War

Appears in collection : An introduction to dynamics on surfaces and random walks / Géométrie et Dynamiques sur les surfaces

The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study,this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.

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Citation data

  • DOI 10.24350/CIRM.V.20207703
  • Cite this video War, Khadim Mbacke (23/07/2024). Geodesic flow on surfaces without conjugate points - part 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20207703
  • URL https://dx.doi.org/10.24350/CIRM.V.20207703

Bibliography

  • CLIMENHAGA, Vaughn, KNIEPER, Gerhard, et WAR, Khadim. Closed geodesics on surfaces without conjugate points. Communications in Contemporary Mathematics, 2022, vol. 24, no 06, p. 2150067. - https://doi.org/10.1142/S021919972150067X
  • CLIMENHAGA, Vaughn, KNIEPER, Gerhard, et WAR, Khadim. Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points. Advances in Mathematics, 2021, vol. 376, p. 107452. - https://doi.org/10.1016/j.aim.2020.107452

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