Entropy Equipartition along almost Geodesics in Negatively Curved Groups by Amos Nevo

PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID)

ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi)
DATE : 05 December 2022 to 16 December 2022
VENUE : Ramanujan Lecture Hall and Online

The programme will have an emphasis on the many recent exciting breakthroughs in the ergodic theory of group actions on homogeneous spaces. This subject, which also goes by the name "homogeneous dynamics" has seen dramatic advances in the last few decades. Homogeneous dynamics comprises the study of group actions on homogeneous spaces of Lie groups. The dynamics of these actions are extremely rich and have surprising connections to diverse parts of mathematics. An early example of such a connection is Margulis's proof of the long standing conjecture of Oppenheim regarding values taken by quadratic forms at integer points. The programme will feature mini courses as well as research level talks exploring recent advances in homogeneous dynamics and their connections to number theory and geometry. India has a strong tradition in this area and the programme will provide an occasion to celebrate the fundamental contributions of S. G. Dani, a pioneer in the subject who turns 75 this year. Young mathematicians, especially those with an interest in these areas are encouraged to apply.

ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.

0:00:00 Entropy Equipartition along almost Geodesics in Negatively Curved Groups
0:00:14 Convergence of entropy along geodesics in negatively-curved groups, SMB theorem, Rokhlin entropy
0:02:04 Plan of the talk
0:03:10 Shannon entropy and information function
0:04:06 Dynamics of partitions in measure-preserving dynamical system
0:06:42 Convergence of normalized Shannon entropies
0:07:59 Entropy theory for amenable groups
0:09:59 Convergence of entropy for amenable groups
0:11:31 Kolmogorov-Sinai entropy
0:13:18 Entropy equipartition : Shannon-McMillan-Breiman theorem for amenable groups
0:16:25 Free groups : Ornstein-Weiss example, '87
0:18:35 Bernoulli factors with higher base entropy
0:20:11 Entropy theory for free groups : Bowen's f-invariant
0:23:26 Beyond amenable groups : Bowen's sofic entropy
0:25:06 Beyond amenable groups : Rokhlin entropy
0:27:52 Finitary entropy and Seward's inequality
0:30:08 Negatively curved groups
0:32:24 Geodesics in M and almost geodesics in Gamma
0:35:52 Convergence along almost geodesics
0:42:11 Shannon-McMillan-Breiman along almost geodesics
0:46:18 Orbital Rokhlin entropy
0:49:25 Naturalist of the approximation process
0:51:00 Method of proof : Theorem A
0:56:03 Some comments
0:59:12 Q&A
1:04:31 Wrap Up