Asymptotically harmonic manifolds are first introduced by F. Ledrappier in the ergodic-theoretical context.
We shall recall the importance of asymptotically harmonic manifolds with negative curvature from a dynamical viewpoint, with their relationship with Katok’s rigidity conjecture on Liouville measures.
We also verify characterizations of asymptotically harmonic manifolds, which reveal their geometric, dynamical, and stochastic aspects.