Volume Entropy - Properties

Properties

  • Volume entropy h is always bounded above by the topological entropy htop of the geodesic flow on M. Moreover, if M has nonpositive sectional curvature then h = htop. These results are due to Manning.
  • More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed Riemannian manifold without conjugate points (Freire and Mañé).
  • Locally symmetric spaces minimize entropy when the volume is prescribed. This is a corollary of a very general result due to Besson, Courtois, and Gallot (which also implies Mostow rigidity and its various generalizations due to Corlette, Siu, and Thurston):
Let X and Y be compact oriented connected n-dimensional smooth manifolds and f: Y → X a continuous map of non-zero degree. If g0 is a negatively curved locally symmetric Riemannian metric on X and g is any Riemannian metric on Y then
h^n(Y,g)\operatorname{vol}(Y,g)\geq |\operatorname{deg}(f)| h^n(X,g_0)\operatorname{vol}(X,g_0),
and for n ≥ 3, the equality occurs if and only if (Y,g) is locally symmetric of the same type as (X,g0) and f is homotopic to a homothetic covering (Y,g) → (X,g0).

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