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).

Read more about this topic:  Volume Entropy

Famous quotes containing the word properties:

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)