Smooth measure

In mathematics , smooth measures are mainly used in the theory of dynamic systems .

definition

A measure on a smooth manifold is called smooth if there is a Riemannian metric with a volume form and a measure defined by it , so that is absolutely continuous with respect to . ${\displaystyle \mu }$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle g}$ ${\displaystyle vol_{g}}$ ${\displaystyle m}$ ${\displaystyle \mu }$ ${\displaystyle m}$

Applications

In dynamics, invariant smooth measures have different properties than any invariant measures. For example, in the Margulis-Ruelle inequality, invariant smooth measures always have equality (Pesin's entropy formula).

literature

• Chapter 5 in Katok-Hasselblatt: Introduction to the modern theory of dynamical systems , Cambridge University Press 1995

Web links

• Pesin entropy formula (Scholarpedia)