Margulis-Ruelle inequality
In the mathematical theory of dynamic systems , the Margulis-Ruelle inequality states that the entropy of a dynamic system can be estimated upwards by the sum of its positive Lyapunov exponents .
Mathematical formulation
Let a - diffeomorphism of a compact manifold with an invariant probability measure . Then the inequality applies to the Kolmogorow-Sinai entropy
- ,
where the Lyapunov exponents are in the point and on the right-hand side it is only summed over the positive Lyapunov exponents.
For smooth invariant probability measures, Pesin's stronger entropy formula applies :
- .
However, for non-smooth invariant probability measures, a strict inequality can hold. This is the case, for example, when the non-migrating set is finite and hyperbolic . The Ledrappier-Young theorem gives precise conditions as to when equality applies in the Margulis-Ruelle inequality .
literature
- Min Quian, Jian-Sheng Xie, Shu Zhu: Smooth ergodic theory for endomorphisms , Lecture Notes in Mathematics 1978, Springer Verlag, 2009.
Web links
- Pesin entropy formula (Scholarpedia)