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 ${\displaystyle f\colon M\to M}$ ${\displaystyle C^{1+\alpha }}$ ${\displaystyle \nu }$ ${\displaystyle h_{\nu }(f)}$

${\displaystyle h_{\nu }(f)\leq \int _{M}\sum _{\lambda _{i}(x)>0}\lambda _{i}(x)d\nu (x)}$,

where the Lyapunov exponents are in the point and on the right-hand side it is only summed over the positive Lyapunov exponents. ${\displaystyle \lambda _{1}(x),\ldots ,\lambda _{n}(x)}$ ${\displaystyle x\in M}$ ${\displaystyle x}$

For smooth invariant probability measures, Pesin's stronger entropy formula applies :

${\displaystyle h_{\nu }(f)=\int _{M}\sum _{\lambda _{i}(x)>0}\lambda _{i}(x)d\nu (x)}$.

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)