8
$\begingroup$

How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic if "averaging" over "states" of the physical system equals the "average" over time)?

I am sorry maybe the question is a bit unspecific for now, but I guess it's still of interest also for others who are about to dive into the subject.

1 Answers 1

3

They couldn't be related more closely: they're exactly the transformations for which the physical average equals the time average almost everywhere.

Specifically, they observe Birkhoff's ergodic theorem, that $\int_X fd\mu=\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1}f(T^kx), \mu$-almost everywhere, for any integrable $f$. A nice special case is if $f$ is the characteristic function of some subspace of $A$, in which case this says that the sequence $T^kx$ gets into $A$ proportionally often to the measure of $A$ in $X$ for just about every $x$.

  • 0
    Right, I think I see what you were going for in your first comment, and if you'll replace $X$ with $x$ you'll get the translation of the end of my answer.2012-08-03