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.


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 $x$.

  • $\begingroup$ You mean $\mathbb{P}((T^k \circ X) \in A) = \mathbb{P}(A)$? And with $k \to X_k = T^k X$, $\mathbb{N}$ is interpreted as (discrete) time? $\endgroup$ – Suedklee Aug 2 '12 at 19:31
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    $\begingroup$ If $f = \chi_{A}$ is the characteristic function of subspace $A$, the Birkhoff sum $\frac{1}{n}\sum_{k=0}^{n-1}f(T^kx)$ is precisely $|\{k \in [\![ 0, n-1 ]\!] \, | \, T^k(x) \in A\}|/n$, that is: the proportion of time the orbit of $x$ spent in $A$ during the first $n$ seconds. That this proportion tends to the relative measure of $A$ is the physical meaning of ergodicity, IIRC. $\endgroup$ – PseudoNeo Aug 2 '12 at 20:18
  • $\begingroup$ @Suedklee: if by $X$ you mean the whole space, then $\mathbb{P}(T^K\circ X)=\mathbb{P}(A)$ is just measure invariance, so yes, if that's what you were going for. My formulation does have $\mathbb{N}$-value time, which you could generalize to flows by taking an integral instead of a sum. $\endgroup$ – Kevin Carlson Aug 2 '12 at 21:06
  • $\begingroup$ The whole space should $(\Omega, \mathcal{F}, \mathbb{P})$ as usual in probability theory - I interpreted your small-x $x$ as a random variable which is usually denoted in capital letters, so I mean a random variable $X$. What do you mean by $x$? $\endgroup$ – Suedklee Aug 3 '12 at 7:09
  • $\begingroup$ With apologies for notational differences, I had $x$ as a point in $\Omega$, and $f$ as an $L^1$ function (complex or real valued) on that space. Your statement might hold, but you need an interpretation of "almost all" random variables. $\endgroup$ – Kevin Carlson Aug 3 '12 at 12:20

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