Does ergodic theorem apply to trajectories outside of attractor? Ergodic theorem says that $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{t=1}^nf(T^tx) = \displaystyle\int f\,\mathrm{d}\mu$ for $\mu$-almost every $x$. In many cases, the support of $\mu$ has zero Lebesgue measure.
My question is that, what if $f$ is uniformly continuous on a neighborhood of the trajectory, and $x$ is outside of the support of $\mu$? When can we have similar identities? Thanks a lot.
 A: Axiom A diffeomorphisms have this property. The following result is due to Bowen and Ruelle.
Theorem
Let $X$ be a connected compact manifold and $T : X \rightarrow X$ be an Axiom A $C^2$ diffeomorphism: the non wandering set $\Omega$ of $T$ is an hyperbolic set and the periodic orbits of $T$ are dense in $\Omega$. Then there is a finite number of disjoint compact attractors $K_i$ included in $\Omega$, each of them supporting an ergodic probability measure $\mu_i$ called a Sinai-Ruelle-Bowen measure, such that the union of the bassins of attraction $B(K_i) = \{x\in X \mid d(T^n(x, K_i)\rightarrow 0\}$ is of full Lebesgue measure.
Moreover, for lebesgue almost $x\in B(K_i)$, for all $f : X\rightarrow \bf R$ uniformly continuous,
$$
{1\over n}\sum_{k=1}^n f \circ T^k(x) \rightarrow \int f d\mu_i.
$$
The set of Axiom A flows is open in the $C^2$ topology but is far from dense in the set of all $C^2$ diffeomorphisms on $X$ in general. There are  extensions to non-uniformly hyperbolic systems of that result but also many counterexamples.
The theorem follows from the fact that a point in the bassin of $K_i$ is on the stable leaf of a point on $K_i$ so that the two Birkhoff means are asymptotic. As a result, the asymptotic behavior of the trajectories of a set of positive measure of points is dictated by the measure $\mu_i$ on the attractor, even if that measure is singular with respect to the Lebesgue measure.
A: The key word you are looking for is "physical measures", sometimes known as SRB measures (because of the result.that coudy mentions).
See https://link.springer.com/article/10.1023/A:1019762724717
A related notion you may find interesting is given here: https://arxiv.org/abs/1106.4074
A nice paper discussing related notions is the following https://link.springer.com/article/10.1007/BF01212280
