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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.

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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.

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  • $\begingroup$ Is there any theory giving a definite answer, i.e. the identity holds for every x, instead of for almost every x? $\endgroup$
    – ycz
    Commented May 10, 2021 at 12:12
  • $\begingroup$ If $x$ is a periodic point of $T$, the mean alors its orbit converges to the integral with respect to the Dirac measure along the orbit, not to the integral with respect to the Lebesgue measure. Convergence for all x is related to unique ergodicity, not ergodicity. $\endgroup$
    – coudy
    Commented May 10, 2021 at 20:27
  • $\begingroup$ Can we claim that the system is uniquely ergodic if the maximal attractor can be approximated by a single trajectory? $\endgroup$
    – ycz
    Commented May 11, 2021 at 8:11
  • $\begingroup$ If the Birkhoff means converge for all x, then the system is uniquely ergodic. See e.g. the book of Walters, ergodic theory. $\endgroup$
    – coudy
    Commented May 11, 2021 at 8:22
  • $\begingroup$ Yes that is a direct result by the definition of unique ergodicity. But this statement does not help me because what I want to know is when the BIrkhoff converge for all x... So I need to know when a system is uniquely ergodic in a straightforward/intuitive way... Can we say that the system is uniquely ergodic if the maximal attractor can be approximated by a single trajectory? $\endgroup$
    – ycz
    Commented May 11, 2021 at 8:26
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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

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