Birkhoff ergodic theorem and the measure of the bad points

Question

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My question is: what is, given $\epsilon>0,$ the estimation of $\mu\left(x:\left|\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)-\int f \, d\mu\right|>\epsilon\right)$ when $N\in \mathbb{N}$ is big? I am looking for a proof or reference that I have not yet found.

Answer 1

The key words here are "large deviations"; large deviations theory addresses exactly this question. The answer depends quite a bit on the specific measure and system in question, but roughly speaking one may say the following: if the system displays a sufficient amount of hyperbolic behaviour (for example, an Axiom A system, or a system with the specification property, which is a sort of uniform topological mixing) and if the measure $\mu$ has some sort of Gibbs property relative to a potential function $\phi$, then the measure of the set you describe decays exponentially in $N$, and the rate of exponential decay depends in a precise manner on the topological pressure function.

Let me state a concrete series of theorems to make the above more precise. Let $(X,T)$ be a transitive Axiom A system, and let $\phi\colon X\to \mathbb{R}$ be Hölder continuous. Then by a result of Bowen ("Some systems with unique equilibrium states", 1974/5, or if you prefer, "Equilibrium states and the ergodic theory of Axiom A diffeomorphisms"), there is a unique invariant measure $\mu$ that maximises the quantity $h_\mu(T) + \int\phi\,d\mu$. (This maximum value is the topological pressure $P(\phi)$.) Moreover, $\mu$ has the following Gibbs property: if $B(x,n,\delta)$ denotes the set of points $y$ such that $d(f^kx,f^ky)\leq \delta$ for all $0\leq k\leq n$, then there is a constant $K(\delta)$ such that $$ (*) \qquad \frac 1{K(\delta)} \leq \frac{\mu(B(x,n,\delta))}{e^{-nP(\phi) + S_n\phi(x)}} \leq K(\delta) $$ for every $(x,n)\in X\times \mathbb{N}$.

Now a 1990 result of Lai-Sang Young on large deviations shows that for a system $(X,T)$ as above and a measure $\mu$ satisfying $(*)$, one has $$ \lim_{N\to\infty} \frac 1N \log \mu\{x\mid |\frac 1N S_N f(x) - \int f\,d\mu| \geq \epsilon\} = \sup \{ h_\nu(T) + \int \phi \,d\nu - P(\phi) \mid \nu\in \mathcal{M}_T(X), \left|\int f\,d\nu - \int f\,d\mu\right| \geq \epsilon\} < 0. $$

This behaviour is typical in the uniformly hyperbolic setting. In the non-uniformly hyperbolic setting, there are also examples where the rate of decay is slower than exponential. For example, this occurs in the case of the absolutely continuous invariant measure for the Manneville-Pomeau map.

Answer 2

There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\limsup_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(T^ix)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)

As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.