Local stability of ergodic averages

Abstract:

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of $[0,1]$ and a characteristic function $f = \chi _A$ such that the ergodic averages $A_n f$ do not converge to a computable element of $L^2([0,1])$. In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator $T$ on a separable Hilbert space and any element $f$, it is possible to compute a bound on the rate of convergence of $\langle A_n f \rangle$ from $T$, $f$, and the norm $\| f^* \|$ of the limit. In particular, if $T$ is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space $\mathcal {X}$ and $f$ is any computable element of $L^2(\mathcal {X})$, then there is a computable bound on the rate of convergence of the sequence $\langle A_n f \rangle$.

The mean ergodic theorem is equivalent to the assertion that for every function $K(n)$ and every $\varepsilon > 0$, there is an $n$ with the property that the ergodic averages $A_m f$ are stable to within $\varepsilon$ on the interval $[n,K(n)]$. Even in situations where the sequence $\langle A_n f \rangle$ does not have a computable limit, one can give explicit bounds on such $n$ in terms of $K$ and $\| f \| / \varepsilon$. This tells us how far one has to search to find an $n$ so that the ergodic averages are “locally stable” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov.

Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of “proof mining.”