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Let ${(X,T,\mu)}$ be a measure-preserving system – a probability space ${(X,\mu)}$ equipped with a measure-preserving translation ${T}$ (which for simplicity of discussion we shall assume to be invertible). We will informally think of two points ${x,y}$ in this space as being “close” if ${y = T^n x}$ for some ${n}$ that is not too large; this allows one to distinguish between “local” structure at a point ${x}$ (in which one only looks at nearby points ${T^n x}$ for moderately large ${n}$) and “global” structure (in which one looks at the entire space ${X}$). The local/global distinction is also known as the time-averaged/space-averaged distinction in ergodic theory.

A measure-preserving system is said to be ergodic if all the invariant sets are either zero measure or full measure. An equivalent form of this statement is that any measurable function ${f: X \rightarrow {\bf R}}$ which is locally essentially constant in the sense that ${f(Tx) = f(x)}$ for ${\mu}$-almost every ${x}$, is necessarily globally essentially constant in the sense that there is a constant ${c}$ such that ${f(x) = c}$ for ${\mu}$-almost every ${x}$. A basic consequence of ergodicity is the mean ergodic theorem: if ${f \in L^2(X,\mu)}$, then the averages ${x \mapsto \frac{1}{N} \sum_{n=1}^N f(T^n x)}$ converge in ${L^2}$ norm to the mean ${\int_X f\ d\mu}$. (The mean ergodic theorem also applies to other ${L^p}$ spaces with ${1 < p < \infty}$, though it is usually proven first in the Hilbert space ${L^2}$.) Informally: in ergodic systems, time averages are asymptotically equal to space averages. Specialising to the case of indicator functions, this implies in particular that ${\frac{1}{N} \sum_{n=1}^N \mu( E \cap T^n E)}$ converges to ${\mu(E)^2}$ for any measurable set ${E}$.

In this short note I would like to use the mean ergodic theorem to show that ergodic systems also have the property that “somewhat locally constant” functions are necessarily “somewhat globally constant”; this is not a deep observation, and probably already in the literature, but I found it a cute statement that I had not previously seen. More precisely:

Corollary 1 Let ${(X,T,\mu)}$ be an ergodic measure-preserving system, and let ${f: X \rightarrow {\bf R}}$ be measurable. Suppose that $\displaystyle \limsup_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \mu( \{ x \in X: f(T^n x) = f(x) \} ) \geq \delta \ \ \ \ \ (1)$

for some ${0 \leq \delta \leq 1}$. Then there exists a constant ${c}$ such that ${f(x)=c}$ for ${x}$ in a set of measure at least ${\delta}$.

Informally: if ${f}$ is locally constant on pairs ${x, T^n x}$ at least ${\delta}$ of the time, then ${f}$ is globally constant at least ${\delta}$ of the time. Of course the claim fails if the ergodicity hypothesis is dropped, as one can simply take ${f}$ to be an invariant function that is not essentially constant, such as the indicator function of an invariant set of intermediate measure. This corollary can be viewed as a manifestation of the general principle that ergodic systems have the same “global” (or “space-averaged”) behaviour as “local” (or “time-averaged”) behaviour, in contrast to non-ergodic systems in which local properties do not automatically transfer over to their global counterparts.

Proof: By composing ${f}$ with (say) the arctangent function, we may assume without loss of generality that ${f}$ is bounded. Let ${k>0}$, and partition ${X}$ as ${\bigcup_{m \in {\bf Z}} E_{m,k}}$, where ${E_{m,k}}$ is the level set $\displaystyle E_{m,k} := \{ x \in X: m 2^{-k} \leq f(x) < (m+1) 2^{-k} \}.$

For each ${k}$, only finitely many of the ${E_{m,k}}$ are non-empty. By (1), one has $\displaystyle \limsup_{N \rightarrow \infty} \sum_m \frac{1}{N} \sum_{n=1}^N \mu( E_{m,k} \cap T^n E_{m,k} ) \geq \delta.$

Using the ergodic theorem, we conclude that $\displaystyle \sum_m \mu( E_{m,k} )^2 \geq \delta.$

On the other hand, ${\sum_m \mu(E_{m,k}) = 1}$. Thus there exists ${m_k}$ such that ${\mu(E_{m_k,k}) \geq \delta}$, thus $\displaystyle \mu( \{ x \in X: m_k 2^{-k} \leq f(x) < (m_k+1) 2^{-k} \} ) \geq \delta.$

By the Bolzano-Weierstrass theorem, we may pass to a subsequence where ${m_k 2^{-k}}$ converges to a limit ${c}$, then we have $\displaystyle \mu( \{ x \in X: c-2^{-k} \leq f(x) \leq c+2^{-k} \}) \geq \delta$

for infinitely many ${k}$, and hence $\displaystyle \mu( \{ x \in X: f(x) = c \}) \geq \delta.$

The claim follows. $\Box$ Jeff on Almost all Collatz orbits atta… Johan Aspegren on Sharp bounds for multilinear c… Matthias Hübner on 245B, Notes 8: A quick review… Tom on Almost all Collatz orbits atta… Anonymous on Almost all Collatz orbits atta… Buzzman on Almost all Collatz orbits atta… Buzzman on The parity problem obstruction… Med-94 on Hamidoune’s Freiman-Knes… Buzzman on The parity problem obstruction… Jeff on Almost all Collatz orbits atta… Buzzman on Almost all Collatz orbits atta… Jeff on Almost all Collatz orbits atta… Buzzman on Almost all Collatz orbits atta… Jeff on Almost all Collatz orbits atta… Robert Frost on Almost all Collatz orbits atta…