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In Universal Characteristic Factors and Furstenberg Averages, Tamar Ziegler talks about "unconventional ergodic averges" which were used in Furstenberg's proof of Szemeredi's theorem. I had trouble understanding the notion of "characteristic factor" in dynamical system, so I focused on some of the earlier examples in her paper.

Let $X = (X, \mathcal{B}, \mu, T)$ be a measure-preserving dynamical system. I will not worry about the σ-algebra $\mathcal{B}$ or the measure $\mu$ (which is Lebesgue). So we can just write the dynamical system $T: X \to X$ preserving $\mu$ on $\mathcal{B}$.

This induces a map on the function spaces taking $T:\psi(x) \mapsto \psi(Tx)$. Ziegler looks at averages $$ \frac{1}{N} \sum_{n=1}^N f(T^n x)g(T^{2n} x)h(T^{3n}x) \to \int f \, d\mu \cdot \int g \, d\mu \cdot \int h \, d\mu.$$ If $X$ is weak mixing the left and right sides are equal.

In general, $f,g,h$ at the three points $x,T^n x, T^{2n}x$ are not ''independent" and this average is another dynamical system based on $T$. One obstruction to these averages becoming constant is an eigenfunction $\psi(Tx) = \lambda \psi(x)$. Then $$\frac{1}{N} \sum_{n=1}^N \psi(T^n x)^2\psi^{-1}(T^{2n} x) = \psi(x).$$ In this case, $T$ restricted to eigenfunctions behaves like rotations $z \mapsto z + \alpha$ in an abelian group.

Zieglier says that if you have a second-order eigenfunction $ T \phi = \psi \phi, T\psi = \lambda \psi $ the average becomes a nilsystem. Then you get a relation $$\phi(T^n x)^3 \phi(T^{2n} x)^{-3} \phi(T^{3n}x) = \phi(x).$$ (This is a bit like the identity $n^2 - 3(n+1)^2 + 3(n+2)^2 - (n+3)^2 = 0$. Finally $$ \frac{1}{N} \sum_{n=1}^N \phi(T^n x)^3 \phi(T^{2n} x)^{-3} \phi(T^{3n}x) = \phi(x).$$


I guess I am wondering to what extent these 'characteristic factors' generalize the notion of 'eigenfunction' for the operators $T$ and how they are related to nilsystems. Is it possible to find invariant nilsystems in dynamical systems arising from ordinary differential equations in several variables?

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The first important thing to note about a characteristic factor is that it's not uniquely defined. In particular, if you expand a characteristic factor to a larger factor, you still have a characteristic factor. So the interesting question is to find a characteristic function which is small enough that its elements are easy to work with. One of the natural ways of looking at a characteristic factor is as the factor generated by certain canonical functions.

The classic case (and the one which motivated the more recent work) is where the eigenfunctions are the canonical functions, the factor is the collection of functions with pre-compact orbit, and the factor is characteristic for the average you describe in your question.

Eigenfunctions can be viewed as coming from nilsystems where the group is abelian (that is, the step is 1), so a generalization is to work with functions coming from nilsystems of higher step. These functions generalize the eigenfunctions, and are characteristic for more complicated averages. In that sense these particular characteristic factors generalize the eigenfunctions.

More generally, the notion of a characteristic factor captures, relative to arbitrary averages, what the factor generated by the eigenfunctions captures relative to a particular average.

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