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I'm trying to solve an exercise in "Ergodic Theory with a view towards Number Theory".

The exercise is suppose $T$ is an ergodic automorphism on an compact abelian group, show that it is mixing of all orders.

(The idea is to use the dual basis of characters and ergodicity being equivalent to no finite periods).

I am unable to show the mixing of all orders.

Thus we need to show for character $\chi_i$ that unless they're all trivial, then as $n_i \to \infty$, we can't have $\chi_0 \times \chi_1(T^{n_1}) \times \ldots \times \chi_k(T^{n_1 + \ldots +n_k}) = 1$.

I can't even refute $\chi_0 (T^{2^n}) \chi_1(T^n) = \chi_2$. Basically I would love to have a repitition of some sort so that I can cancel and induct.

Please give me a hint.

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Begin by thinking about the dual automorphism to a hyperbolic toral automorphism. You can project the dual group onto an "expansive" direction. Any relation of the sort you are looking at can be shifted by the automorphism so that all powers are negative except for one large positive one. In the projection, this would mean that a huge number is the sum of very small numbers, a contradiction. The kind of argument works whenever there is some sort of expansiveness. Then you need to show all ergodic group automorphisms do have some sort of expansiveness.

Much more is true, see my paper Ergodic group automorphisms are exponentially recurrent, Israel J. Math. 41 (1982), 313-320.

What is more interesting is that this fails for commuting group automorphisms (or algebraic ${\Bbb Z}^d$-actions), the classic example due to Ledrappier. Amazingly, it does remain true in the case of commuting toral automorphisms, however the proof uses some very deep number theory about additive relations in fields (see Cor. 27.6 in Dynamical Systems of Algebraic Origin by Klaus Schmidt).

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  • $\begingroup$ Thanks for the hint! I'm trying to implement it for the hyperbolic toral case. For simplicity I started by assuming the matrix diagonalizable- your argument gives that the only problem can appear on eigenspaces of eigenvalues of modulus 1. Since they are difficult to handle, I hoped I can avoid handling them by using Kronkers theorem to get a nonnorm 1 eigenvalue, and hope no integer vector is orthogonal to it. Are there any properties of the dual group in general I should be aware of ? I would love for it to be finitely generated for example $\endgroup$
    – Andy
    Commented Mar 23, 2020 at 19:27
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    $\begingroup$ All you need is an eigenvalue outside the unit circle, and you're right that by Kronecker there must be one since otherwise you violate ergodicity. But there are situations in which there is no geometric hyperbolicity at all, for automorphisms of solenoids rather than tori. For these, there is an arithmetic hyperbolicity that serves as a replacement. This is part of a beautiful "global" view, based on treating all completions of $\Bbb Q$ equally, both $p$-adic and real, and each plays a cooperative role. $\endgroup$
    – Douglas Lind
    Commented Mar 23, 2020 at 20:58
  • $\begingroup$ My problem is even if I have an eigenvalue $\lambda$ outside the unit circle with eigenvector $e$, my vectors $v_i$ can be orthogonal to it, thus not letting me project along it. I think there is no choice but to try to understand what is going for nonroots of unity on the circle. Regarding your second point that sounds fascinating, do you have a good source? $\endgroup$
    – Andy
    Commented Mar 23, 2020 at 21:54
  • $\begingroup$ Use the eigenspace decomposition, where "project" doesn't mean orthogonal projection but rather projection along the complementary eigenspace. There's a very short paper with Schmidt, Bernoullicity of solenoidal automorphisms and global fields, Israel J. Math 87 (1994), 33-35, which has this point of view. Also a paper with Ward Automorphisms of solenoids and p-adic entropy, Ergod. Theory & Dynam. Syst. 8 (1988), 411-419, where the p-adic expansion and contraction using p-adic eigenvalues is essential to getting the complete picture. $\endgroup$
    – Douglas Lind
    Commented Mar 24, 2020 at 1:50
  • $\begingroup$ A more thorough understanding of this happens when considering the joint action of $d$ commuting automorphisms. See my paper with Einsiedler, Algebraic $\BbbZ ^d$-actions of entropy rank one, Trans. AMS 356 (2004), 1799-1832. It's really a beautiful "global" picture of what's going on. $\endgroup$
    – Douglas Lind
    Commented Mar 24, 2020 at 1:52

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