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If a continuous map of a compact metric space has positive topological entropy, then it has a Li-Yorke pair. In general, the converse is not true, and a zero entropy continuous map can have a Li-Yorke pair.

Can a zero entropy automorphism of the torus have a Li-Yorke pair? More generally, can a zero entropy endomorphism of a compact Lie group have a Li-Yorke pair?

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I'm assuming a Li-Yorke pair for $T$ is a pair $x,y$ such that $\limsup_{n\to \infty} d(T^n(x), T^n(y)) > 0 $ and $\liminf_{n\to \infty} d(T^n(x), T^n(y)) = 0$.

For toral automorphisms, I think the answer to your question is no. The idea is this: if $T$ is ergodic, then it has positive entropy (every ergodic linear toral automorphism does, see for instance here). If $T$ is not ergodic, you can find a factor of (some power of) $T$ on a torus of lower dimension which is ergodic, which means again that $T$ has positive entropy.

Indeed, being non-ergodic implies that there is an eigenvalue which is a root of the unity. We may replace $T$ by some power of $T$ and assume that $1$ is an eigenvalue. Moreover you can find an integer eigenvector $v$ with eigenvalue $1$ (because the coefficients of the matrix corresponding to $T$ are integer), so passing to the quotient by $\mathbb{R}v$ you should get a torus of dimension $n-1$. Since $T$ had a Li-Yorke pair, so does the automorphism induced by $T$ on the quotient, so you have reduced the dimension. If the new map is still not ergodic, we may repeat this reduction. But this process has to stop before reaching dimension $1$ since you cannot have a Li-Yorke pair in dimension $1$.

Note: I edited this answer since the previous one had a big mistake (the claim was false)

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  • $\begingroup$ I'm a bit confused when you say that your claim answers the question negatively. Your claim suggests that the question might have a negative answer, but it leaves open the possibility that a non-linear automorphism of the torus might have zero entropy and have a Li-Yorke pair. Correct? $\endgroup$
    – Will Brian
    Aug 29, 2016 at 18:21
  • $\begingroup$ I thought linear automorphisms are all there is. Am I missing something? $\endgroup$ Aug 30, 2016 at 0:11
  • $\begingroup$ Oh, I see. I'm looking at the torus as a topological space, and as such it has lots of "automorphisms" (which I took to mean self-homeomorphisms). You're looking at it as a Lie Group, where all the "automorphisms" are translations by a group element (linear). Looking at the question again, I think your reading might be the intended one, and your answer now makes sense to me. (Interestingly, I think one gets the opposite answer under my reading of the question.) $\endgroup$
    – Will Brian
    Aug 30, 2016 at 13:46
  • $\begingroup$ Yes, if you consider just homeomorphisms, you can have all sorts of weird things. For example a transitive homeomorphism with a fixed point would be an example with a Li-Yorke pair, and this can have zero entropy (for example it can be the time-one map of a flow, as the example described in this thread) $\endgroup$ Aug 30, 2016 at 19:29

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