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Let $A$ be an $n \times n$ matrix (let's say hyperbolic, but this might be irrelevant). Consider the action of $A$ on $(\mathbb{R} \backslash \mathbb{Z})^n$. Does this action always have periodic orbits with minimal period being an even number $k\geq2$?

It is easy to exhibit such orbits explicitly in specific cases. E.g. if $n=2$ and $c \neq 0 \ mod \ a+d $, taking the orbit of $(\frac{1}{a+d},0)$. However I've been unable to find a proof in general.

EDIT: the general answer is no, however the question still holds under the assumption that $A$ is a hyperbolic matrix (no eigenvalues of absolute value $1$).

mkk
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  • Perhaps I'm missing something, but isn't $x\mapsto \alpha,x\mod 1$ a counter-example for $n=1$ when $\alpha$ is transcendental? – Mark McClure Feb 14 '15 at 20:08
  • @MarkMcClure: That matrix doesn't act on $\Bbb R/\Bbb Z$, only on $\Bbb R$. – Greg Martin Feb 14 '15 at 20:28
  • @GregMartin Thanks. However, my understanding of $\mathbb R / \mathbb Z$, or the real numbers modded out by the subgroup of integers, is that every action on $\mathbb R$ induces one on $\mathbb R / \mathbb Z$. If the intention is to stick with integer matrices (as I suspect that it is), then my example is not a counter-example. I just don't see where that's stated. It's really minor, though, just a notational curiosity. – Mark McClure Feb 15 '15 at 00:28
  • An action on $\Bbb R$ only extends to an action on $\Bbb R/\Bbb Z$ if it sends integers to integers. For example, in your proposed action $x\mapsto \alpha x$: where is $\frac12+\Bbb Z=\frac32+\Bbb Z$ sent? Is it sent to $\frac12\alpha+\Bbb Z$, or $\frac32\alpha+\Bbb Z$? – Greg Martin Feb 15 '15 at 00:40

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Not in general, no. If $A$ has odd order - that is, if $A^k=I_n$ for some odd number $k$ - then all orbits are periodic with periods dividing $k$, hence all odd. For example, $A=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}$ satisfies $A^3=I_2$, so every orbit has size $3$ (except the orbit of the fixed point $(0,0)$).

Greg Martin
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