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Given the linear Anosov map $F_L$ of $\mathbb{T}^2$: $$z\mapsto \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix} z \quad\text{mod }1$$ it's already proved that there exists $P\in\mathbb N$ such that $F_L^P|_{M_N}=Id$, where $M_N$ denotes the set of points of the torus whose coordinates are rational with denominator $N\in\mathbb N$.

I want to prove that the minimum of such number is $P(N)$ and we have that $P(N)\leq N^4$. My ideas stop at the point where I note that any rational point with common denominator $N$ is a sum of the vectors $(1/N, 0)$ and $(0,1/N)$ with integer coefficients. Your help is much appreciated.

FunnyBuzer
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  • For related questions have a look at this: https://www.sciencedirect.com/science/article/pii/S0166864115000590 – John B Oct 11 '19 at 21:23
  • @JohnB thanks for your comment! It's a pertinent article, indeed. But I couldn't find any reference that deal with the problem mentioned above. Any other ideas? – FunnyBuzer Oct 12 '19 at 10:07
  • Not from my part, but we do find less likely to be an easy question after reading the paper, right? Overall, it should amount to find some residues. It is indeed a good idea to start with prime numbers! – John B Oct 12 '19 at 18:10

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