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I cannot seem to figure this problem out.

I know that the hyperbolic toral automorphisms $A$ is just an integer hyperbolic matrix with determinant $\pm 1$ that has eigenvalues $0<|\mu|<1<|\lambda|$. I know that we can find a basis of $R^2$ in terms of two eigenvectors corresponding to each eigenvalue say $v_1,v_2$. I also know that is $x=a_1 v_1+ a_2 v_2$ then $A^n(x)=a_1 \lambda^n v_1+ a_2 \mu^n v_2$ and we can estimate its Euclidean norm using the norm $||x||=\max\{|a_1|,|a_2|\}$. I feel like this is most of the puzzle pieces but I am not sure how to put this all together to show $A$ mod 1 is expansive. Any help would be appreciated.

In case there is many definitions out there a map is expansive if $\exists$ a $\delta$ s.t for any $x,y$ there $\exists$ $n$ for which $d(A^n(x),A^n(y))>\delta$

2132123
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1 Answers1

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Consider the equation $$d(A^n(x),A^n(y)) = \| A^n(x) - A^n(y) \| = \|A^n(x-y)\| $$ Substituting $u=x-y$, the problem is reduced to understanding the sequence $\|A^n(u)\|$ for any vector $u \ne \vec 0$. Setting $u = av_1 + bv_2$ you have $$\|A^n(av_1 + bv_2)\| = \|a A^n(v_1) + b A^n(v_2)\| = \|a \mu^n v_1 + b \lambda^n v_2\| $$ Now there are two cases: $a \ne 0$ and $b \ne 0$.

If $b \ne 0$ then we can use the triangle inequality estimate $$\|a \mu^n v_1 + b \lambda^n v_2\| \ge \|bv_1\| \cdot |\lambda|^n - \|av_1\| \cdot |\mu|^n $$ The first term goes to $+\infty$ as $n \to +\infty$, and the second term goes to $0$.

If $a \ne 0$ then one similarly uses the estimate $$\|a \mu^n v_1 + b \lambda^n v_2\| \ge \|av_1\| \cdot |\mu|^n - \|bv_1\| \cdot |\lambda|^n $$ and one lets $n \to -\infty$.

Lee Mosher
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  • @Lee_Mosher So I actually got this far but deemed this a dead end as I could not related the euclidean distance to the distance on the torus. Yes $||A^n(u)||$ is unbounded for $n \in \Z$ but where do we go from here? How does this show expansiveness? – 2132123 Oct 24 '20 at 02:43
  • Replace the original $u$ by something really tiny. Locally the metric on the torus is Euclidean. – Lee Mosher Oct 24 '20 at 02:49
  • I am afraid i do not follow. – 2132123 Oct 24 '20 at 02:49
  • There is a number $r>0$ such that every $r$-ball on the torus is isometric to an $r$-ball on the Euclidean plane. Now choose $x,y$ on the torus to have distance $< r/2$, hence $y$ is contained in the $y$-ball centered on $x$. Transfer over to a ball in Euclidean space of radius $r$, and iterate until the two points have distance between $r/2$ and $r$, and hence are in some (other) Euclidean ball of radius $r$. Transfer back to the torus. And now I've gotta get some sleep. – Lee Mosher Oct 24 '20 at 02:54