Setup: Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.

For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} f(\phi^n(x))$. Let $M = \sup_{x \in T^2} \underline{f}(x)$.

Let $N_\epsilon(x) = \min\{n \geq 0 \colon f(\phi^n(x)) < M + \epsilon\}$, and let $N_\epsilon = \sup_{x \in T^2} N_\epsilon(x)$.

Notice that $N_\epsilon < \infty$ for all $\epsilon > 0$. Indeed, if we let $U_n = \{x \in T^2\colon N_\epsilon(x) \leq n\}$ ($n \geq 0$), this forms an open cover of $T^2$ by definition of $M$, so by compactness some $U_n$ must be all of $T^2$.

Question: Is there a way of estimating $M$? And, what I most want to know, how does the growth of $N_\epsilon$ as $\epsilon \to 0$ depend on $\phi$ and $f$? Could we write down an algorithm to upper-bound $N_\epsilon$?

The $f$ I'm considering is given by a piecewise formula, real-analytic on each of the (finitely many) pieces.

Another way of putting it would be, given a "nice" open set $U$ in the torus (e.g. $U = \{x\colon f(x) < K\}$ for $f$ piecewise real-analytic, so $\partial U$ consists of piecewise real-analytic curves) is there a good way of telling whether every orbit intersects $U$, and if so estimating the minimum $N$ such that for every $x$, there exists $0 \leq n \leq N$ with $\phi^n(x) \in U$?

Context: My current interest in this has to do with the geodesic flow on the space $X = \{A\mathbb{Z}^2\colon A \in SL_2(\mathbb{R})\}$ of unimodular lattices in the plane. It's given by $g_t(A\mathbb{Z}^2) = \left(\begin{array}{l}e^t&0\\0&e^{-t}\end{array}\right)A\mathbb{Z}^2$.

This flow has some periodic orbits. For instance, if $A_0 = \left(\begin{array}{l}\sqrt{2}&1\\-\sqrt{2}&1\end{array}\right)$, then we get $$\left(\begin{array}{l}3 + 2\sqrt{2}&0\\0&3 - 2\sqrt{2}\end{array}\right)\left(\begin{array}{l}\sqrt{2}&1\\-\sqrt{2}&1\end{array}\right)\mathbb{Z}^2 = \left(\begin{array}{l}\sqrt{2}&1\\-\sqrt{2}&1\end{array}\right)\left(\begin{array}{l}3&2\\4&3\end{array}\right)\mathbb{Z}^2 = \left(\begin{array}{l}\sqrt{2}&1\\-\sqrt{2}&1\end{array}\right)\mathbb{Z}^2.$$

There's a natural torus bundle $Y$ over $X$, with the fiber over $A\mathbb{Z}^2$ being $\mathbb{R}^2/A\mathbb{Z}^2$, which represents translates of unimodular lattices. And there's a flow in $Y$ that projects down to the geodesic flow in $X$.

Under this flow, for instance, the torus $\mathbb{R}^2/A_0\mathbb{Z}^2$ ($A_0$ as above) gets mapped onto itself via $\left(\begin{array}{l}3&2\\4&3\end{array}\right)$. I'm really trying to understand certain things about the flow in $Y$. But the question I ask about toral automorphisms and functions/open sets seems basic enough that it might also be important for other things.

  • $\begingroup$ The geodesic flow on rank-$1$ spaces is Bernoulli, and so is hyperbolic toral automorphisms (Adler-Weiss). One cannot estimate $M$ based on any "finite information" (you can simply extend the orbit a bit more and change the index where this value is "attained"), so one cannot get any uniformly "pointwise" result. One can take the associated Markov partition (Adler-Weiss, again), and refine it to covering of this $U$ and in the end it boils down to counting "allowed" patterns in cylinders. And one might have some quantitative information from a mixing estimate (+Chebyshev's inequality or so). $\endgroup$ – Asaf Mar 18 '15 at 18:12
  • $\begingroup$ Thanks for your comment. But there is uniformity in the sense that, as mentioned above, after some finite time every point will have visited a point where $f < M + \epsilon$. I'm most interested in knowing what this finite time will be. Suppose we know $\phi$, and have a (piecewise defined) formula for $f$; is there an algorithm that, given $\epsilon$, will give an upper bound for $N_\epsilon$? You may have tried to address this, so I'd be grateful for an elaboration on mixing estimates etc. $\endgroup$ – Kiran Parkhe Mar 19 '15 at 8:45
  • 3
    $\begingroup$ In the generality you've posed I find it difficult to get clear intuition for $N_\varepsilon$. Have you tried out this problem for a simpler function $f$ (e.g. $f(x):=\mathrm{dist}(x,x_0)$ for some special point $x_0$) and a simpler dynamical system such as $T(x):=2x \mod 1$ on the circle $\mathbb{R}/\mathbb{Z}$? Or perhaps you could investigate what happens when the underlying system is the full shift on two symbols? Perhaps this will help to build intuition. $\endgroup$ – Ian Morris Mar 19 '15 at 11:38

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