I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who knows what those words mean together) could easily explain it.

Consider the cat map $A$ and the Markov partition $\mathcal{R} =$ {$R_1,\dots,R_5$} shown below: alt text The rectangles in the partition are numbered from 1 (darkest) to 5 (lightest).

Now for a given initial point $x$ with rational coordinates (so that the period $t(x)$ of the sequence $A^\ell x$ is finite) consider the matrix $T(x)$ with entries $T_{jk}(x)$ equal to the cardinality of {$\ell < t(x): A^\ell x \in R_j \land A^{\ell + 1}x \in R_k$}, i.e., the number of times per period that the trajectory goes from the $j$th rectangle to the $k$th rectangle. Clearly the sparsity pattern of $T(x)$ is inherited from the matrix defining the corresponding subshift of finite type.

Let $L_q$ denote the set of rational points in $[0,1)^2$ with denominator $q$. When I compute the sum $T_{(q)} := \sum_{x \in L_q} T(x)$ I get some surprising near-equalities. For instance, with $q = 240$ I get

  301468           0      301310      186567           0
  186567           0      186407      114903           0
  301310           0      301251      186407           0
       0      301470           0           0      186407
       0      186407           0           0      115060

and when $q = 322$ I get

  262625           0      262624      162291           0
  162291           0      162312      100312           0
  262624           0      262632      162312           0
       0      262603           0           0      162312
       0      162312           0           0      100312

The entries of each matrix are bunched around 3 values. What's more, the stochastic matrices obtained by adding unity to each entry and then row-normalizing agree to one part in a thousand.

Is there a (simple) explanation for this?


The reason that the matrices are alike is that, up to a multiplicative factor, each is approximately encoding $Area(T(R_i) \cap R_j))$. This is because the rational points with denominator $q$ are close being distributed across the $R_i$ in proportion to their areas. (I think you're counting these points with multiplicity equal to the period, but I don't think this makes a difference.) In particular, notice that the ratio of the three values that the entries of each matrix bunch around is about $\phi^2 : \phi : 1$, where $\phi$ is the golden ratio.

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  • $\begingroup$ Nice, I figured it would have a simple answer. Many thanks. I guess it would be straightforward to extend this to generic hyperbolic toral automorphisms since their SRB measures are all the same (viz. Lebesgue measure). $\endgroup$ – Steve Huntsman Dec 15 '09 at 0:14
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    $\begingroup$ Sketch of generalization: let $f$ be an Anosov diffeomorphism on a compact manifold $M$ with SRB measure $\mu$ and Markov partition $\mathcal{R} =$ {$R_1,\dots,R_n$}. Define a stochastic matrix $P_{jk} := \mu(f(R_j) \cap R_k)$. If $X_\ell$ is a set of cardinality $\ell$ uniformly distributed w/r/t $\mu$ then the row-normalization of $T_{(X)} := \sum_{x \in X_\ell} T(x)$ converges to $P$ (presumably in measure). $\endgroup$ – Steve Huntsman Dec 15 '09 at 0:28

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