3 Removed number theory tag based on answer

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: 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?

(PS/FYI: I've put the number theory tag on because the arithmetic properties of the cat map rely on such things as the theory of quadratic fields.)

2 fixed two minor typos
1