In some article describes the method of generating a family of sequences of integers (chaotic mixing) using Toral Automorphisms theory and integer-lattices. The main mathematical structure in this process is a $\ 2х2 $ matrix A:

$$ A =\left [ \begin{matrix} 1&1\\1&2 \end{matrix} \right] $$

The iterated application of A on a point r (belonging to integer lattice L size of N and having coordinates as x and y), result in dynamical system, that can be represent as:

$$ \left [ \begin{matrix} x_{n+1}\\y_{n+1} \end{matrix} \right] = \left [ \begin{matrix} 1&1\\1&2 \end{matrix} \right] \left [ \begin{matrix} x_{n}\\y_{n} \end{matrix} \right] mod(N)$$

The article provides the following example:

Consider a two-dimensional diagonal integer-lattice (i.e., the diagonal points are the only ones considered in the iterated application of A).

$\ S = [1, 2, ... N-1]$ - some sequence of integers.

Let N = 26. After the $\ 8^{th}$ iteration from $\ S_0 = [1,2,3,4,5,6,7,8,...,24,25,26]$ we get $\ S_8 = [1,14,9,22,4,17,25,12,...,24,6,19]$.

I do not understand how exactly the two-dimensional diagonal integer-lattice looks like for this example, and how the$\ S_8 $ sequence came out...


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