An SO(2,2) Iterated Function System Part 1

 This is the first, introductory part of a short series of posts explaining the meaning of the picture below:

The last part of "Notes on Clifford algebra Cl(2,0)" ended with two one-parameter subgroups of SL(2,R): LT(a) and RT(a). They are left- and right-triangular matrices with1 on the diagonal and parameter a in the lower left (resp. upper right) corner. Cf. Eqs. (47) and (48) in the notes. LT and RT can act on Cl(2) matrices from the left or from the right, with left and right actions commuting.

Now let us collect together several facts.

1. Our work area is R^2,2 - the Möbius extension of R^1,1, where  R^1,1 is the Minkowski space-time with only one space dimension. The signature of  R^2,2 is (++--).
2. The we restrict our attention to the null cone N in R^2,2. It consists of points with coordinates  (x¹,x²,x³,x⁴) which satisfy (x¹)² + (x²)² - (x³)² - (x⁴)² = 0, or (x¹)² + (x²)² = (x³)² + (x⁴)².
3. The note Tuesday Special - Tetractys and Lattice Infinity we have discussed "balanced tetrads: 
   a² + b² = c² + d², (a,b,c,d integers),           (1)
   and the algorithm of generating all of them:
   

   Proposition 1. Every primitive solution of  (1) is of the form

   a = (mp+nq)/2,
   b = (np-mq)/2,
   c = (mp-nq)/2,
   d = (mq+np)/2,

   where m,n,p,q are integers. Conversely, for any integers m,n,p,q such that a,b,c,d are integers, the formula above provides a solution of  a² + b² =c² + d².

4. Points on N with all four integer coordinates  are "balanced tetrads" as in 3.
5.  Each SL(2,R) transformation induces an SO(2,2) transformation of R2,2, that maps the null cone N into itself.
6. SO(2,2) matrices with integer coefficients map points of N with integer coefficients to other points of N with integer coefficients.  
   
7. Sl(2,R) matrices RT(2a), LT(2a), for a = 1 and a = -1, induce SO(2,2) transformations with integer coefficients.
   

The last observation follows from the explicit formulas (45) in Notes. Denoting by I₂ the 2x2 identity matrix, we have:

For left actions:

Λ(I₂, LT(2a)) = {{-1, -a, -a, 0}, {a, -1, 0, -a}, {-a, 0, -1, a}, {0, -a, -a, -1}};
Λ(I₂, RT(2a)) = {{-1, a, -a, 0}, {-a, -1, 0, -a}, {-a, 0, -1, -a}, {0, -a, a, -1}};

For right actions:

Λ(LT(2a), I₂) = {{-1, -a, a, 0}, {a, -1, 0, -a}, {a, 0, -1, -a}, {0, -a, a, -1}};
Λ(RT(2a), I₂) = {{-1, a, a, 0}, {-a, -1, 0, -a}, {a, 0, -1, a}, {0, -a, -a, -1}};

We choose a=1 and a = -1 and obtain altogether 8 SO(2,2) matrices with integer coefficients. We can use these eight matrices to construct an iterated function system on the torus as follows.

Iterated function system (IFS) from eight SO(2,2) matrices.

We have obtained eight SO(2,2) matrices, let us call them M₁,...,M₈:

M₁ = {{1, 1, -1, 0}, {-1, 1, 0, 1}, {-1, 0, 1, 1}, {0, 1, -1, 1}},

M₂ = {{1, -1, 1, 0}, {1, 1, 0, -1}, {1, 0, 1, -1}, {0, -1, 1, 1}},

M₃ = {{1, -1, -1, 0}, {1, 1, 0, 1}, {-1, 0, 1, -1}, {0, 1, 1, 1}},

M₄ = {{1, 1, 1, 0}, {-1, 1, 0, -1}, {1, 0, 1, 1}, {0, -1, -1, 1}},

M₅ = {{1, 1, 1, 0}, {-1, 1, 0, 1}, {1, 0, 1, -1}, {0, 1, 1, 1}},

M₆ = {{1, -1, -1, 0}, {1, 1, 0, -1}, {-1, 0, 1, 1}, {0, -1, -1, 1}},

M₇ ={{1, -1, 1, 0}, {1, 1, 0, 1}, {1, 0, 1, 1}, {0, 1, -1, 1}},

M₈ = {{1, 1, -1, 0}, {-1, 1, 0, -1}, {-1, 0, 1, -1}, {0, -1, 1, 1}}.

We can start now the IFS-game. We select an initial point x₀ on N with integer coordinates. For instance x₀ = (0,1,0,1) is a good candidate. We apply to x0 each of the matrices Mi to obtain 8 new points x_i = M_i x₀. To each of the new point we apply each of Mi. We obtain 64 points x_ji = M_jx_i. And so on. At step n we obtain 8ⁿ points. Each of them is a point on N with integer coordinates, thus defining a "balanced tetrad" of the type a² + b² = c² + d².

To be continued...