Consider a commutative 3D algebra $T$ where the nonreal units $x,y$ satisfy
$$x^2 = A_1 + A_3 y $$ $$xy = 1 + B_2 x + B_3 y$$ $$y^2 = C_1 + C_3 y$$
where all the parameters $A_1,A_3,B_2,..$ are real.
Also we want this algebra to be a non-associative algebra.
Squaring a number in this algebra is done by applying the above to arrive at this formula :$[*]$
$$(a + b x + cy)^2 =$$
$$a^2 + b^2 A_1 + 2bc + c^2 C_1 +$$ $$(2ab + 2bc B_2) x + $$ $$(2ac + b^2 A_3 + 2bc B_3 + c^2 C_3 ) y$$
($a,b,c$ are also real)
Now consider the sequence
$z(0) = a + bx + cy = a(0) + b(0) x + c(0) y$
$z(1) = (a+bx+cy)^2 = a(1) + b(1) x + c(1) y$
$z(2) = z(1)^2 = a(2) + b(2) x + c(2) y$
$z(n) = z(n-1)^2 = a(n) + b(n)x + c(n)y$
Where the squares are computed by the formula $*$ above.
Now we wonder about the 3D julia set created by this sequence.
Main question :
So for a given $a,b,c$ and the parameters $A_1,A_3,B_2,...$ when does the sequence converge to infinity ?
Is there some kind of " norm formula " that determines this ?
( compare with complex numbers : if $a^2 + b^2 > 1 $ repeatly squaring converges to infinity, otherwise it does not. )
Secondary question :
When is the Julia set connected ?
Pictures are very much appreciated.
Keep in mind we only consider the non-associative algebra's of this type. Otherwise it is just isomophic to squaring complex numbers anyway.
I assume some julia sets turn out to be fractals.
Will we get pictures like the Mandelbulb ?
Closely related :
Is this 3D algebra $T$ power-associative?
Solving a specific polynomial system of equations of degree $2$