Can one define the analogs of Mandelbrot and Julia sets in TGD framework?

The stimulus to this contribution came from the question related to possible higher-dimensional analogs of Mandelbrot and Julia sets (see this). The notion complex analyticity play a key role in the definition of these notions and it is not all clear whether one can define these analogs.

I have already earlier considered the iteration of polynomials in the TGD framework (see this) suggesting the TGD counterparts of these notions. These considerations however rely on a view of M⁸-H duality which is replaced with dramatically simpler variant and utilizing the holography=holomorphy principle(see this) so that it is time to update these ideas.

This principle states that space-time surfaces are analogous to Bohr orbits for particles which are 3-D surfaces rather than point-like particles. Holography is realized in terms of space-time surfaces which can be regarded as complex surfaces in H=M⁴× CP₂ in the generalized sense. This means that one can give H 4 generalized complex coordinates and 3 such generalized complex coordinates can be used for the 4-surface. These surfaces are always minimal surfaces irrespective of the action defining them as its extermals and the action makes itself visible only at the singularities of the space-time surface.

Ordinary Mandelbrot and Julia sets

Consider first the ordinary Mandelbrot and Julia sets.

1. The simplest example of the situation is the map f:z→ z²+c. One can consider the iteration of f by starting from a selected point z and look for various values of complex parameter c whether the iteration converges or diverges to infinity. The interface between the sets of the complex c-plane is 1-D Mandelbrot set and is a fractal. One can generalize the iteration to an arbitrary rational function f, in particular polynomials.
2. For polynomials of degree n also consider n-1 parameters c_i, i=1,…,n, to obtain n-1 complex-dimensional analog of Mandelbrot set as boundaries of between regions where the iteration lead or does not lead to infinity. For n=2 one obtains a 4-D set.
3. One can also fix the parameter c and consider the iteration of f. Now the complex z-plane decomposes to two a finite region with a finite number of components and its complement, Fatou set. The iteration does not lead out from the finite region but diverges in the complement. The 1-D fractal boundary between these regions is the Julia set.

Holography= holomorphy principle

The generalization to the TGD framework relies heavily on holography=holomorphy principle.

1. In the recent formulation of TGD, holography required by the realization of General Coordinate Invariance is realized in terms of two functions f₁,f₂ of 4 analogs of generalized complex coordinates, one of them corresponds to the light-like (hypercomplex) M⁴ coordinate for a surface X²⊂ M⁴ and the 3 complex coordinates to those of Y² orthogonal to X² and the two complex coordinates of CP₂.

Space-time surfaces are defined by requiring the vanishing of these two functions: (f₁,f₂)=(0,0). They are minimal surfaces irrespective of the action as long it is general coordinate invariant and constructible in terms of the induced geometry.

In the number theoretic vision, the functions f_i are naturally rational functions or polynomials of the 4 generalized complex coordinates. I have proposed that the coefficients of polynomials are rationals or even integers, which in the most stringent approach are smaller than the degree of the polynomial. In the most general situation one could have analytic functions with rational Taylor coefficients.

The polynomials f_i=P_i form a hierarchy with respect to the degree of P_i, and the iteration defined is analogous to that appearing in the 2-D situation. The iteration of P_i gives a hierarchy of algebraic extensions, which are central in the TGD view of evolution as an increase of algebraic complexity. The iteratikon would also give a hierarchy of increasingly complex space-time surface and the approach to chaos at the level of space-time would correspond to approach of Mandelbrot or Julia set.

One could consider the iteration as (f₁,f₂)→ (f₁• f₁,f₂• f₂) continued indefinitely. One could also iterate only f₁ or f₂. Each step defines by the vanishing conditions a 4-D surface, which would be analogous to the image of the z=0 in the 2-D iteration. The iterates form a sequence of 4-surfaces of H analogous to a sequence of iterates of z in the complex plane.

The sequence of 4-surfaces also defines a sequence of points in the “world of classical worlds” (WCW) analogous to the sequence of points z,f(z),…. This conforms with the idea that 3-surface is a generalization of point-like particles, which by holography can be replaced by a Bohr orbit-like 4-surface.

What the WCW analogy of the Mandelbrot and Julia sets could look like?

Do the analogs of Mandelbrot and Julia sets exist at the level of space-time?

Could one identify the 3-D analogs of Mandelbrot and Julia sets for a given space-time surface? There are two approaches.

1. The parameter space (c₁,c₂) for a given initial point h of H for iterations of f₁-c₁,f₂-c₂) defines a 4-D complex subspace of WCW. Could one identify this subset as a space-time surface and interpret the coordinates of H as parameters? If so, there would be a duality, which would represent the complement of the Fatou set (the thick Julia set) defined as a subset of WCW as a space-time surface!
2. One could also consider fixed points of iteration for which iteration defines a holomorphic map of space-time surface to itself. One can consider generalized holomorphic transformations of H leaving X⁴ invariant locally. If they are 1-1 maps they have interpretation as general coordinate transformations. Otherwise they have a non-trivial physical effect so that the analog of the Julia set has a physical meaning. For these transformations one can indeed find the 3-D analog of Julia set as a subset of the space-time surface. This set could define singular surface or boundary of the space-time surface.

Could Mandelbrot and Julia sets have 2-D analogs?

What about the 2-D analogs of the ordinary Julia sets? Could one identify the counterparts of the 2-D complex plane (coordinate z) and parameter space (coordinate c).

1. Hamilton-Jacobi structure defines what the generalized complex structure is (see this) and defines a slicing of M⁴ in terms of integrable distributions of string world sheets and partonic 2-surfaces transversal or even orthogonal to each other. Partonic 2-surface could play the role of complex plane and string world sheet the role of the parameter space or vice versa.

Partonic 2-surfaces resp. and string world sheet having complex resp. hyper-complex structures would therefore be in a key role. M⁸-H duality maps these surfaces to complex resp. co-complex surfaces of octonions having Minkowskian norm defined as number theoretically as Re(o²).

The natural proposal is that the light-like 3-surfaces defining boundaries between Euclidean and Minkowskian regions of the space-time surface define light-like orbits of the partonic 2-surface. And string world sheets are minimal surfaces having light-like 1-D boundaries at the partonic 2-surface having physical interpretation as world-lines of fermions.

One could also iterate only f₁ or f₂ allow the parameter c of the initial value of f₁ to vary. This would give the analog of Mandelbrot set as a set of 2-D surfaces of H and it might have dual representation as a 2-surface.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.