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 M^{8}-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=M^{4}× CP_{2} 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.

- The simplest example of the situation is the map f:z→ z
^{2}+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. - 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. - 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.

- In the recent formulation of TGD, holography required by the realization of General Coordinate Invariance is realized in terms of two functions f
_{1},f_{2}of 4 analogs of generalized complex coordinates, one of them corresponds to the light-like (hypercomplex) M^{4}coordinate for a surface X^{2}⊂ M^{4}and the 3 complex coordinates to those of Y^{2}orthogonal to X^{2}and the two complex coordinates of CP_{2}.Space-time surfaces are defined by requiring the vanishing of these two functions: (f

_{1},f_{2})=(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 of TGD, M
^{8}-H-duality (see this) maps the space-time as a holomorphic surface X^{4}⊂ H is mapped to an associative 4-surface Y^{4}⊂ M^{8}. The condition for holography in M^{8}is that the normal space of Y^{4}is quaternionic.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. - In the TGD context, there are 4-complex coordinates instead of 1 complex coordinate z. The iteration occurs in H and the vanishing conditions for the iterates define a sequence of 4-surfaces. The initial surface is defined by the conditions (f
_{1},f_{2})=0. This set is analogous to the set f(z)=0 for ordinary Julia sets.One could consider the iteration as (f

_{1},f_{2})→ (f_{1}• f_{1},f_{2}• f_{2}) continued indefinitely. One could also iterate only f_{1}or f_{2}. 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.

- Also in this case, one can see whether the iteration converges to a finite result or not. In the zero energy ontology (ZEO), convergence could mean that the iterates of X
^{4}stay within a causal diamond CD having a finite volume.

**The counterparts of Mandelbrot and Julia sets at the level of WCW**

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

_{1},f

_{2})= (c

_{1},c

_{2}) equivalent with the roots of (f

_{1}-c

_{1}, f

_{2}-c

_{2}) =(0,0). Here c

_{1}and c

_{2}define complex parameters analogous to the parameter c of the Mandelbrot sent. One can iterate the two functions for all pairs (c

_{1},c

_{2}). One can look whether the iteration converges or not and identify the Mandelbrot set as the critical set of parameters (c

_{1},c

_{2}). The naive expectation is that this set is 3-D dimensional fractal.

_{1},f

_{2})=0 fixes the starting point (not necessarily uniquely since 3-D surface does not fix the Bohr orbit uniquely: this is the basic motivation for ZEO). The analogy with the initial point of iteration suggests that we can assume (f

_{1},f

_{2})=(c

_{1},c

_{2}) but this leads to the analog of the Mandelbrot set. The notions coincide at the level of WCW.

**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.

- The parameter space (c
_{1},c_{2}) for a given initial point h of H for iterations of f_{1}-c_{1},f_{2}-c_{2}) 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! - 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 X
^{4}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 in TGD?**

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).

- Hamilton-Jacobi structure defines what the generalized complex structure is (see this) and defines a slicing of M
^{4}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. M^{8}-H duality maps these surfaces to complex*resp.*co-complex surfaces of octonions having Minkowskian norm defined as number theoretically as Re(o^{2}). - In the case of Julia sets, one could consider generalized holomorphic transformations of H mapping X
^{4}to itself as a 4-surface but not reducing to 1-1 maps. If f_{2}(f_{1}) acts trivially at the partonic 2-surface Y^{2}(string world sheet X^{2}), the iteration reduces to that for f_{1}(f_{2}). Within string world sheets and partonic 2-surfaces the iteration defines Julia set and its hyperbolic analog in the standard way. One can argue that string world sheets and partonic 2-surfaces should correspond to singularities in some sense. Singularity could mean this fixed point property.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 f

_{1}or f_{2}allow the parameter c of the initial value of f_{1}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. - The 2-D analog of the Mandelbrot set could correspond to a set of 2-surfaces obtained by fixing a point of the string world sheet X
^{2}. Also now one could consider holomorphic maps leaving the space-time surface locally but not acting 1-1 way. The points of Y^{2}would define the values of the complex parameter c remaining invariant under these maps. The convergence of the iteration of f_{1}in the same sense as for the Mandelbrot fractal would define the Mandelbrot set as a critical set. For the dual of the Mandelbrot set X^{2}and Y^{2}would change their roles.

For a summary of earlier postings see Latest progress in TGD.

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

## No comments:

Post a Comment