p-Adic length scale hypothesis and Mandelbrot fractals

This post (see this) is a continuation to the previous suggesting a definition of generalized p-adic numbers in which functional powers of map g₁: C²→ C² for which g₁ is a prime polynomial P_p with coefficients in extension E of rationals with degree low than p of P would define the analogs of powers of p-adic prime. For simplicity, one can restrict E to rationals. The question was whether it might be possible to understand p-adic primes satisfying p-adic length scale hypothesis as ramified primes appearing as divisors of the discriminant of the iterate of g₁=P.

Generalized p-adic numbers as such are a very large structure and the systems satisfying the p-adic length scale hypothesis should be physically and mathematically special. Consider the following assumptions.

1. Consider generalized p-adic primes associated restricted to the case when f₂ is not affected in the iteration so that one has g=(g₁,Id) and g₁= g₁(f₁) is true. This would conform with the hypothesis that f₂ defines the analog of a slowly varying cosmological constant. If one assumes that the small prime corresponds to q=2, the iteration reduces to the iteration appearing in the construction of Mandelbrot fractals and Julia sets. If one assumes g₁= g₁(f₁,f₂), f₂ defines the analog of the complex parameter appearing in the definition of Mandelbrot fractals. The values of f₂ for which the iteration converges to zero would correspond to the Mandelbrot set having a boundary, which is fractal.
2. For the generalized p-adic numbers one can restrict the consideration to mere powers g₁ⁿ as analogs of powers pⁿ. This would be a sequence of iterates as analogs of abstractions. This would suggest g₁(0)=0.
3. The physically interesting polynomials g₁ should have special properties. One possibility is that for q=2 the coefficients of the simplest polynomials make sense in finite field F₂ so that the polynomials are P₂(z== f₁,ε) =z² +ε z= z(z+ε), ε= +/- 1 are of special interest. For q>2 the coefficients could be analogous to the elements of the finite field F_q represented as phases exp(i2π k/3).

Consider now what these premises imply.

1. Quite generally, the roots of P° ⁿ(g₁) are given R(n)= P°^(-n)(0). P(0)=0 implies that the set R_n of roots at the level n are obtained as R_n= R_n(new)∪ R_n-1, where R_n(new) consist of q new roots emerging at level n. Each step gives q^n-1 roots at the previous level and q^n-1 new roots.
2. It is possible to analytically solve the roots for the iterates of polynomials with degree 2 or 3. Hence for q= 2 and 3 (there is evidence for the 3-adic length scale hypothesis) the inverse of g₁ can be solved analytically. The roots at level n are obtained by solving the equation P(r_n)= r_n-1,k for all roots r_n-1,k at level n-1. The roots in R_n-1(new) give q^n-1 new roots in R_n(new).
3. For q=2, the iteration would proceed as follows:

   0→ {0, r₁} → {0,r₁} ∪ {r₂₁, r₂₂} → {0,r₁} ∪ {r₂₁, r₂₂}∪ {r₁₂₁, r₂₂₁, r₁₂₂, r₂₂₂} → ... .

4. The expression for the discriminant D of g₁°ⁿ can be deduced from the structure of the root set. D satisfies the recursion formula D(n)= D(n,new)× D(n-1) × D(n,new;n-1). Here D(n,new) is the product

   ∏_{ri,rj ∈ D(n,new)}(r_i-r_j)²

   and D(n,new;n-1) is the product

   ∏_{ri∈ D(n,new),rj ∈ D(n-1)}(r_i-r_j)².

5. At the limit n→ ∞, the set R_n(new) approaches the boundary of the Fatou set defining the Julia set.

As an example one can look at the iteration of g₁(z)= z(z-ε).

1. The roots of z(z-ε)=0 are {0,r₁}={0,ε}. At second level, the new roots satisfy z(z-ε)=r₁=ε given by {(ε/2)(1 +/- (1+4r₁)^1/2}. At the third level the new roots satisfy z(z-ε)=r₂ and given by {(ε/2)(1 +/- (1+4r₂)^1/2}.
2. The points z=0 and z=ε are fixed points. Assume ε=1 for definiteness. The image points w(z)= z(z-ε) satisfy the condition |w(z)/z|=|z-1|. For the disk D(1,1):|z-1| ≤ 1 the image points therefore satisfy |w| ≤ |z| ≤ 2 and belong to the disk D(0,2): |z |≤ 2.

   For the points in D(0,2)\ D(1,1) the image point satisfies |w|=|z-1||z| giving |z|-1 ≤ |w| ≤ |z|+1. Inside D(0,2)\ D(1,1) this gives 0 ≤|w| ≤ 3. Therefore w can be inside D(2,0) including D(1,1) also inside disk D(0,3).

   For the points z outside D(2,0)|w|=|z-1||z| ≥ 2. So that the iteration leads to infinity here.

3. For the inverse of the iteration relevant for finding the roots of f^°(-n) leads from the exterior of D(2,0) to its interior but cannot lead from interior to the exterior since in this case f would lead to exterior to interior. Hence the values of the roots w_n in ∪_n f°^(-n) belong to the disc D(2,0).

The conjecture deserving to be killed is that the discriminant D for the iterate has Mersenne primes as factors for primes n defining Mersenne primes M_n= 2ⁿ-1 and that also for other values of n D contains as a factor ramified primes near to 2ⁿ.

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter About Langlands correspondence in the TGD framework.

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.