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p-Adic length scale hypothesis and Mandelbrot fractals

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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 Pp 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 Fq 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 Rn of roots at the level n are obtained as Rn= Rn(new)∪ Rn-1, where Rn(new) consist of q new roots emerging at level n. Each step gives qn-1 roots at the previous level and qn-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(rn)= rn-1,k for all roots rn-1,k at level n-1. The roots in Rn-1(new) give qn-1 new roots in Rn(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} → … .

  • 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)(ri-rj)2

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

    ri∈ D(n,new),rj ∈ D(n-1)(ri-rj)2.

  • At the limit n→ ∞, the set Rn(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). 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}.
  • 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 Mn= 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.


    Source: https://matpitka.blogspot.com/2025/03/p-adic-length-scale-hypothesis-and.html


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