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The Tao of Abel and Schroeder
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The Tao of Abel's and Schroeder's Functional Equations

Sums of Geometric Progressions
When it come to dynamics, everything is build on the sums of geometric progressions including . With geometric growth we approach exponential growth. Unless the base of the geometric progression is 1, then then sums are linear. The Abel Functional Equation manifest as the shift transformation, the simplest transformation in dynamics as it has no fixed point. This has been noted that the fractal for tetration's Abel's equation, , that the infinite spines neither curve inward or outward. It approaches being invariant under translation which imposes symmetry on the system. 

You can't get there from here
  • Abel's functional equation
  • Schroeder's Functional Equation  
Physics
The Shell Thron boundary points associated with the  roots of unity impose a n-fold symmetry with the Abel 1-fold symmetry. The good news is that this allows up to conceptualize algebraic structures as the monster group existing on a higher dimensional Shell Thron boundary. In the Shell Thron boundary is likely equivalent to being measure preserving which is important because in the mathematical physics of dynamics a dynamical system is an iterated measure preserving function.
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