The Analytic Continuation of the Ackermann Function

What lies beyond exponentiation?

Extending the arithmetic operations beyond addition, multiplication, and

exponentiation to the complex numbers.

Overview

Very high level overview because of the amount of material in multiple branches of mathematics.

• Complex Systems – A New Kind of Science

• Arithmetic

• Dynamics of the Complex Plane

• Combinatorics

New Kind of Science

• Chaos beyond exponentiation.• Vertical catalog of complex systems.• Based on iterated functions.• Arithmetic and physics are two major roles

played by iterated functions.• Iterated functions as a candidate for a

fundamental dynamical system in both mathematics and physics.

Dynamics and Combinatorics

Arithmetic

• Arithmetic is part of the Foundations of Mathematics.

• Ackermann function is a recursive function that isn’t primitively recursive.

• Different definitions of the Ackermann function.

• Transfinite mathematics

Operator Spiral Ackermann Knuth Conway

Addition a+b ack(a,b,0)    

Multiplication a*b ack(a,b,1)    

Exponentiation ab ack(a,b,2) a ↑ b a→b→1

Tetration ba ack(a,b,3) a ↑↑ b a→b→2

Pentation ba ack(a,b,4) a ↑↑↑ b a→b→3

Hexation   ack(a,b,5) a↑↑↑↑b a→b→4

...     ... ...

Circulation   ack(a,b,∞) a ↑∞ b a→b→∞

Systems of Notation for Arithmetic Operators

Definition of Ackermann Function

Let f(x) ≡ a → x → k and f(1) = a → 1 → k = a;

then

f2(1) = f(a) = a → a → k

= a → 2 → (k+1)

f3(1) = f(a → a → k) = a → (a → a → k) → k

= a → 3 → (k+1)

Negative Integers a → 1 → 2 = a a → 0 → 2 = 1 This differs from the historical Ackermann function where a → 0 → 2 = a

The next two are multivalued so the values on the principle branch are shown. a → -1 → 2 = 0 a → -2 → 2 = -∞

Period three behavior -1 → 0 → 3 = 1 -1 → 1 → 3 = -1 -1 → 2 → 3 = -1 → -1 → 2 = 0 -1 → 3 → 3 = -1 → (-1 → 2 → 3) → 2 = -1 → 0 → 2 = 1

The first indication that for negative integer the Ackermann function can be very stable.for n>2 -1→k→n = - 1 → (k+3) → n.

-1 → 0 → n = 1 -1 → 1 → n = -1 -1 → 2 → n = 0 -1 → 3 → n = 1

Hypothesis

Tetration Through Octation

For 1 ≤ a < 2a → ∞ → ∞ = a

a → ∞ → k useful for creating a series of “interesting” transfinite number.

Transfinite nature of circulation:

2 → 2 → ∞ = 4

2 → 3 → ∞ = 2 → (2 → 2 → ∞) → ∞

= 2 → 4 → ∞

= 2 → (2 → (2 → 2 → ∞) → ∞) → ∞

= 2 → (2 → 4 → ∞) → ∞

= ∞

3 → 2 → ∞ = 3 → 3 → ∞

= 3 → (3 → 3 → ∞) → ∞

= 3 → (3 → (3→3→∞) → ∞) → ∞

= ∞

Transfinite Ackermann Expansions

Tetration

• First objective is understanding tetration.

• What if tetration and beyond is vital for mathematics or physics?

• With so many levels of self organization in the world, tetration and beyond likely exists.

Julia set for the map of ez

Tetration by period

Tetration by escape

Dynamical Systems

• Iterated function as a dynamical system.• Analytic continuation can be reduced to a problem

in dynamics.• Taylor series of iterated function. Most

mathematicians believe this is not possible, but my research is consistent with other similar research from the 1990’s.

• Iterated exponents for single valued and iterated logarithms for multi-valued solutions.

Fa di Bruno formula

• Hyperbolic case

• Maps are Flows

nk

n

k

nn

knn n

gg

kk

fnfgfgY

!!1!!

!),,(

1

)(1

11

nm

knmk

m

n

m

knm fD

m

fDDf

kk

fDmfD

m

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!1!1!!

! 1

1)(1

1

• Derivatives of composite functions.

• Fa di Bruno difference equation.

Classification of Fixed Points

• Topological Conjugancy and Functional Equations – Multiple Cases for Solution

• Fixed Points in the Complex Plane– Superattracting– Hyperbolic (repellors and attractors)– Irrationally Neutral– Rationally Neutral– Parabolic Rationally Neutral

Combinatorics• OEIS – On Line Encyclopedia of Integer Sequences

• Umbral calculus and category theory.

• Bell polynomials as derivatives of composite functions.

Dm f(g(x))

• Schroeder summations.

• Hierarchies of height n and the combinatorics of tetration.

Exponential Generating Functions

Hierarchies of 2 or Bell Numbers

Hierarchies of 3

Hierarchies of 4

xe - Tetration as phylogenetic trees of width x

Schroeder Diagrams & Summations

Validations• Deeply consistent with dynamics. • f a (f b(z)) - f a+b(z) = 0 verified for a number of

solutions.• Software validates for first eight derivatives and

first eight iterates. Mathematica software reviewed by Ed Pegg Jr.

• A number of combinatorial structures from OEIS computed correctly including fractional iterates.

NKS Summary

• Wolfram’s main criticism is inability of continuous mathematics to deal with iterated functions.

• CAs are mathematics not physics, many non-physical solutions.

• “Physics CA” needs OKS for validation.• CAs appear incompatible with Lorenz

transforms and Bell’s Theorem.

Summary• Subject is in protomathematics stage, but becoming acceptable areas

of research; numerous postings on sci.math.research lately.

• Arithmetic → Dynamics → Combinatorics → Arithmetic

• If maps are flows, then the Ackermann function is transparently extended.

• Suggests time could behave as if it is continuous regardless of whether the underlying physics is discrete or continuous.

• Continuous iteration connects the “old” and the “new” kinds of science. Partial differential iterated equations

• Tetration displays “sum of all paths” behavior, so logical starting place to begin looking for tetration in physics is QFT and FPI. Tetration and many other iterated smooth functions appear compatible with the Lorenz transforms and Bell’s Theorem.