Tetration:Dynamics

From Tetration.net

1 Dynamics - Extending Tetration
- 1.1 Symbolic Dynamics - The First Hurtle
- 1.2 Topological Conjugacy - The Second Hurtle
  - 1.2.1 Functional Equations
  - 1.2.2 Linearization

Dynamics - Extending Tetration

The following approach laid out for extending tetration is based on continuously iterated functions and is therefore is not only valid for tetration but for pentation, hexation and through out the Ackermann Function. An additional benefit is that continuous iteration of necessity yields the Lagrange Inversion which provides the inverse of functions where $f (0) = 0$ . If the Taylor series of $f n (z)$ can be constructed in general, then setting $n = 1$ gives $f - 1 (z)$ . So continuous iteration should be as useful for deriving the different inverses of the Ackermann function like the super-logarithm. Riordan's Combinatorial Identities chapter on Partition Polynomials has a stiff amount umbral calculus but he begins by taking on the problem of inverses of functions using Bell polynomials to find the Taylor series of $g (z) = f - 1 (z)$ by solving for the succesive derivitives using $f (g (z)) = z$ , $D f (g (z)) = 1$ , $D k f (g (z)) = 0, k > 1$ .

Symbolic Dynamics - The First Hurtle

The first problem for finding a simple extension for tetration is that while the exponential function is single-valued; its inverse the logarithmic function is infinitely multivalued. Therefore while $n a$ will be single-valued for $n \in \mathbb{N}\$ , $- 1 a$ will be infinitely multivalued for $n \in \mathbb{Z}\$ .

Consider the symbolic dynamics of the $- n e$ for $n \in \mathbb{N}\$

$\begin{matrix}\sum_\omega \end{matrix} = \{s=(s_0s_1s_2 \ldots)|s_n \in \mathbb{Z}\ \}$

${}^{-1}e = 2 \pi i j = \{\ldots, -4 \pi i, -2 \pi i, 0, 2 \pi i, 4 \pi i, \ldots \} = \bigcup {}^{-1}e_{(j)}\,$ for $j \in \mathbb{Z}\$ .

${}^{-2}e = \log(2 \pi i j) + 2 \pi i k = \bigcup {}^{-2}e_{(jk)}$ for $j,k \in \mathbb{Z}\$ .

${}^{-2}e_{(00)}= -\infty \,$

When people talk about extending tetration to the real or complex numbers, they are typically talking about defining $b a (0...)$ . My position is that any complete extension to the domain of tetration must account for the fact that $b a$ may take an uncountable infinity of values. A function $g$ can be defined on $s = (s 0 s 1 s 2 ...)$ such that $g : s \rightarrow x, x \in \mathbb{R}\$

${}^{b}a = \bigcup {}^{b}a_{(s_0s_1s_2 \ldots)}$ where $b \notin \mathbb{N}\$

Let $h (x)\equiv x a$ be the function for the Riemann branch where $- 2 a = - \infty$ . Then the region of convergence would prohibit the use of $h (x)$ for calculating $2 a$ . Of course there are always other Riemann branches, but ironically $h (x)$ is the only branch where $x a$ maps the positive reals into the reals.

Since my work in extending tetration is based on using fixed points, the technique I use applies to ${}^{b}a = \bigcup {}^{b}a_{(s_0s_1s_2 \ldots)}$ where $s 0 = s 1 = s 2 = ...$ ; in other words ${}^{b}a_{(s_0s_0s_0 \ldots)}$ .

Topological Conjugacy - The Second Hurtle

Falling back to a more humble goal, we ask if we can define ${}^{x}a_{(0 \ldots)} \,$ ? Unfortunately, even by restricting ourselves to the single Riemann branch with ${}^{x}a_{(0 \ldots)} \,$ , we still must contend with topological conjugacy. $x a (0...)$

Functional Equations

Just because we don't have a commonly accepted quantitative explanation of tetration doesn't mean that we don't already have an excellent qualitative explanation of tetration. Symbolic dynamics and topological conjugacy are part of the discipline of complex dynamics. While complex dynamics may not be of immediate help in extending tetration to the complex numbers, complex dynamics is directly useful for debugging or completely ruling out proposed extensions of tetration to the complex numbers. Topological conjugacy is a particularly good tool because it addresses the problem of linearization. In the late Eighteen Hundreds several different functional equations were found to be helpful to those trying to understand the dynamics of iterated functions in the complex plane. An alternative perspective is that if you are primarily interested in studying functional equations, then iterated functions can become a useful tool when trying to solve certain types of functional equations.

$f (h (x)) = c f (x)$ Schröder equation

$f (h (x)) = f (x) + 1$ Abel equation.

Linearization

Any attempt to extend tetration must ultimately have to propose some scheme for linearization, for moving from discretely iterated functions or maps to continuously iterated functions or flows. It is particularly useful to study the dynamics of functions in the neighborhood of their fixed points for two reasons. The dynamics are simplest at the fixed point and the dynamics at the fixed point are the paramount influence on the global dynamics of the iterated function. The ramification is that since there are several different types of fixed points present in tetration including the infinite family of rational fixed points. In order for tetration to be extended it must be shown that each type of fixed point can be linearized in a manner consistent with its specific functional equation. The specific form of linearization associated with each type of fixed point then provides the formula for each specific case of tetration.

Tetration Mandelbrot by Period	Tetration Mandelbrot by Escape
Hyperbolic Tetration - Julia set of $1.5 z$	Irrationally Neutral Tetration - Julia set of $(.4515 + 1.9629 i) z$
Parabolic Tetration - Julia set of $1.444667861 z$	Rationally Neutral Tetration - Julia set of period 2 bifurcation for $0.0659880358 z$
Rationally Neutral Tetration - Julia set of period 3 bifurcation for $(0.03095355 + 1.7392241043 i) z$	Rationally Neutral Tetration - Julia set of period 4 bifurcation for $(1.98933207 + 1.193282199 i) z$