Intervals of Tetration

[Daniel]

There are two subjects which have caused me a great deal of confusion, so I will try and make sense of them to myself, and if I get anything wrong, hopefully someone will correct me of my error. The subjects are Classification of fixed points / periodic points / cycles, and the Lyapunov characteristic / number / exponent. Since both of these subjects are properties of self-maps, they will vary with the base of the exponential in question.

I recommend A History of Complex Dynamics, From Schroder to Fatou and Julia by Daniel S. Alexander and Complex Dynamics by Lennart Carleson and Theodore W. Gamelin for an explaination of the classification of fixed points.

The Lyapunov characteristic is the most important concept here. There are linearization theorems in both complex dynamics and higher dimensional dynamics that show that the Lyapunov characteristic number or multiplier controls the dynamics of the map. Historically it found that certain cases of dynamical systems obeyed functional equations, but that understanding all cases of a given dynamical system like tetration involves understanding several different functional equations. Further more, the dynamics of these cases had more in common with the cases of other functions obeying the same functional equation than the different cases of the same function. The classification of fixed points is about proving that certain Lyapunov characteristic numbers are associated to certain functional equations.

The Lyapunov characteristic number is the first derivative of the function being mapped at the function's fixed point. Let \( f(x_0)=x_0 \), then \( f'(x_0) \). The Lyapunov characteristic numbers are also known as the multipliers \( f'(x_0)=\lambda \) because \( \lambda \) is the multiplier in Schoeder's functional equation \( f(h(x))=\lambda f(x) \). The Lyapunov exponent is just the log of the Lyapunov characteristic number \( \ln( \lambda) \) and is handy because if it's real value is negative then the fixed point is an attractor and if the real value is positive then the fixed point is a repellor. The nicely generalizes to matrices. If the Lyapunov exponent is a positive semidefinite matrix, then the fixed point is a repellor. A number of approaches to extending the definition of tetration are attempts to linearize exponential dynamics.

The Lyapunov characteristic number in tetration is just the log of the fixed point therefore \( e^{e^{2 \pi i x}} \) is the location of rationally neutral fixed points for rational values of \( x \) while \( e^{e^{2 \pi i x - e^{2 \pi i x }}} \) has rationally neutral fixed points for rational values of \( x \).

Most fixed points are hyperbolic fixed points; superattracting, rationally neutral and parabolic rationally neutral fixed points are exceptions. I haven't seen the term elliptic fixed point widely used in dynamics, instead the terms rationally neutral and irrationally neutral are used. The fact that the exponential function \( b^x \) is periodic results in \( b^x \) having a countably infinite number of fixed points for a specific \( b \ne 1 \).

As to the classification of intervals in tetration, in general the different intervals may have different behaviors when exponentially iterated, but they have fixed points in the complex plane under logarithmic iteration. For \( b = e^{-e} \), iteration of \( b^x \) cycles through two values, but iteration of \( b^{b^x} \) has a fixed point. The map of \( b = e^{-e} \) does have fixed points, an infinite number of hyperbolic repelling fixed points in the complex plane.