08/25/2007, 08:43 PM
andydude Wrote:Introduction
In this post we will talk about the Abel Functional Equation, and to some extent, the Schroeder Functional Equation and the Boettcher Functional Equation:
\( A(f(x)) = A(x) + 1 \)\( S(f(x)) = c S(x) \)\( B(f(x)) = B(x)^c \)
Upon inspection we find that each is the exponential of the previous function. In other words: \( S(x) = c^{A(x)} \) and \( B(x) = b^{S(x)} \). So in theory solving any of these functional equations should produce the same results. In practice, however, different methods are used to solve each of these functional equations.
Solving these equations don't give the same results.
Schroeder Functional Equation \( S(f(x)) = c S(x) \) gives the most general solution to dynamical systems which have hyperbolic fixed points.
Abel Functional Equation \( A(f(x)) = A(x) + 1 \) results in the solution for parabolic fixed points and is used for the roots of unity.
Boettcher Functional Equation \( B(f(x)) = B(x)^c \) works for superattracting fixed points. These different solutions end up being topologically different.
Daniel
