Koenig's function is an important concept in the theory of dynamical systems, specifically in the study of one-dimensional iterated function systems. It is a function that solves the Schröder's equation, which is an eigenvalue equation for the composition operator Ch. In other words, Koenig's function is a function that provides insight into the behavior of a dynamical system near a fixed point.

The Schröder's equation is given by:

Ψ(h(x)) = sΨ(x)

Here, Ψ is the Koenig's function, h(x) is a function representing the dynamics of the system, and s is an eigenvalue of the composition operator Ch. The composition operator Ch sends a function f to f(h(.)), meaning it composes the function f with the function h.

A fixed point a of the function h satisfies the condition h(a) = a. If Ψ(a) is finite and Ψ′(a) (the derivative of Ψ with respect to x evaluated at a) does not vanish or diverge, the eigenvalue s is given by:

s = h′(a)

Koenig's function is useful for analyzing the local behavior of dynamical systems near fixed points. By finding the Koenig's function and the eigenvalue s, we can gain insight into the stability and other properties of the fixed point. For instance, if the absolute value of the eigenvalue s is less than 1, the fixed point is stable (attractive), and if it is greater than 1, the fixed point is unstable (repulsive).

See KoenigsFunctionExpansion for information on how to express the Koenigs function of a function as a recursive series expansion

Locally Conjugate Dynamics In Neighborhoods Around Fixed-Points

The Koenig's function Ψ provides a local conjugacy between the dynamics of the given system defined by $f(x)$ and a simple scalar multiplication by x within a certain neighborhood around the fixed point.

The Koenig's function Ψ is an analytic function near the fixed point a, and it conjugates the dynamics of the system to the multiplication by the eigenvalue λ (which is equal to h'(a), the derivative of the function h at the fixed point a):

Ψ(h(x)) = λΨ(x)

This conjugacy is valid within a neighborhood (a disc) around the fixed point, where the function Ψ is well-defined and analytic. This means that, within this disc, the behavior of the system is essentially equivalent to the simple dynamics given by scalar multiplication.

To say that something conjugates the dynamics of a system to multiplication by the eigenvalue λ means that it establishes an equivalence between the original dynamics of the system and a simpler dynamical system that involves only scalar multiplication by λ. This equivalence helps to simplify the analysis of the system's behavior, especially in a neighborhood around a fixed point.

The conjugacy means that within a neighborhood around the fixed point, the behavior of the dynamical system can be understood by considering the simpler scalar multiplication dynamics. Essentially, the function Ψ provides a coordinate transformation that simplifies the dynamics of the system in the vicinity of the fixed point.

It is important to note that this conjugacy is only valid within a certain neighborhood around the fixed point, where the function Ψ is well-defined and analytic. Beyond this neighborhood, the conjugacy may not hold, and the global behavior of the system may be more complex than the simplified scalar multiplication dynamics.

Orbits

The Koenig's function (or Schröder's function) Ψ provides a coordinate transformation that simplifies the analysis of the dynamics of a system by transforming it into a simpler form involving scalar multiplication. In this new coordinate system, the dynamical system is conjugated to a dilation, which means the system's orbit looks like a simple scaling transformation.

As pointed out int the preceeding section, the conjugacy relationship provided by the Koenig's function is given by:

$$h(x) = Ψ^{-1}(\lambda\cdotΨ(x))$$

where $h(x)$ is the original dynamics, λ is the eigenvalue (equal to h'(a) where a is the fixed point), and Ψ^(-1) is the inverse of the Koenig's function. This relationship holds for all functional iterates, i.e., for all real numbers t, not necessarily positive or integer:

$$h_t(x) = Ψ^{-1}(\lambda^t\cdotΨ(x))$$

This conjugacy allows us to reconstruct the smooth orbit (or flow) of the dynamical system and analyze its behavior in the new coordinate system provided by Ψ(x).

The Koenigs function is a Normalized Schlicht Function

A Koenigs function $F(z)$ exists for a holomorphic function $G(z)$ when $G(z)$ has an attracting fixed point at the origin (i.e., $G(0) = 0$) and the multiplier is strictly between 0 and 1 (i.e., $0 < |G'(0)| < 1$).

When the conditions are such that it exists, $F(z)$ satisfies the following properties:

1. $F(0) = 0$
2. $F'(0) = 1$
3. $F(G(z)) = \lambda * F(z)$, where $\lambda = G'(0)$

The Koenigs function $F(z)$ is holomorphic since it is a composition of holomorphic functions, $G(z)$ and $F(z)$. Moreover, it satisfies the normalization conditions required for a schlicht function, $F(0) = 0$, and $F'(0) = 1$. Now, let's demonstrate that $F(z)$ is one-to-one and onto in a neighborhood of the origin.

Given the attracting nature of the fixed point and the fact that $0 < |\lambda| < 1$, the function $G(z)$ brings points closer to the origin upon iteration. Due to this property, $G(z)$ is injective (one-to-one) and surjective (onto) in a neighborhood of the origin. As a consequence, the Koenigs function $F(z)$, which satisfies the functional equation $F(G(z)) = \lambda * F(z)$, is also injective and surjective in a neighborhood of the origin.

Therefore, since $F(z)$ is holomorphic, satisfies the normalization conditions, and is injective and surjective in a neighborhood of the origin, the Koenigs function $F(z)$ is schlicht in a neighborhood of the origin when it exists.