Properties of Fixed Points

Oct 13, 2025 by windailys.com 125 views

Introduction

In the realm of complex analysis, the study of automorphisms of the disk is a fundamental area of research. Automorphisms of the disk are bijective holomorphic functions from the unit disk to itself. In this article, we will focus on the fixed points of automorphisms of the disk, specifically the automorphism $T(z) = \lambda \frac{a-z}{1-\bar{a}z}$ , where $a, \lambda \in \mathbb{C}$ , $|\lambda|=1$ , $0<|a|<1$ , and $\lambda \neq 1$ . The fixed point equation of this automorphism will be derived, and the properties of its fixed points will be explored.

Background

The unit disk, denoted by $\mathbb{D}$ , is the set of complex numbers $z$ such that $|z|<1$ . The automorphism group of the disk, denoted by $\text{Aut}(\mathbb{D})$ , consists of all bijective holomorphic functions from $\mathbb{D}$ to itself. The automorphisms of the disk can be represented by the following formula:

T(z) = \lambda \frac{a-z}{1-\bar{a}z}

where $a, \lambda \in \mathbb{C}$ , $|\lambda|=1$ , $0<|a|<1$ , and $\lambda \neq 1$ . This formula represents a Möbius transformation, which is a bijective holomorphic function from the extended complex plane to itself.

Fixed Point Equation

To find the fixed points of the automorphism $T(z)$ , we need to solve the fixed point equation:

T(z) = z

Substituting the formula for $T(z)$ , we get:

\lambda \frac{a-z}{1-\bar{a}z} = z

Multiplying both sides by $1-\bar{a}z$ , we get:

\lambda (a-z) = z(1-\bar{a}z)

Expanding both sides, we get:

\lambda a - \lambda z = z - \bar{a}z^2

Rearranging the terms, we get:

\lambda z + \bar{a}z^2 = z - \lambda a

Factoring out $z$ , we get:

z(\lambda + \bar{a}z) = z - \lambda a

Dividing both sides by $z$ , we get:

\lambda + \bar{a}z = 1 - \frac{\lambda a}{z}

Multiplying both sides by $z$ , we get:

\lambda z + \bar{a}z^2 = z - \lambda a

Rearranging the terms, we get:

\bar{a}z^2 + (\lambda - 1)z + \lambda a = 0

This is a quadratic equation in $z$ , and we can solve it using the quadratic formula:

z = \frac{-(\lambda - 1) \pm \sqrt{(\lambda - 1)^2 - 4\bar{a}\lambda a}}{2\bar{a}}

Properties of Fixed Points

The fixed points of the automorphism $T(z)$ are the solutions to the quadratic equation:

\bar{a}z^2 + (\lambda - 1)z + \lambda a = 0

The properties of the fixed points can be derived from the quadratic formula:

z = \frac{-(\lambda - 1) \pm \sqrt{(\lambda - 1)^2 - 4\bar{a}\lambda a}}{2\bar{a}}

The fixed points are complex numbers, and they can be represented in the form:

z = x + iy

where $x$ and $y$ are real numbers.

The fixed points are not necessarily distinct, and they can be repeated. The multiplicity of the fixed points can be determined by the discriminant of the quadratic equation:

\Delta = (\lambda - 1)^2 - 4\bar{a}\lambda a

If the discriminant is positive, the fixed points are distinct. If the discriminant is zero, the fixed points are repeated. If the discriminant is negative, the fixed points are complex conjugates.

Conclusion

In this article, we have derived the fixed point equation of the automorphism $T(z) = \lambda \frac{a-z}{1-\bar{a}z}$ , where $a, \lambda \in \mathbb{C}$ , $|\lambda|=1$ , $0<|a|<1$ , and $\lambda \neq 1$ . We have also explored the properties of the fixed points, including their multiplicity and the conditions under which they are distinct, repeated, or complex conjugates.

References

Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill.
Conway, J. B. (1995). Functions of One Complex Variable. Springer-Verlag.
Lang, S. (1999). Complex Analysis. Springer-Verlag.

Q: What is the fixed point equation of the automorphism $T(z) = \lambda \frac{a-z}{1-\bar{a}z}$ ?

A: The fixed point equation of the automorphism $T(z)$ is:

\bar{a}z^2 + (\lambda - 1)z + \lambda a = 0

Q: How do I solve the fixed point equation?

A: To solve the fixed point equation, you can use the quadratic formula:

z = \frac{-(\lambda - 1) \pm \sqrt{(\lambda - 1)^2 - 4\bar{a}\lambda a}}{2\bar{a}}

Q: What are the properties of the fixed points?

A: The properties of the fixed points can be derived from the quadratic formula. The fixed points are complex numbers, and they can be represented in the form:

z = x + iy

where $x$ and $y$ are real numbers. The fixed points are not necessarily distinct, and they can be repeated. The multiplicity of the fixed points can be determined by the discriminant of the quadratic equation:

\Delta = (\lambda - 1)^2 - 4\bar{a}\lambda a

If the discriminant is positive, the fixed points are distinct. If the discriminant is zero, the fixed points are repeated. If the discriminant is negative, the fixed points are complex conjugates.

Q: What is the significance of the fixed points?

A: The fixed points of the automorphism $T(z)$ are significant because they represent the points that remain unchanged under the transformation. In other words, the fixed points are the points that are mapped to themselves under the transformation.

Q: Can the fixed points be used to determine the properties of the automorphism?

A: Yes, the fixed points can be used to determine the properties of the automorphism. For example, the fixed points can be used to determine the type of the automorphism, such as whether it is a rotation or a scaling.

Q: How do I determine the type of the automorphism?

A: To determine the type of the automorphism, you can examine the fixed points. If the fixed points are distinct and have the same argument, the automorphism is a rotation. If the fixed points are repeated, the automorphism is a scaling.

Q: Can the fixed points be used to determine the properties of the disk?

A: Yes, the fixed points can be used to determine the properties of the disk. For example, the fixed points can be used to determine the radius of the disk.

Q: How do I determine the radius of the disk?

A: To determine the radius of the disk, you can examine the fixed points. The radius of the disk is equal to the distance between the fixed points.

Q: What are some applications of the fixed points of automorphisms of the disk?

A: The fixed points of automorphisms of the disk have many applications in mathematics and physics. Some examples include:

Conformal mapping: The fixed points of automorphisms of the disk can be used to determine the conformal mapping of the disk to itself.
Complex analysis: The fixed points of automorphisms of the disk can be used to determine the properties of complex functions.
Physics: The fixed points of automorphisms of the disk can be used to determine the properties of physical systems, such as the behavior of particles in a potential.

Conclusion

In this article, we have answered some common questions about the fixed points of automorphisms of the disk. We have discussed the fixed point equation, the properties of the fixed points, and some applications of the fixed points. We hope that this article has been helpful in understanding the fixed points of automorphisms of the disk.