Relationships between fixed points and eigenvectors in the group
© Demir et al.; licensee Springer 2013
Received: 28 November 2012
Accepted: 21 February 2013
Published: 13 March 2013
is the most frequently studied subgroup of the Möbius transformations. By adding anti-automorphisms
to the group , the group is obtained. The elements of this group correspond to matrices of . In this study, we consider the relationships between fixed points of the elements of the group G and eigenvectors of matrices corresponding to the elements of this group.
where and . Each Möbius transformation is a meromorphic bijection of onto itself and is called an automorphism of .
Möbius transformations form a group with respect to composition. If T is a Möbius transformation, then the composition is called an anti-automorphism of , where . The union of automorphisms and anti-automorphisms also form a group under the composition of functions.
By adding anti-automorphisms to the group , the group is obtained. The elements of this group correspond to the matrices of . If we take , then has the matrix presentation .
The fixed points of automorphisms and anti-automorphisms of the extended complex plane have especially been of great interest in many fields of mathematics, for example, in number theory, functional analysis, theory of complex functions, geometry and group theory (see [1–5] and references therein).
In , Beardon gave some relationships between the fixed points of Möbius maps and the lines of the eigenvectors of their corresponding matrices. So, these studies include the transformations of . In this study, we investigate similar relationships for transformations of . Thus we complete the problem for the group G.
In this section we give brief information about complex lines and fixed points of the transformations of G.
Definition 1 
Theorem 1 
Let f be a Möbius map with corresponding matrix A. Then if and only if is a line of eigenvectors of A.
Here we mention types of the elements in the group G briefly. For each , the point is called a fixed point of T if , and the trace of is defined by . There is a relation between the fixed points and the trace of a transformation of G. Thus we can determine fixed points location in with the trace.
, then there are two fixed points in and is called a hyperbolic element.
, then there is one fixed point in and is called a parabolic element.
, then there are two conjugate fixed points in and is called an elliptic element.
and these points lie on for any with . For any with , the fixed points of form a circle centered at and of radius .
3 Eigenvectors of the matrices corresponding to the transformations in the group G
If , then the connection between fixed points of and lines of eigenvectors for the matrix T corresponding to is explained by Theorem 1. Now we consider the transformations of the group G which belong to .
We use the eigenvector representation for the matrix T. First we begin with the glide reflections.
3.1 Glide reflections
We will show that the fixed points of a glide reflection correspond to the two lines of eigenvectors for the matrix T corresponding to . In the following two lemmas, we determine the eigenvalues and eigenvectors of the matrices which correspond to the glide reflections.
Theorem 2 Let be a glide reflection map in the group G with corresponding matrix T. Then if and only if is a line of eigenvectors of T.
Then T maps to if and only if . Thus, w is a fixed point of if and only if T maps to itself, and so if and only if each non-zero point on is an eigenvector of T. □
respectively. We have the slopes and .
Recall that we have for any reflection transformation.
Proof By (3.1), if we use the condition , the result is obtained. □
First we begin the case . For this case, the set of fixed points is a circle with radius ∞ (that is, a line on the complex plane).
Proof The proof is easy by the facts that () and . □
- (i)For the matrix , we have the eigenvalues and and the eigenvectors(3.5)
- (ii)For the matrix , we have the eigenvalues and and the eigenvectors(3.6)
If we choose as a parameter, we find the eigenvectors as and . The second part of the proof can be obtained similarly. □
In the first part of Lemma 4, we have the slopes as and . In the second part, we have and .
The matrix represents the reflection . The set of the fixed points of this reflection is a circle with radius ∞, that is, the line .
The matrix represents the reflection . The set of the fixed points of this reflection is the circle .
Proof The proof follows by straightforward computations. □
In the following theorem, we explain the relationship between fixed points of the reflections with and eigenvectors of the matrices corresponding to those reflections.
Theorem 3 Let be a reflection map in the group G with and let T be the matrix corresponding to . Then and are the lines of the eigenvectors of the matrix T and the set of the fixed points of the reflection is the line .
Proof The proof follows by Lemma 3, Lemma 4 and Lemma 5. □
Finally, we consider the reflections with . Lemma 6 can be proven in a similar way as Lemma 4.
In Lemma 6, we have the slopes as and . In the following theorem, we explain the relationship between the set of fixed points of the reflections with and eigenvectors of matrices corresponding to those reflections.
Theorem 4 Let be a reflection map in the group G with and let T be the corresponding matrix of . If and are the lines of the eigenvectors of the matrix T, then the set of the fixed points of the reflection is the circle centered at and of radius .
Then the proof follows by Lemma 6. □
Then we have and . Thus the fixed point set is a circle centered at and of radius .
Dedicated to Professor Hari M Srivastava.
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