Topological conjugacy classes of affine maps
📝 Abstract
A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b $, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n $. The affine maps over $\ff=\rr$ or $\cc$ are investigated. We prove that affine maps having fixed points are topologically conjugate if and only if their linear parts are topologically conjugate. If affine maps have no fixed points and $n=1$ or 2, then they are topologically conjugate if and only if their linear parts are either both singular or both non-singular. Thus we obtain classification up to topological conjugacy of affine maps from $\ff^n$ to $\ff^n $, where $\ff=\rr$ or $\cc $, $n\leq 2 $.
💡 Analysis
A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b $, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n $. The affine maps over $\ff=\rr$ or $\cc$ are investigated. We prove that affine maps having fixed points are topologically conjugate if and only if their linear parts are topologically conjugate. If affine maps have no fixed points and $n=1$ or 2, then they are topologically conjugate if and only if their linear parts are either both singular or both non-singular. Thus we obtain classification up to topological conjugacy of affine maps from $\ff^n$ to $\ff^n $, where $\ff=\rr$ or $\cc $, $n\leq 2 $.
📄 Content
The main result of the paper is the following theorem: Theorem 1.1. Let f , g : F n → F n , f (x) = Ax + b, g(x) = Cx + d be affine maps and
If each of f and g has at least one fixed point, then f t ∼ g if and only if their linear parts are topologically conjugate.
If f and g have no fixed points and n = 1 or 2, then f t ∼ g if and only if det A and det C are either simultaneously equal to 0 or simultaneously different from 0.
2 Topological classification of linear maps from R n to R n , n ≥ 1.
We will use the classification of linear maps up to topological conjugacy for the classification of affine maps, therefore we recall some known results about linear maps.
Let f : R n → R n , f (x) = Ax be a linear map. Then the real canonical form (written R A or RCF) of the matrix A (see [4]) may be written in the form:
, where the matrices A α , α = +, -, ∞, 0, satisfy conditions listed in the following table:
The eigenvalues λ of this matrix
For a linear map f : C n → C n , f (z) = Az by using the Jordan canonical form (written J A ) of the matrix A, the matrices A α , α = +, -, ∞, 0 are defined similarly.
The maps f , g : R n → R n are said to be linearly conjugate (written f ℓ ∼ g), if there exists a linear bijection h :
A map f : R n → R n is called periodic if there is k ∈ N such that f k = id R n . The smallest possible k is called the period of map f .
Kuiper and Robbin [12,14] gave the topological classification of those linear maps from R n to R n , whose matrices have no eigenvalues which are the roots of unity.
Theorem 2.1. [12,14] Let f , g : R n → R n , f (x) = Ax, g(x) = Cx be linear maps, whose matrices have no eigenvalues which are the roots of unity.
Then f t ∼ g if and only if
rank(A + ) = rank(C + ), sign( det(A + ) ) = sign( det(C + ) ), rank(A -) = rank(C -), sign( det(A -) ) = sign( det(C -) ),
Remark 2.1. The equalities A ∞ = C ∞ and A 0 = C 0 hold up to the order of diagonal blocks of the matrices.
Kuiper and Robbin [12] showed that the general problem reduced to the case when A and C were periodic matrices. They conjectured that for all periodic linear maps topological conjugacy is equivalent to linear conjugacy, and they proved this when periods of maps are equal to s = 1, 2, 3, 4 or 6.
Later, other conditions for which topological conjugacy is equivalent to linear conjugacy were found. The problem of the topological classification of matrices with eigenvalues which are the roots of unity was partially solved by Kuiper and Robbin [12,14], Cappell and Shaneson [6,7,8,9,10], Hsiang and Pardon [11], Madsen and Rothenberg [13], Schultz [15].
Among these papers we will consider the works of Cappell and Shaneson. They found counterexample to the conjecture of Kuiper and Robbin [6] and they proved that for periodic linear maps from R n to R n , n ≤ 5 topological conjugacy is equivalent to linear conjugacy [7,8,10].
Since necessary and sufficient condition for linear conjugacy of two linear maps is equivalent to the equality of RCF’s of their matrices, it follows that two periodic
The following proposition is a conclusion of the results of Kuiper and Robbin, Cappell and Shaneson.
Proposition 2.1. Let f , g : R n → R n , n ≤ 5, f (x) = Ax, g(x) = Cx be linear maps.
Then
Classification up to topological conjugacy of all periodic linear maps has not been solved completely yet, and so it is still unsolved for all linear maps.
3 Topological classification of affine maps from R n to R n , n ≥ 1.
3.1 Classification of affine maps from R n to R n , n ≥ 1, such that each of them has at least one fixed point.
We begin with classification up to topological conjugacy of affine maps from
It is based on the existence of a fixed point of the affine map. For the large class of affine maps this problem was completely solved.
The following theorem is one of the most important results in the topological classification of affine maps.
= Ax, therefore the maps f and g have the same number of fixed points.
Since g(0) = 0, it follows that g(x) = Ax has at least one fixed point, therefore f (x) = Ax+b also has at least one fixed point.
By assumption there is q ∈ R n such that f (q) = q.
Let h(x) = x + q, then it is easy to see that
= Cx + d be affine maps such that:
each of f and g has at least one fixed point;
the matrices A and C have no eigenvalues which are the roots of unity.
T hen f t ∼ g if and only if
Proof. By assumption each of f and g has at least one fixed point.
By Theorem 3.1:
Employing Theorem 2.1 to the maps r(x) = Ax and s(x) = Cx we obtain the result.
Consequently, necessary and sufficient conditions of topological conjugacy of affine maps such that each of them has at least one fixed point coincide with the necessary and sufficient conditions of topological conjugacy of their linear parts.
3.2 Classification of affine maps from R to R.
In this paper we completely solve the problem of topological classification of affine maps from
At first consider
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