On the Arnold’s Classification Conjecture on Dynamics Of Complexity of Linear Intersections

Abstract

Let X,Y be two linear subspaces of the m-dimensional complex space $\mathbb {C}^{m}$ with m > 1. The dimensions of the subspaces X and Y are k and m − k respectively and let $F:\mathbb {C}^{m}\to \mathbb {C}^{m}$ be a non-degenerate linear operator. In this work, we study the properties of the intersection between the subspace Y and the n-iteration of the subspace X under F. In the case when the dimension of the subspace X is either one or two, we give some results about a geometrical classification when we obtain an infinite set of moments n of no transversality between the space Y and the n-iteration of X under F.

Appendix: A

1.1 A.1 Proof of Lemma 9

Proof Some details

For each 1 ≤ l ≤ m,

$$ \begin{array}{@{}rcl@{}} {\Omega}_{w}(\boldsymbol{x}_{l}) & = & \boldsymbol{x}_{l}\wedge\bigg(\sum\limits_{1\le i<j\le m}a_{i,j}E_{i,j}\bigg)\\ & = & \sum\limits_{l< i<j\le m}a_{i,j}E_{l,i,j}-\sum\limits_{1\le i<l<j\le m}a_{i,j}E_{i,l,j}+\sum\limits_{1\le i<j<l}a_{i,j}E_{i,j,l}, \end{array} $$

and the rank of its associated matrix $[{\Omega }_{w}]_{B}^{B_{3}}$ is m − 2. We consider the m rows E_1,2,3, E_1,2,4, E_1,3,4, E_2,3,4 and E_1,2,l, with 5 ≤ l ≤ m, that is, we have

where C is a matrix (m − 4) × 4

$$ C=\left( \begin{array}{cccc} a_{2,5} & -a_{1,5} & 0 & 0\\ {\vdots} & {\vdots} & {\vdots} & \vdots\\ a_{2,m} & -a_{1,m} & 0 & 0 \end{array}\right) $$

and B is a diagonal matrix (m − 4) × (m − 4)

$$ B=\left( \begin{array}{ccc} a_{1,2} & {\ldots} & 0\\ {\vdots} & {\ddots} & \vdots\\ 0 & {\ldots} & a_{1,2} \end{array}\right). $$

Since the rank of the matrix $[{\Omega }_{w}]_{B}^{B_{3}}$ is m − 2, then the determinant of the matrix A is zero, thus

$$ 0=det(A)=det\left( \begin{array}{cccc} a_{2,3} & -a_{1,3} & a_{1,2} & 0\\ a_{2,4} & -a_{1,4} & 0 & a_{1,2}\\ a_{3,4} & 0 & -a_{1,4} & a_{1,3}\\ 0 & a_{3,4} & -a_{2,4} & a_{2,3} \end{array}\right)det(B), $$

then

$$ (a_{1,2}a_{3,4}-a_{1,3}a_{2,4}+a_{1,4}a_{2,3})^2a_{1,2}^{m-4}=0. $$

□

1.2 A.2 Proof of Lemma 10

Proof

We consider the case when I = {1, 2} and J = {3, 4} as in the proof of Lemma 9. In the general case, we consider an adequate bijection σ : {1,…,m}→{1,…,m}.

There exists an (m − 2)-vector w decomposable by W such that $\varphi ^{*}=\varphi ^{*}_{w}$. Thus,

$$ w=\sum\limits_{1\le i<j\le m}c_{(i,j)^{\prime}}E_{(i,j)^{\prime}} $$

where

$$ (i,j)^{\prime}=\{i_1,\ldots,i_{m-2}\}=\{1,\ldots,m\}\setminus\{i,j\}, i_1<\cdots<i_{m-2}. $$

Since $\{i,j\}\cap (i,j)^{\prime }=\emptyset$, by the previous lemma, we have

$$ b_{i,j}E_{1,\ldots,m}=c_{(i,j)^{\prime}}E_{(i,j)^{\prime}}\wedge E_{i,j}, $$

then

$$ \begin{array}{@{}rcl@{}} b_{1,2}&=&c_{3,4,\ldots,m}, b_{2,3}=c_{1,4,\ldots,m},\\ b_{1,3}&=&-c_{2,4,\ldots,m}, b_{2,4}=-c_{1,3,5,\ldots,m},\\ b_{1,4}&=&c_{2,3,5,\ldots,m}, b_{3,4}=c_{1,2,5,\ldots,m}. \end{array} $$

The rank of ${\Omega }_{w}:\mathbb {C}^{m}\to \bigwedge ^{m-1}\mathbb {C}^{m}$ is 2 and

$$ \begin{array}{@{}rcl@{}} {\Omega}_{w}(\boldsymbol{x}_{l}) & = & \boldsymbol{x}_{l}\wedge\bigg(\sum\limits_{1\le i<j\le m}c_{(i,j)^{\prime}}E_{(i,j)^{\prime}}\bigg)\\ & = & \sum\limits_{i=l<j\le m}(-1)^{l-1}c_{(i,j)^{\prime}}E_{(j)^{\prime}}+\sum\limits_{1\le i<j=l}(-1)^{l}c_{(i,j)^{\prime}}E_{(i)^{\prime}} \end{array} $$

with 1 ≤ l ≤ m. The rank of the matrix $[{\Omega }_{w}]_{B}^{B_{m-1}}$ is 2. We consider the minor for the first four columns and the rows $E_{(1)^{\prime }},E_{(2)^{\prime }},E_{(3)^{\prime }}$ y $E_{(4)^{\prime }}$, then the determinant of the matrix

$$ A=\left( \begin{array}{cccc} 0 & c_{(1,2)^{\prime}} & -c_{(1,3)^{\prime}} & c_{(1,4)^{\prime}}\\ c_{(1,2)^{\prime}} & 0 & -c_{(2,3)^{\prime}} & c_{(2,4)^{\prime}}\\ c_{(1,3)^{\prime}} & -c_{(2,3)^{\prime}} & 0 & c_{(3,4)^{\prime}}\\ c_{(1,4)^{\prime}} & -c_{(2,4)^{\prime}} & c_{(3,4)^{\prime}} & 0 \end{array}\right) $$

equals zero. Thus

$$ c_{(1,2)^{\prime}}c_{(3,4)^{\prime}}-c_{(1,3)^{\prime}}c_{(2,4)^{\prime}}+c_{(1,4)^{\prime}}c_{(2,3)^{\prime}}=0, $$

then

$$ b_{1,2}b_{3,4}-b_{1,3}b_{2,4}+b_{1,4}b_{2,3}=0. $$

(6)

For m = 4, the matrix $[{\Omega }_{w}]_{B}^{B_{m-1}}$ coincides with the above matrix. If φ^∗ is decomposable by W, the rank of matrix $[{\Omega }_{w}]_{B}^{B_{m-1}}$ is 2 and (3) holds. Conversely, if (3) holds, then all the minors of order 3 vanish and there always exists a minor of order 2 different from zero. □

1.3 A.3 Proof of Lemma 12

Proof

First, we suppose w is a 2-vector decomposable by the 2-dimensional subspace W. We have

$$ w=\sum\limits_{1\le i<j\le m}a_{i,j}\boldsymbol{x}_i\wedge\boldsymbol{x}_j=\sum\limits_{I=\{i_1,i_2\}\subseteq\{1,\ldots,m\}}a_IE_I $$

and

$$V(supp(w))=\{i\in I:I\subseteq\{1,\ldots,m\},a_{I}\ne 0\}.$$

We suppose that there exists l ∈{1,…,m} which is not a vertex of the support of w, this is, if l ∈ I, then a_I = 0. Since w is decomposable by W, then v ∧ w = 0 for all v ∈ W.

If we express v as a linear combination of the elements of B, $\boldsymbol {v}={\sum }_{i=1}^{m}v_{i}\boldsymbol {x}_{i}$, then in v ∧ w = 0 we get a linear combination of the elements of B₃ (a basis of 3-vectors). This implies that a_Jv_l = 0 for all J, thus v_l = 0 and $W\subseteq \langle \boldsymbol {x}_{1},\ldots \boldsymbol {x}_{l-1},\boldsymbol {x}_{l+1},\ldots ,\boldsymbol {x}_{m}\rangle$, this is a contradiction.

Now, let i₁ and j₁ be two vertices of the support of w, with i₁ < j₁. If L = {i₁,j₁} satisfies a_L≠ 0, then there exists the edge i₁j₁. If a_L = 0, we consider I = {i₁,i₂} and J = {j₁,j₂} such that a_Ia_J≠ 0. In the case when i₂ = j₂, the vertices i₁ and j₁ are connected by the path {i₁,i₂,j₁}. If i₂≠j₂, then I ∩ J = ∅ and there exist the edges i₁j₂ and i₂j₁, by Lemma 11. Thus, the vertices i₁ and j₁ are connected by the path {i₁,i₂,j₁}. Therefore, the support of w is a connected graph and for any pair of vertices either they are adjacent or there exists a path of length 2 connecting them.

On the other hand, if φ^∗ is decomposable by the (m − 2)-dimensional subspace W, then there exists an (m − 2)-vector ω decomposable by W such that $\varphi ^{*}=\varphi ^{*}_{\omega }$. Thus,

$$ \begin{array}{@{}rcl@{}} \varphi^{*} & = & \sum\limits_{I=\{i_{1},i_{2}\}\subseteq\{1,\ldots,m\}}b_{I}\varphi^{*}_{I},\\ \omega & = & \sum\limits_{J=\{i_{1},\ldots,i_{m-s}\}\subseteq\{1,\ldots,m\}}a_{J}E_{J}. \end{array} $$

We suppose that there exists l ∈{1,…,m} which is not a vertex of the support of φ^∗, this is, if l ∈ L, then b_L = 0. For each I = {i₁,i₂} we consider $I^{\prime }=\{1,\ldots ,m\}\setminus I$, by Lemma 8, we have

$$ b_IE_{1,\ldots,m}=a_{I^{\prime}}E_{I^{\prime}}\wedge E_I $$

and l∉I for all b_I≠ 0. Thus, $l\in I^{\prime }$ for all $a_{I^{\prime }}\ne 0$. This implies that v ∧ ω = 0 for all v = cx_l with $c\in \mathbb {C}$, therefore, $\langle \boldsymbol {x}_{l}\rangle \subseteq W$, this is a contradiction.

We prove that the support of φ^∗ is a connected graph in an analogous way to the proof about the support of w is a connected graph. □

1.4 A.4 Proof of Lemma 14

Proof

We suppose b ∈ N(a) and c∉N(a). By Lemma 12, there exists either the edge bc or a path of length 2 connecting the vertices b and c.

If we have a path {b,d,c}, d≠a, then {a,b}∩{d,c} = ∅, thus, there exist the edges ad and bc, by Lemma 11. Therefore, there always exists the edge bc.

Now, we suppose b,c∉N(a). Since a is a vertex of the support, then there exists at least a vertex d such that d ∈ N(a).

If there exists the edge bc, by Lemma 11, there exist either the edges ab and dc, or the edges ac and db. This is a contradiction, therefore, if b,c∉N(a), then there is no edge connecting them. □