I'm reading a book on symplectic geometry because I want to learn about the generalisations of Calabi Yaus (yes, things get messier still....). I want to run my attempt at an answer past someone who'll know like Ben or QH or Temur or Guest or Prom, so : \(\phi : V \to V\) is a morphism (and a linear map) such that \(\omega(\phi(v),\phi(v')) = \omega(v,v')\) where \(\omega\) is a symplectic form. In terms of matrices and vectors, \(\omega(v,v') = v^{t}J v' \) where \(J = \left( \begin{array}{cc} 0 & \mathbb{I}_{n} \\ -\mathbb{I}_{n} & 0 \end{array} \right)\). Therefore \(\phi\) has the matrix representation M satisfying \(M^{t}J M = J\) (just as the Lorentz matrices satisfy \(\Lambda^{t}\eta \Lambda = \eta\) define the transformations). This group of M is known as \(Sp_{n}(K)\) where K is the field, like the K in \(GL_{2n}(K)\) (which \(Sp_{n}(K)\) is a subgroup of). We find that, when K=R, if \(\lambda \in \mathbb{C}\) is an eigenvalue of M then so is \(\bar{\lambda}\), \(\lambda^{-1}\) and \(\bar{\lambda}^{-1}\). A morphism on V is called stable if for each \(\epsilon > 0\) there is a \(\theta > 0\) such that \(||\phi^{N}v|| < \epsilon\) for all \(N \in \mathbb{N}\) as soon as \(||v|| < \theta\). Right, that's all ground work. Now the questions : 1. If \(\phi \in Sp(V)\) has eigenvalue \(|\lambda| \not= 1\) then it's not stable. 2. Whenever all the eigenvalues are distinct and with norm 1, \(\phi\) is stable. The first one is pretty straight forward. If there's a \(|\lambda|<1\) then \(|1/\lambda|>1\) and any component of v in that direction will be enlarged and enlarged with repeated applications of M, so it'll be unbounded. Hence the eigenvalues all need to be norm 1. I could make this rigorous but it's not really needed for my plans. However, I don't get the second questions requirement the eigenvalues are distinct? This seems to be the caveat which prevents it from being an 'if and only if' setup, because it implies if the eigenvalues aren't distinct, then it fails. Is it something to do with writing v in terms of a Gram-Schmidt basis (only for symplectic forms, the book does mention such a thing) and this having trouble being compatible with stability? A general discussion on this would be nice too, since I'm going for more of a conceptual understanding in some places, the detailed calculations are confined to very specific things. It's just good to know how things fit together.
This is not particularly because of the symplectic structure; it has to do with the Jordan normal form, algebraic multiplicity, elementary divisor etc. When there is off-diagonal 1's in the Jordan normal form of a matrix, then the corresponding eigenvalue does not exactly correspond to a scaling factor; it indicates some “twisting” of the subspace. Here is an example to illustrate this. Take \(M=\left(\begin{array}{cc}1&1\\0&1\end{array}\right).\) It has the eigenvalue 1 with algebraic multiplicity 2. So the norms of both eigenvalues are 1. But \(M^n=\left(\begin{array}{cc}1&n\\0&1 \end{array}\right)\) for any integer \(n\geq0\), which means that the norm of \(M^n\) will grow unboundedly. Although symplecticity did not play a role here, to show that symplecticity will not remove the problem, we picked (in the beginning) M to be symplectic: \(M^tJM=J\) with \(J=\left(\begin{array}{cc}0&1\\-1&0\end{array}\right).\)
Well hello to another excellent thread! Clearly "distinct eigenvalues" is not necessary for stability - take \(M=\mathbf{I}\).
I sorta got that when \(\lambda_{n}=e^{i\theta_{n}}\) then you get some kind of 'phase shift' which is different for each eigenvector of M, so it doesn't alter the mod so much as shuffle it around, like the Schrodinger equation gives you a wave function whose coefficients evolve in time but whose squared total is constant, so they 'shuffle around' their relative weights, if you see what I mean. Doh! When in doubt, go for the simplest one! Please Register or Log in to view the hidden image! I'll have more questions but I'm a little scattered at the moment due to a lengthy weekend. More to come Please Register or Log in to view the hidden image! That and some Kahler geometry, since I found a book which is a very nice introduction to it which follows on a lot from Nakahara but doesn't give me too much credit when it comes to particular terms and concepts, which is always nice since textbooks often assume too much about me :bawl:
