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Aug 18, 2011 at 13:40 history edited user17119 CC BY-SA 3.0
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Aug 18, 2011 at 13:10 comment added Donu Arapura Initially, I thought you meant pointwise invariance $\forall s\in S, T(s)=s$ , but I guess you mean $T(S)\subseteq S$. You can disregard my previous comment.
Aug 18, 2011 at 12:58 history edited user17119 CC BY-SA 3.0
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Aug 18, 2011 at 12:46 comment added user17119 Donu, why would the linear span of any invariant set have to lie in V? Take the eigenspace spanned by the eigenvector corresponding to some other eigenvalue. It is invariant under T, but does not lie in V.
Aug 18, 2011 at 2:51 comment added Donu Arapura Or do you mean sets such that $T(S)\subset S$?
Aug 18, 2011 at 2:42 comment added Donu Arapura It's clear isn't it? Let $V$ be the $+1$-eigenspace (which might be $0$) of $T$. Then any subset of $V$ is invariant. Conversely, the linear span of an invariant set would have to lie in $V$.
Aug 18, 2011 at 2:29 history edited Yemon Choi
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Aug 18, 2011 at 2:28 comment added Yemon Choi But what kinds of set are you looking for? It seems like you want algebraic sets of some kind (e.g. solutions to systems of equations) and not just any old set.
Aug 18, 2011 at 1:49 comment added user17119 What I'm looking for is a reference to the mathematical area/technique that deal with such problems.
Aug 18, 2011 at 1:39 comment added Yemon Choi Without further conditions on $T$ or some maximality condition on your non-linear set $S$, this seems tricky to answer sensibly. (Case in point: take T to be the identity and S to be anything very like a whale.) Is there a more precise version of this question, specifying domains etc, that you coudl pose?
Aug 18, 2011 at 1:11 history asked user17119 CC BY-SA 3.0