Closing lemma

In the rough case an analysis of the structure of such a limit set (called a quasiminimal set, which is defined as the closure of an unclosed Poisson-stable trajectory) may be performed using Pugh s closing lemma. The main conclusion that follows from this analysis (see Sec. 7.3) is that periodic orbits are dense in a rough quasiminimal set. In particular, we will see that the number of periodic orbits is infinite. Systems possessing such limit sets are called systems with complex dynamics. 

Theorem 7.7. (Closing lemma, Pugh) Let xq be a non-wandering point of a smooth flow. Then, arbitrarily close in -topology, there exists a smooth flow which has a periodic orbit passing through the point xq... 

For more than two almost invariant sets one has to consider all eigenmea-sures corresponding to eigenmodes for eigenvalues close to one. In this case, the following lemma will be helpful. 

The previous discussion has shown us how to calculate the total number of possible cyclic states. We also know, from Lemma 2, that all cycle lengths must divide the maximal cycle length Hiv obtain the exact number of distinct cycles and their lengths takes a little bit more work. If flw prime, we know that the only possible cyclic lengths are 1 and It can then be shown that only the null configuration is a fixed point unless N is some multiple of 3, it which case there are exactly four distinct cycles of length one. If Hat i ot prime, there can exist as many cycles as there are divisors of Although there is no currently known closed form... 

Let k be an algebraically closed field. In this section we review the methods of [Ellingsrud-Str0mme(l)] for the determination of a cell decomposition of and modify them in order get a cell decomposition and thus (for k = C) the homology of the strata Zt and Gt of Hilb"(fc[[x, y]]). Let R = fc[[ ,y]]. Let Hilb (A2,0) be the closed subscheme with the induced reduced structure of (A2) parametrizing subschemes with support 0. By lemma 2.1.4 we have... 

Here we use again the notations we have introduced in definition 3.1.1. In the same way as in lemma 3.1.2 we can show that 2 (A) is represented by a closed subscheme... 

Proof. The implication from left to right is trivial. To prove the other implication, by Lemma 8.2, we may assume that k is perfect. Let T = Spec(Wn+i(fc)) <- 5 be the closed subscheme of S constructed in Lemma 8.2. Remark that G ObCs0 <=> Gt0 ObCy0 and similar for Ms(G). Hence, by Proposition 7.2, the result follows from the classical Dieudonne module theory. ... 

Since the G-action on // 1(C) is free, the slice theorem implies that the quotient space gTl((,)/G has a structure of a C°°-manifold such that the tangent space TaAtt-HO/G) at the orbit G x is isomorphic to the orthogonal complement of Vx in Txg 1( ). Hence the tangent space is the orthogonal complement of Vx IVX JVX KVX in TxX, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient g 1(()/G. In order to show that these define a hyper-Kahler structure, it is enough to check that the associated Kahler forms u>[, u) 2 and co z are closed by Lemma 3.32. 

The first section of this chapter is a collection of general observations on closed subsets most of which are straightforward generalizations of facts on subgroups. For instance, we show that the set of all double cosets of two closed subset of S is a partition of S cf. Lemma 2T.3. We also introduce transversals (as they have been introduced in [45]), and, at the end if this section, we show that closed subsets give rise to subschemes. 

Lemma 2.1.1 LetT and U be closed subsets of S. ThenTU is closed if and only ifTU = UT. 

The first equation follows from Lemma 1.3.2(iii).) Therefore, TU is closed. 

Lemma 2.1.2 LetT and U be closed subsets of S. Then we have 1 = TCiU if and only if, for each element s in TU, there exist uniquely determined elements t in T and u in U such that s tu. 

Since y G zR, yR C zR. (This follows from Lemma 1.3.8 together with the hypothesis that R is closed.) From y G zR we also obtain c G yR = yR. Thus, we conclude, as before, that zR C yR. 

Lemma 2.3.1 Let p and q be elements in S, let T be a closed subset of S, and assume that T has finite valency. Then the following hold. 

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