Let be a diffeomorphism. A point is **non-wandering** if for all neighborhood of , there is increasing sequence where . We write .

**Closing lemma:** For any diffeomorphism , for any . For all there exists diffeomorphism s.t. and for some .

Suppose , is compact, then for any , there exists , s.t. .

First we apply a selection process to pick an appropriate almost-orbit for the closing. Set .

If there exists where

then we replace the origional finite sequence by or . Iterate the above process. since the sequence is at least one term shorter after each shortening, the process stops in finite time. We obtain final sequence s.t. for all ,

.

Since the process is applied at most times, , after the first shortening, .

i.e. both initial and final term of the sequence is at most . Along the same line, we have, at the -th shortening, the distance between the initial and final sequence and is at most . Hence for the final sequence .

There is a rectangle where

(i.e. shrunk by a factor of w.r.t. the center) and for all .

Next, we perturb in i.e. find with and id. Hence .

Suppose are the lengths of , .

By main value theorem, for all .

On the other hand, since , it's at least away from the boundary of . i.e. there exists bump function satisfying the above condition and .

Hence in order to move a point by a distance , we need about such bump functions, to move a distance , we need about bumps.

For simplicity, we now suppose is a surface. By starting with an (and hence ) very small, we have for all is contained in a small neighbourhood of . Hence on is close to the linear map . Hence mod some details we may reduce to the case where is linear in a neighborhood of .

By choosing appropiate coordinate system in , we can have preserving the horizontal and vertical foliations and the horizontal vectors eventually grow more rapidly than the vertical vectors.

It turns out to be possible to choose to be long and thin such that for all , has height greater than width. (note that bumps will be able to move the point by a distance equal to the width of the original rectangle . Since horizontal vectors eventually grow more rapidly than the vertical vectors, there exists s.t. for all , has width greater than its height.

For small enough , the boxes are disjoint for . Construct to be identity outside of

For the first boxes, we let preserve the horizontal foliation and move along the width so that has the property that lies on the same vertical fiber as .

On the boxes , we let pushes along the vertical direction so that

Since iterates of the rectangle are disjoint, for , .

Hence .

Therefore we have obtained a periodic point of .

Since , we may further perturb to move to . This takes care of the linear case on surfaces.

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