The Connecting Lemma(s)

Following Hayashi, Wen&Xia, Arnaud

Pugh’s Closing Lemma

• If an orbit comes back very close to itself

Pugh’s Closing Lemma

• If an orbit comes back very close to itself

•Is it possible to close it by a small pertubation of the system ?

Pugh’s Closing Lemma

• If an orbit comes back very close to itself

•Is it possible to close it by a small pertubation of the system ?

An orbit coming back very close

A C0-small perturbation

The orbit is closed!

A C1-small perturbation: No closed orbit!

For C1-perturbation less than , one need a safety distance, proportional to the jump:

Pugh’s closing lemma (1967)

If x is a non-wandering point of a diffeomorphism f on a compact manifold, then there is g, arbitrarily C1-close to f, such that x is a periodic point of g.

•Also holds for vectorfields

•Conservative, symplectic systems (Pugh&Robinson)

What is the strategy of Pugh?

• 1) spread the perturbation on a long time interval, for making the constant very close to 1.

For flows: very long flow boxes

For diffeos

2) Selecting points:

The connecting lemma• If the unstable

manifold of a fixed point comes back very close to the stable manifold

•Can one create homoclinic intersection by C1-small perturbations?

The connecting lemma (Hayashi 1997)

By a C1-perturbation:

Variations on Hayashi’s lemma

x non-periodic point

Arnaud,Wen & Xia

Corollary 1: for C1-generic f,H(p) = cl(Ws(p)) cl(Wu(p))

Other variation

x non-periodic

in the closure of

Wu(p)

Corollary 2: for C1-generic fcl(Wu(p)) is Lyapunov stable

Carballo Morales & Pacifico

Corollary 3: for C1-generic fH(p) is a chain recurrent class

30 years from Pugh to Hayashi : why ?

Pugh’s

strategy :

This strategy cannot work for connecting lemma:

• There is no more selecting lemmas

Each time you select one red and one blue point,There are other points nearby.

Hayashi changes the strategy:

Hayashi’s strategy.

• Each time the orbit comes back very close to itself, a small perturbations allows us to shorter the orbit:

one jumps directly to the last return nearby, forgiving the intermediar orbit segment.

What is the notion of « being nearby »?Back to Pugh’s argument For any C1-neighborhood of f and any

>0 there is N>0 such that:

For any point x there are local

coordinate around x such that

Any cube C with edges parallela to the axes

and Cf i(C)= Ø

0<iN

Then the cube C verifies:

For any pair x,y

There are x=x0, …,xN=y such that

The ball B( f i(xi), d(f i(xi),f i(xi+1)) ) where is the safety distance

is contained in f i( (1+)C )

Perturbation boxes1) Tiled cube : the ratio between adjacent tiles is bounded

The tiled cube C is a N-perturbation box for (f,) if:

for any sequence (x0,y0), … , (xn,yn),

with xi & yi in the same tile

There is g -C1-close to f,

perturbation in Cf(C)…fN-1(C)

There is g -C1-close to f,

perturbation in Cf(C)…fN-1(C)

There is g -C1-close to f,

perturbation in Cf(C)…fN-1(C)

The connecting lemma

Theorem Any tiled cube C,

whose tiles are Pugh’s tiles

and verifying Cf i(C)= Ø, 0<iN

is a perturbation box

Why this statment implies the connecting lemmas ?

x0=y0=f i(0)(p)x1=y1=f i(1)(p)…xn=f i(n)(p); yn=f –j(m)(p)xn+1=yn+1=f -j(m-1)(p)…xm+n=ym+n=f –j(0)(p)

By construction, for any k,

xk and yk belong to

the same tile

For definition of perturbation box, there is a g C1-close to f

Proof of the connecting lemma:

Consider (xi,yi) in the same tile

Consider the last yi in the tile of x0

And consider the next xi

Delete all the intermediary points

Consider the last yi in the tile

Delete all intermediary points

On get a new sequence (xi,yi) with at most 1 pair in a tile

x0 and yn

are the original

x0 and yn

Pugh gives sequences of points joining xi to yi

There may have conflict between the perturbations in adjacent tiles

Consider the first conflict zone

One jump directly to the last adjacent point

One delete all intermediary points

One does the same in the next conflict zone, etc, until yn

Why can one solve any conflict?

There is no m other point nearby!the strategy is well defined