Pugh closing lemma

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Christian Bonatti (2008), Scholarpedia, 3(6):5072. doi:10.4249/scholarpedia.5072 revision #91675 [link to/cite this article]
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Curator: Christian Bonatti

If the positive orbit of a point comes back very close to the initial point, is it possible to close it, creating a periodic orbit, by a small perturbation of the system? The answer depends on the meaning of small perturbation. The closing lemma is the positive answer to this question for the \(C^1-\)topology

This article will briefly cover:

  • a presentation of the problem
  • the classical version of the \(C^1\)-Closing Lemma
  • a detailed scheme of proof
  • some generalizations of the Closing Lemma
  • discussion on the use of the \(C^1-\)topology
  • some open questions

Contents

Introduction

Periodic orbits are the simplest recurrent orbit of a dynamical system, and they have been important for the understanding of the global dynamics. For example:

  • Smale [S] splits the non-wandering set of every Axiom A diffeomorphisms in finitely many transitive hyperbolic basic pieces: for this he defines an equivalence relation on the set of hyperbolic periodic orbits (two orbits are equivalent if the stable manifold of each of them cuts transversely the unstable manifold of the other) and the basic pieces are the closure of the equivalence classes (called homoclinic classes).
  • [BC] shows that, for \(C^1\)-generic diffeomorphisms, every chain recurrent class containing a periodic point \(x \) is the homoclinic class of \(x\ .\)
  • The Henon attractors and the Lorenz attractors are homoclinic classes of periodic orbits.

More generally, one tries to organize the global dynamics of a system by considering the sets where the orbits present some kind of recurrence. The most usual notions of recurrence are

  • The set of period points is denoted by \(Per(f)\)
  • recurrent points : a point is recurrent if its belongs to its own limit set
  • The set \(\Omega(f)\) of non-wandering points (introduced by Birkhoff): a point is wandering if it has a neighborhood disjoint from all its positive iterates.
  • The set \(R(f)\) of chain recurrent points: a point \(x\) is chain recurrent if, for every \(\delta>0\) there is a \(\delta-\)pseudo orbit \(x=x_0,x_1,\dots,x_n=x \) (with \(d(f(x_i),x_{i+1})< \delta\)). Conley in [C] organizes the global dynamics of every homeomorphisms or flow using the chain recurrence classes defined on \(R(f)\ :\) two points \(x\) and y are equivalent if one can go from \(x\) to \(y\) and from \(y\) to \(x\) by \(\delta-\)pseudo orbits for all \(\delta>0\ .\)

These sets satisfy the inclusions \(Per(f)\subset \Omega(f) \subset R(f)\ .\) In general they are distinct sets. There are very complicated dynamical systems without periodic orbits: for instance, consider the time \(t\) map of an Anosov flow, where \(t\) is not one of the periods.

However, one can hope that the lack of periodic orbits is an exceptional behavior. This leads to a perturbative point of view on Dynamical System: to avoid exceptional behavior, one allows some perturbation of the initial system. This notion of perturbation of a dynamical system depends on the topology on the space of dynamical systems.

Given a compact manifold \(M\ ,\) one consider the space \(Diff^r(M)\) of all the \(C^r-\)diffeomorphisms of \(M\) endowed with the \(C^r-\)topology. One says that a \(C^r-\)generic diffeomorphism satisfies a property \(P\) if the property \(P\) is satisfied on a residual subset of \(Diff^r(M)\) (a subset of a Baire space is residual if it contains the intersection of a countable family of open and dense subsets; the complement of a residual subset is a meager subset). The study of generic diffeomorphisms is strongly related with the perturbations lemmas which produce some dynamical property by performing a small perturbation of the original system. For instance Kupka and Smale proved that the periodic orbits of a \(C^r\)-generic diffeomorphism are all hyperbolic and their invariant manifolds are everywhere transverse.

The study of \(C^1\)-generic diffeomorphisms has got many important results, due to perturbations lemmas using specifically the \(C^1\)-topology. One of the first and certainly most famous of these perturbations lemmas is Pugh \(C^1\)-closing lemma.

The Pugh closing lemma

Consider the space of diffeomorphisms \(Diff^1(M)\) of a compact manifold \(M\ ,\) endowed with the \(C^1\)-topology. The closing lemma tells that every non-wandering orbit can be closed by a \(C^1\)-small perturbation of the diffeomorphism.

Theorem(Pugh [P], [PR]) Let \(f\) be a diffeomorphisms of a compact manifold M and let \(x\in M\) be a non-wandering point of \(f\ .\) Then any \(C^1\)-neighborhood \(U\) of \(f\) contains a diffeomorphism \(g\in U\) such that \(x\) is a periodic point of \(g\ .\)

This result remains true among conservative (either volume preserving or symplectic) diffeomorphisms, and also holds for vector fields (in both dissipative and conservative settings, including Hamiltonian vector fields).

It remains mostly unknown for the \(C^2\)-topology, the few partial results [H][G] leading to the general feeling that it should be wrong.

Question: Let \(M\) be a compact manifold, and \(Aper^r(M)\subset Diff^r(M)\) be the set of diffeomorphisms having no periodic orbits. Is the \(C^r-\)interior of \(Aper^r(M)\) empty, for every \(r>1\ ,\) and every manifold \(M \ ?\)

The answer to this question remains unknown for \(r=2\) and \(M\) is the torus \(T^2\ .\)

Difficulty of the C1-closing lemma

Let \(x\) be a non-wandering point of a diffeomorphisms \(f\) of a compact manifold. By definition of non-wandering points, there are \(n>0\) and \(y\) arbitrarily close to \(x\) such that \(f^n(y)\) is arbitrarily close to \(x\ .\) It is tempting to close the orbit of \(y\) just by pushing the point \(f^n(y)\) on \(y\ .\) This section explains that this naïve idea leads to the \(C^0\)-closing lemma, but does not hold for the \(C^1\)-topology. The difficulty for the \(C^1\)-topology comes from a safety distance needed for performing a local \(C^1\)-perturbation.

Closing an orbit in 1 step: the C0-closing lemma

Definition A perturbation \(g\) of \(f\) is closing the orbit of \(y\) in one step at time \(n\) if \(g^i(y)=f^i(y)\) for \(i\in\{1,\dots,n-1\}\) and \(g^n(y)=y\ .\)

Proposition (the \(C^0-\)closing lemma) Given a non-wandering point \(x\) of a diffeomorphism \(f\ ,\) there are

  • a sequence of diffeomorphism \(g_n\) converging to \(f\) in the \(C^0-\)topology,
  • a sequence of point \(y_n\) converging to \(x\ ,\)
  • a sequence of positive integer \(t_n>0\ ,\)

such that \(g_n\) is closing the orbit of \(y_n\) in one step at the time \(t_n\ .\) Furthermore, for large \(n\ ,\) \(g_n\) coincides with \(f\) out of an arbitrarily small neighborhood of \(f^{-1}(x)\ .\)

The perturbation \(g\) in the proposition is equal to \(f\) on \(y,f(y),\dots, f^{n-2}(y)\) but satisfies that \(g(f^{n-1}(y))=y\ .\) If there is some intermediary time \(i \in \{1,\dots,n-2\}\) for which the distance \(d(f^i(y),f^n(y))\) is smaller than the distance \(d(y, f^n(y))\) then the derivative of \(g\) will be very different from the derivative of \(f\) a some point between \(f^{n-1}(y)\) and \(f^{i-1}(y)\ .\) For this reason, \(g\) is not a \(C^1\)-small perturbation of \(f\) and we cannot get the \(C^1-\)closing lemma by closing some orbit in one step.

Safety distance for C1-small one step perturbations

We consider perturbations of \(f\) of the form \(g= f\circ h\) where \(h\) is a map which is the identity map out of a small disk \(D\ ,\) and such that \(h(f^{n-1}(y)) = f^{-1}(y)\ .\) Then the \(C^1-\)distance from \(h\) to the identity map is comparable with the ratio \(\frac{d(f^{-1}(y),f^{n-1}(y))}{r}\) where \(r\) is the radius of \(D\ .\)

If we want that the \(C^1-\)distance \(d^1(h,id)\) is less than \(\varepsilon> 0\ ,\) we need that the points \(f^i(y)\ ,\) \(i \in \{1,\dots,n-2\}\ ,\) remain at a distance larger than the safety distance \(\delta =\frac{d(f^{n-1}(y),f^{-1}(y))}{\varepsilon}\) from the point \(f^{n-1}(y)\ .\)

If \(\varepsilon\) is small, this safety distance is much larger than the return distance \(d(f^{n-1}(y), f^{-1}(y))\ .\) In other words, we need that the orbit of \(y\) remains far from \(y\) until \(n-1\ ,\) and suddenly come back very close to \(y\ .\) This condition is not satisfied in general. This implies that, in general, you cannot close in one step a recurrent orbit \(x\) by a \(C^1-\)small perturbation \(g\) of \(f\ .\)

Remark: if we want the \(C^r\) distance \(d^r(h,id) < \varepsilon\) for \(r>1\ ,\) then the safety distance \(\delta\) satisfies that \(\delta^r\) is comparable to \(\frac{d( f^{n-1}(y), f^{-1}(y))}{\varepsilon}\ .\) This safety distance is much larger than \(d( f^{n-1}(y), f^{-1}(y))\) when \(d( f^{n-1}(y), f^{-1}(y))\) tends to \(0\ .\) This explains that the \(C^r-\)closing lemma remains an open question.

Ideas of Pugh’s argument

Pugh’s closing lemma is considered as a very difficult result. This section presents the main ideas in its proof, but cannot avoid some technicality. Next section is independent to this one.

Pugh’s argument splits in two steps.

The first step : spreading the perturbation along a segment of orbit.

We have seen that, closing the segment of orbit \(y,f(y),\dots,f^n(y)\) just by moving the point \(f^n(y)\) is possible by an \(C^1\)-perturbation of size \(\varepsilon\ ,\) only if the point \(f^i(y)\ ,\) \(i<n\ ,\) remains at the safety distance \(C_1\cdot d(f^n(y),y)\) with \(C_1= \frac 1\varepsilon\ .\)

What happens with this safety distance if we allow moving not only the point \(f^n(y)\ ,\) but the two points \(f^{n-1}(y)\) and \(f^n(y)\ ?\)

Intuitive idea: pushing \(f^{n-1}(y)\) half the way in the direction of \(f^{-1}(y)\) makes that its image by \(f\) has been also pushed more or less half the way in the direction of \(y\ ,\) so that the jump we need for closing the orbit has been divided by 2: the safety distance we need has been also divided by 2 , that is, the safety distance for this new strategy is \(C_2 \cdot d( f^n(y),y)\ ,\) where \(C_2\) is more or less \(\frac{C_1}2\ .\)

Dividing the safety distance by 2 is not enough. So, we fix a number \(N>0\ ,\) and we allow moving the points \(f^{n-N}(y),\dots,f^n(y)\ :\) the safety distance will be of the form \(C_N.d( f^n(y),y)\) and one can hope that \(C_N\) will be arbitrarily small for \(N\) large.

Let us see how Pugh performed rigorously this rough idea:

We need to perform a perturbation of \(f\) supported on arbitrarily small disks. Hence we may assume that the restrictions of \(f\) to each of these disks are linear maps. The key lemma of the \(C^1-\)Closing Lemma is the following technical lemma on linear algebra.

Lemma 1: Given \(\varepsilon>0\ ,\) \(\eta>0\) and \(K>1\) there is \(N\) with the following property.

Given any matrices \(A_0,\dots,A_{N-1}\) in \(GL(\mathbb{R}, d)\) such that \(||A_i||<K\) and \(||A_i^{-1}||<K\ ,\) there is an orthogonal basis \(E= (e_1,..,e_d)\) of \(\mathbb{R}^d\) with the following property:

Denote by \(B_r\) the cube of \(\mathbb{R}^d\) consisting in the points whose coordinates in the basis \(E\) belongs to \([-r,r]\ .\) Then, for every pair of points \(a,b\) in the cube \(B_1\ ,\) there is a sequence \(g_i\ ,\) \(i=0,\dots,N-1\ ,\) of diffeomorphisms with :

  • \(d^1(g_i,A_i)< \varepsilon\) (i.e. the \(g_i\) are \(\varepsilon-C^1-\) perturbations of the \(A^i\) for the \(C^1\)-topology),
  • \(g_i\) is equal to \(A_i\) out of the image \(A_{i-1}\circ\cdots\circ A_0(B_{1+\eta})\) of the cube \(B_{1+\eta}\ ,\)
  • \( g_{N-1}\circ\cdots\circ g_0(a)=A_{N-1}\circ\cdots \circ A_0(b)\)

This lemma says that, in the pattern given by the basis \(E\ ,\) you can push the orbit through \(a\) on the orbit through \(b\) in \(N\) steps, without changing \(f\) out of the \(N\) first iterates of the cube which is \(1+\eta\) times the smallest cube containing \(a\) and \(b\ .\)

The second step: selecting two returns close to x

We fix the size \(\varepsilon>0\) of the perturbations we allow and we fix some small number \(\eta>0\ .\) Lemma1 gives us the time \(N\) we need for performing the perturbation.

As \(x\) is a non-wandering point there a points \(y\) arbitrarily close to \(x\) having a return \(f^n(y)\ ,\) \(n>0\ ,\) arbitrarily close to \(x\ .\) The points \(y\) and \(f^n(y)\) are so close to \(x\) that we can consider that the iterates \(f^i\ ,\) \(i=1,\dots,N\ ,\) are linear maps on a disk \(D\) containing \(x\) and the points \(y \) and \(f^n(y)\ .\) Lemma 1 provides local coordinates on the disk \(D\ ,\) (corresponding to the basis E in the lemma).

We look at the set \(I\subset\{0,\dots,n\}\) of all the return times of \(y\) in the disk \(D\ :\) \(i\in I \Leftrightarrow f^i(y)\in D\ .\) So we get a finite set of points \(y_i\ ,\) \(i\in I\ ,\) in the disk \(D\ .\)

Pugh shows that there exists two of this points \(y_i, y_j\ ,\) \(i<j\ ,\) and a cube \(C\) containing \(y_i\) and \(y_j\) and such that the homothetic cube \((1+ \eta)C\) does not contain any other point \(y_k\ .\)

Now Lemma 1 build a \(\varepsilon-C^1-\)small perturbation \(g\) of \(f\) which is equal to \(f\) out of the \(N\) first iterates of the cube \((1+ \eta)C\) and such that \(g^N(y_j) = f^N(y_i)\ .\) As \(f\) has not been changed on \(f^{N+i}(y)=f^N(y_i), f^{N+i+1}(y)\dots f^{j-1}(y)=f^{-1}(y_j)\ ,\) one gets that \(y_j\) is a periodic orbit of \(g\) of period \(j-i\ ,\) ending the proof.

Consequences and generalizations

Generic density of the periodic orbits in the non-wandering set

The closing lemma asserts that every non- wandering point can be made periodic by a small perturbation. This periodic point can be made hyperbolic by a new perturbation, and now it persists under small perturbations. This proves

Theorem for \(C^1-\)generic diffeomorphisms, the set of hyperbolic periodic points is dense in the non-wandering set.

This fact is related to the definition of “Axiom A” diffeomorphisms. Axiom A means that the non-wandering set is hyperbolic, and is the closure of the set of periodic points. This second requirement always looks somewhat strange. However there are examples of diffeomorphisms whose non-wandering set is hyperbolic and is not the closure of the set of periodic orbits.

Mañé ergodic closing lemma

Consider a recurrent point \(x\) of a diffeomorphism \(f\ .\) Pugh closing lemma build a diffeomorphism \(g\) close to \(f\) such that \(x\) is periodic and let \(t>0\) be its period for \(g\ .\) But nothing asserts that the orbit \(g^i(x)\) remains close to \(f^i(x)\) for \(i\in\{0,\dots,t\}\ .\)

We will say that \(x\) is well closable if we can build diffeomorphisms \(g^n\) converging to \(f\) such that \(x\) is periodic of period \(t_n\) and \(g_n^i(x)\) remains close to \(f^i(x)\) for \(i\in\{0,\dots,t_n\}\ .\)

This property is important because it implies that the periodic orbit of \(x\) for \(g_n\) keeps many property of the orbit of \(x\) for \(f\ :\) for instance, the Birkhoff averages of some continuous function \(\varphi\) along the periodic orbits of \(x\) for \(g_n\) will converge to the Birhoff averages of \(\varphi\) along the positive orbit of \(x\ .\)

Mañé ergodic closing lemma : The set of well closable point has full measure for every invariant probability measure.

As a consequence, one gets

Theorem The set of Dirac measures along the periodic orbits is dense in the set of invariant ergodic measures, for every \(C^1-\)generic diffeomorphism.

The ergodic closing lemma is one of the key argument in the proof by Mañé of the stability conjecture (every \(C^1-\)structurally stable diffeomorphism satisfies the Axiom A and the strong transversality condition).

Hayashi connecting lemma

For ending the proof of the stability conjecture for flows, Hayashi proved a new perturbation lemma using Lemma 1 above of Pugh’s argument:

The connecting lemma (Hayashi [Ha]): let \(p\) and \(q\) be two hyperbolic periodic saddle points. Assume that there is a sequence of point \(x_n\) converging to a point \(x\) in the unstable manifold of \(p\) and a sequence \(i_n>0\) such that \(f^{i_n}(x_n)\) converges to a point \(y\) in the stable manifold of \(q\ .\) Then there are diffeomorphisms \(g_n\) converging to \(f\) in the \(C^1\)-topology and number \(j_n>0\) such that \(x\) belongs to the unstable manifold of \(p\ ,\) \(y\) belongs to the stable manifold of \(q\ ,\) and \(g_n^{j_n}(x)=y.\)

This connecting lemma has been generalized and used by many authors for understanding the dynamics of the \(C^1-\)generic diffeomorphisms.

The connecting lemma for pseudo orbits

Bonatti and Crovisier [BC] gave the following generalization of Hayashi connecting lemma

Theorem: Let \(f\) be a diffeomorphisms whose periodic orbits are all hyperbolic. Consider points \(x, y\) such that there are \(\delta-\)pseudo orbits starting at \(x\) and ending at \(y\) for every \(\delta\ .\) Then there are arbitrarily small \(C^1\) perturbations \(g\) of \(f\) such that \(y\) belong to the positive orbit of \(x\ .\)

A direct \(C^1-\)generic consequence of this result is that, for every \(C^1-\)generic diffeomorphisms, the closure of the set of periodic orbits is the chain recurrent set \(R(f)\ .\) This result has many other consequences on the dynamics of \(C^1\) generic diffeomorphisms, see [BC]. It also holds for conservative (volume preserving or symplectic) diffeomorphisms, see [ABC].

References

  • [ABC] Arnaud, M.-C., Bonatti, Ch., and Crovisier, S.Dynamiques symplectiques génériques. Ergodic Theory Dynam. Systems 25 (2005), no. 5, 1401--1436.
  • [BC] Bonatti, Ch. and Crovisier, S. Récurrence et généricité. Invent. Math. 158 (2004), no. 1, 33--104.
  • [C] Conley, C. Isolated invariant sets and Morse index, CBMS Regional Conference Series in Mathematics, 38, AMS Providence, R.I., (1978)
  • [G] Gutierrez, C. A counter-example to a \(C^2\) closing lemma. Ergodic Theory Dynam. Systems 7 (1987), no. 4, 509--530.
  • [Ha] Hayashi, Sh. Connecting invariant manifolds and the solution of the \(C^1\) stability and \(\Omega\)-stability conjectures for flows. Ann. of Math. (2) 145 (1997), no. 1, 81--137.
  • [H] Herman, M. Différentiabilité optimale et contre-exemples à la fermeture en topologie \(C^\infty\) des orbites récurrentes de flots hamiltoniens. [Optimal differentiability and counterexamples to the \(C^\infty\) closing lemma for Hamiltonian vector fields] C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 1, 49--51.
  • [P] Pugh, C. The closing lemma. Amer. J. Math. 89 1967 956--1009.
  • [PR] Pugh, C.and Robinson, C. The \(C^{1}\) closing lemma, including Hamiltonians. Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261--313.
  • [S] Smale, S. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 1967 747--817.


Internal references

  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Ernest Barreto (2008) Shadowing. Scholarpedia, 3(1):2243.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.


See also

Chain recurrent set, Ergodic theory, Periodic orbit, Shadowing

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