Pugh's closing lemma

In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:

Let

f:M\to M

be a

C^{1}

diffeomorphism of a compact smooth manifold

M

. Given a nonwandering point

x

of

f

, there exists a diffeomorphism

g

arbitrarily close to

f

in the

C^{1}

topology of

\operatorname {Diff} ^{1}(M)

such that

x

is a periodic point of

g

.^[1]

Interpretation

Pugh's closing lemma means, for example, that any chaotic set in a bounded continuous dynamical system corresponds to a periodic orbit in a different but closely related dynamical system. As such, an open set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically; this is the basis of some autonomous convergence theorems.

References

^ Pugh, Charles C. (1967). "An Improved Closing Lemma and a General Density Theorem". American Journal of Mathematics. 89 (4): 1010–1021. doi:10.2307/2373414. JSTOR 2373414.

Pugh's closing lemma

Contents

Interpretation

See also

References

Further reading