# Charles C. Pugh

Charles Chapman Pugh (born 1940) is an American mathematician who researches dynamical systems. Pugh received his PhD under Philip Hartman of Johns Hopkins University in 1965, with the dissertation The Closing Lemma for Dimensions Two and Three.[1] He has since been a professor, now emeritus, at the University of California, Berkeley.

Charles C. Pugh
Charles Pugh, Berkeley, 1993
Born1940 (age 81–82)
United States
NationalityAmerican
CitizenshipUnited States
Alma materJohns Hopkins University (PhD)
Known forWork in dynamical systems
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
ThesisThe Closing Lemma for Dimensions Two and Three (1965)
Notable studentsAmie Wilkinson
Websitehttps://math.berkeley.edu/people/faculty/charles-c-pugh

In 1967 he published a closing lemma named after him in the theory of dynamical systems.[2][3] The lemma states: Let f be a diffeomorphism of a compact manifold with a nonwandering point x.[4] Then there is (in the space of diffeomorphisms, equipped with the ${\displaystyle C^{1}}$ topology) in a neighborhood of f a diffeomorphism g for which x is a periodic point. That is, by a small perturbation of the original dynamical system, a system with periodic trajectory can be generated.

In 1970 he was an invited speaker at the International Congress of Mathematicians in Nice, delivering a talk on Invariant Manifolds.

Mary Cartwright (left) with Charles Pugh, Nice, 1970

## Books

• Real Mathematical Analysis, Springer-Verlag, 2002

## Notes

1. ^
2. ^ Bonatti, Christian (June 10, 2008). "Pugh closing lemma". Scholarpedia. 3 (6): 5072. doi:10.4249/scholarpedia.5072. ISSN 1941-6016.
3. ^ 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. ISSN 0002-9327. JSTOR 2373414.
4. ^ Wandering points were introduced by George Birkhoff to describe dissipative systems (with chaotic behavior). In the case of a dynamical system given by a map f, a point wanders if it has a neighborhood U which is disjoint to all of the iterations of the map on it: ${\displaystyle f^{n}(U)\cap U=\varnothing .\,}$