# Closing Lemma for Diffeomorphism (Dynamical Systems)

\begin{conjecture} Let $f\in Diff^{r}(M)$ and $p\in\omega_{f} $. Then for any neighborhood $V_{f}\subset Diff^{r}(M) $ there is $g\in V_{f}$ such that $p$ is periodic point of $g$ \end{conjecture} There is an analogous conjecture for flows ( $C^{r}$ vector fields . In the case of diffeos this was proved by Charles Pugh for $r = 1$. In the case of Flows this has been solved by Sushei Hayahshy for $r = 1$ . But in the two cases the problem is wide open for $r > 1$

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

## Bibliography

Dynamics beyond uniform hyperbolicity :\Springer [Encyclopaedia of Mathematical Sciences Volume 102, Mathematical Phisics,2005]

* indicates original appearance(s) of problem.