Some more questions about regularity of homeomorphisms of foliations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:40:19Z http://mathoverflow.net/feeds/question/31480 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31480/some-more-questions-about-regularity-of-homeomorphisms-of-foliations Some more questions about regularity of homeomorphisms of foliations Ainu 2010-07-11T23:19:42Z 2010-07-12T01:37:55Z <p>This is a continuation of this question answered by Dmitri.</p> <p>Let $F$ and $F'$ be smooth ($C^\infty$) foliations of a manifold $M$. Assume that there is a homeomorphism $h$ that takes $F$ to $F'$.</p> <ol> <li>Assume $F$ has a compact leaf, can one conclude there is a diffeomorphism that takes $F$ to $F'$?</li> <li>Assume that $h$ is $C^1$, can one conclude there is a ($C^\infty$) diffeomorphism that takes $F$ to $F'$?</li> <li>Assume that $M$ is a torus and $F'=L$ is a Diophantine-irrational-line foliation of the torus. Then, in case of $M=T^2$ following the outline of Sam Nead one can show that there's a diffeomorphism that takes $F$ to $L$, does this hold for higher dimensional tori?</li> <li>Dmitri gives us certain obstruction to existence of a diffeomorphism. Are there other interesting obstructions? </li> </ol> http://mathoverflow.net/questions/31480/some-more-questions-about-regularity-of-homeomorphisms-of-foliations/31491#31491 Answer by Ainu for Some more questions about regularity of homeomorphisms of foliations Ainu 2010-07-12T01:37:55Z 2010-07-12T01:37:55Z <p>After some pondering, here is a negative answer to 1.</p> <p>Consider the foliation from Figure 2 of these notes <a href="http://www.crm.cat/Conferences/0910/Acfoli/Hurder.pdf" rel="nofollow">http://www.crm.cat/Conferences/0910/Acfoli/Hurder.pdf</a></p> <p>The return map on the parallel is $C^1$ only if the derivative of the return map at the unique fixed point is 1. However if we draw a similar picture with two compact leaves (circles) then we can get a smooth foliation such that the return map has two fixed points with derivatives a>1 and b&lt;1. And it is possible to perturb this foliations so that a and b are perturbed to different numbers a' and b'. This perturbation can be carefully designed so that the new foliation is continuously conjugate to the old one.</p>