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Let $W$ be a (generally noncompact) 3-manifold, $\pi:W\to S^1$ a smooth fibration and $\mathcal{L}_f$ a 1-dimensional foliation by flow lines of a smooth flow on $W$, having each fiber as a cross-section and inducing monodromy diffeomorphism $f:L\to L$ on a specific fiber $L$. By abuse, I will refer to the foliation $\mathcal{L}_f$ as a flow. I need to replace $f$ by a generally not smooth homeomorphism $h$ in the same isotopy class. One easily constructs a topological flow $\mathcal{L}_h$ inducing this monodromy. There is considerable latitude in the construction of this flow. For instance, one can make it smooth on $W\setminus L$, but with possibly wild discontinuity of the velocity whenever a flow line crosses $L$. I would like to be able to construct $\mathcal{L}_h$ so that it is integral to a $C^0$ line field. I see no obvious obstruction to this. I have an approach that I cannot make work. Any ideas?

The main importance of this is to be able to use the Schwartzmann-Sullivan theory of asymptotic cycles for a certain compact sublamination of $\mathcal{L}_h$.

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Do I understand it correctly that the problem can be reduced to an essentially local problem? Like this. Consider a surface L and a homeo g that is $C^0$ close to identity. Consider the product $L\times[0,1]$. Then the difficulty is to construct a $C^0$ integrable line field in this strip that gives $g:L\times 0->L\times 1$. –  Andrey Gogolev Oct 10 '10 at 20:20
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Yes, that is correct.

I suspect that, in the generality I have asked this question, there may be a counterexample. It would probably involve a homeomorphism $h$ with really bad fractal behavior. I am most interested in a Handel-Miller endperiodic homeomorphism $h:L\to L$ (see an ArXiv posting by John Cantwell and myself: arXiv:1006.4525 Handel-Miller theory and finite depth foliations).

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