Currently I'm studying "Introduction to dyamical systems" by Stuck and Brin. In chapter 6 they define absolutely continuous and transversely absolutely continuous foliations. By proposition 6.2.2 if $W$ is transversely absolutely continuous then it is absolutely continuous. But the converse is not true in general. I'm looking for an example of an absolutely continuous foliation which is not transversely absolutely continuous.
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Here is an 'non-dynamical' example. Let $C\subset [0,1]$ be the standard middle-third Cantor set, $f$ be a homeomorphism on $[0,1]$ such that $|f(C)|>0$. Then consider the segments $L_y$ connecting $(0,y)$ with $(1,fy)$ for all $0\le y \le 1$, which form a foliation, say $\mathcal{L}$, of the unit rectangle $R=[0,1]^2$.
- $\mathcal{L}$ is not transversely absolutely continuous: the holonomy map between the two transversals $\{0\}\times [0,1]\to \{1\}\times[0,1]$ is exactly given by $f$.
Suppose $f$ is piecewisely linear on $[0,1]\backslash C$. Then for each $r<1$, the restriction $\mathcal{L}_r$ to $[0,r]\times[0,1]$, is transversely absolutely continuous. In particular, this implies
- $\mathcal{L}$ is absolutely continuous.
