# foliations – Transverse measurements in pseudo-Anosov diffeomorphisms

I have recently started research on pseudo-Anosov diffeomorphisms, which are diffeomorphisms on surfaces $$f: M to M$$ admitting two singular measured foliations $$( mathcal F ^ s, nu ^ s)$$ and $$( mathcal f ^ u, nu ^ u)$$. I have often seen the following notation and I have trouble understanding it:
$$f ( mathcal F ^ s, nu ^ s) = ( mathcal F ^ s, nu ^ s / lambda) quad textrm {and} quad f ( mathcal F u, nu ^ u ) = ( mathcal F ^ u, lambda nu ^ u)$$
for some people $$lambda> 1$$. In this context, both $$mathcal F ^ s$$ and $$mathcal f ^ u$$ are foliations with the same (finite) set of singularities $$S$$, and $$nu s$$ and $$nu ^ u$$ are cross-sectional measures to the leaves of $$mathcal F ^ s$$ and $$mathcal f ^ u$$. In particular, if there is a table of coordinates $$phi: U to mathbb C$$with $$U sub-set M$$ containing no singularity, then the leaves of $$mathcal F ^ s$$ and $$mathcal f ^ u$$ are mapped to sets of the form $${z: mathrm {Re} (z) = mathrm {const} } cap phi (U) quad textrm {and} quad {z: mathrm {Im} (z) = mathrm {const} } cap phi (U)$$
respectively, and $$nu ^ s | _U$$ and $$nu ^ u | _U$$ are withdrawals from $$| mathrm {Re} (dz) |$$ and $$| mathrm {Im} (dz) |$$. A similar formula (with powers of complex variables) is valid in neighborhoods coordinated with singularities.

What I do not understand: These withdrawal expressions imply if $$gamma$$ is a plate of $$mathcal F ^ s$$then since $$nu ^ s ( gamma) = 0$$, we must have $$nu ^ u ( gamma)> 0$$. Previously I had interpreted the notation $$f ( mathcal F ^ u, nu ^ u) = ( mathcal F ^ u, lambda nu ^ u)$$ to signify that $$nu ^ u (f ( gamma)) = lambda nu ^ u ( gamma)$$, for our $$lambda> 1$$. However, in all the articles on the pseudo-Anosov cards that I read, it is clear that points in the same $$mathcal F ^ s$$– uniformly contracted sheet and points in the same $$mathcal f ^ u$$-leaf uniformly expand, which is in direct contradiction with my interpretation. In particular, the exact opposite seems to be the case: $$nu ^ u (f ( gamma)) = nu ^ u ( gamma) / lambda$$ for the plates of $$mathcal F ^ s$$ and $$nu ^ s (f ( gamma)) = lambda nu ^ s ( gamma)$$ for the plates of $$mathcal f ^ u$$.

Can any one explain what the notation $$f ( mathcal F ^ u, nu ^ u) = ( mathcal F ^ u, lambda nu ^ u)$$ and $$f ( mathcal F ^ s, nu ^ s) = ( mathcal F ^ s, nu ^ s / lambda)$$ means? If points in the same stable sheet of $$mathcal F ^ s$$ contract uniformly w.r.t. $$nu ^ u$$ (which is obviously the case in classical Anosov systems) and in the unstable sheet are in uniform expansion w.r.t. $$nu s$$why do we write $$f ( mathcal F ^ s, nu ^ s) = ( mathcal F ^ s, nu ^ s / lambda)$$ and $$f ( mathcal F ^ u, nu ^ u) = ( mathcal F ^ u, lambda nu ^ u)$$?