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 $ swhy 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) $?