The exact anomalous dimensions of the single trace
composite operators in the planar limit of $N=4$ super
Yang-Mills can be studied in the framework of the
Thermodynamic Bethe Ansatz method (TBA). In this
talk, I will discuss the analytic properties of the TBA
solutions -the Y-functions- and the associated set of
functional relations -the Y-system-.
Contrary to the more studied relativistic invariant cases, the
$AdS_5/CFT_4$-related Y-functions live on multi-sheeted
coverings of the complex plane with an infinite number of
square-root branch points.
As a direct consequence of this fact, the TBA equations
and in particular the dressing kernel cannot be obtained
from the Y-system without extra information on the
square-root discontinuities across semi-infinite segments in
the complex plane.
It is shown that the discontinuity functions fulfill an
independent simple set of local functional equations.
The Y-system extended by the discontinuity relations forms
a fundamental set of local non-linear constraints that can
be easily transformed into integral form through Cauchy's