We consider the one-parameter family of billiard tables
which have a common caustic and study the
corresponding family of billiard maps .
The billiard tables are constructed geometrically by the
string construction, where the length of the string is the parameter.
We study the family of circle homeomorphisms obtained by restricting
the billiard map
to the canonical invariant circle belonging to
the caustic and the rotation function .
We prove that for a dense set of curves which
includes curves with flat points and polygons, the function
is a devil-staircase.
We analyze the motion of Aubry-Mather sets passing the
as is increasing. The interesting case are Mather sets with
rational rotation number p/q, which consists of Birkhoff periodic orbits.
The passage of such orbits through the canonical invariant circle
as the parameter changes is accompanied by bifurcations
which can be described by index methods.