Abstract
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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
invariant curve
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.
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