  We consider polygonal billiards with collisions contracting the reflection
angle towards the normal to the boundary of the table. In previous work, we
proved that such billiards has a finite number of ergodic SRB measures
supported on hyperbolic generalized attractors. Here we study the relation of
these measures with the ergodic absolutely continuous invariant probability
measures (acips) of the slap map, the 1-dimensional map obtained from the
billiard map when the angle of reflection is always equal to zero. Our main
result states that for a generic polygon, if the reflection law has a Lipschitz
constant sufficiently small, then there exists a one-to-one correspondence
between the ergodic SRB measures of the billiard map and the ergodic acips of
the corresponding slap map, and moreover that the number of Bernoulli
components of each ergodic SRB measure equals the number of the exact
components of the corresponding ergodic acip. The case of billiards in regular
polygons and triangles is studied in detail.
