  We first construct a genus zero positive allowable Lefschetz fibration over
the disk (a genus zero PALF for short) on the Akbulut cork and describe the
monodromy as a positive factorization in the mapping class group of a surface
of genus zero with five boundary components. We then construct genus zero PALFs
on infinitely many exotic pairs of compact Stein surfaces such that one is a
cork twist of the other along an Akbulut cork. The difference of smooth
structures on each of exotic pairs of compact Stein surface is interpreted as
the difference of the corresponding positive factorizations in the mapping
class group of a common surface of genus zero.
