  We study the behavior of hyperbolic affine automorphisms of a translation
surface which is infinite in area and genus that is obtained as a limit of
surfaces built from regular polygons studied by Veech. We find that hyperbolic
affine automorphisms are not recurrent and yet their action restricted to
cylinders satisfies a mixing-type formula with polynomial decay. Then we
consider the extent to which the action of these hyperbolic affine
automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism.
In particular we study the action of these automorphisms on simple closed
curves and on homology classes. These objects are exponentially attracted by
the expanding and contracting foliations but exhibit polynomial decay. We are
able to work out exact asymptotics of these limiting quantities because of
special integral formula for algebraic intersection number which is attuned to
the geometry of the surface and its deformations.
