  We define a class of dynamical systems by modifying a construction due to
Tao, which includes certain Furstenburg limits arising from the Liouville
function. Most recent progress on the Chowla conjectures and sign patterns of
the Mobius and Liouville functions uses methods that apply to any dynamical
system in this class. Hence dynamical systems in this class with anomalous
local behavior present obstructions to further progress on these problems by
the same techniques. We construct straightforward examples of dynamical systems
in this class based on polynomial phases and calculate the resulting
obstruction. This requires explicit bounds for the number of sign patterns
arising in a certain way from polynomials, which is elementary but not
completely trivial.
