  We construct a short-range potential on a bidimensional full shift and finite
alphabet that exhibits a zero-temperature chaotic behaviour as introduced by
van Enter and Ruszel. A phenomenon where there exists a sequence of
temperatures that converges to zero for which the whole set of equilibrium
measures at these given temperatures oscillates between two ground states.
Br\'emont's work shows that the phenomenon of non-convergence does not exist
for short-range potentials in dimension one; Leplaideur obtained a different
proof for the same fact. Chazottes and Hochman provided the first example of
non-convergence in higher dimensions $d\geq3$; we extend their result for $d=2$
and highlight the importance of two estimates of recursive nature that are
crucial for this proof: the relative complexity and the reconstruction function
of an extension.
