  It follows from a general theorem of Bonk and Eremenko that closed plane
curves which are contractible in the complement to the integral lattice satisfy
a linear isoperimetric inequality. We give an alternative proof of this fact.
Our approach is based on a non-standard combinatorial isoperimetric inequality
which requires a refinement of the small cancellation theory. We present an
application of the isoperimetric inequality for the punctured plane to
Hamiltonian dynamics. Combining it with methods of symplectic topology we show
that every non-identical Hamiltonian diffeomorphism of the 2-torus has at least
linear asymptotic growth of the differential.
