  We have studied the compact phase conformations of semi-flexible polymer
chains confined in two dimensional nonhomogeneous media, modelled by fractals
that belong to the family of modified rectangular (MR) lattices. Members of the
MR family are enumerated by an integer $p$ $(2\leq p<\infty)$, and fractal
dimension of each member of the family is equal to 2. The polymer flexibility
is described by the stiffness parameter $s$, while the polymer conformations
are modelled by weighted Hamiltonian walks (HWs). Applying an exact method of
recurrence equations we have found the asymptotic behavior of partition
function $Z_N$ for closed HWs consisting of $N$ steps. We have established that
$Z_N$ scales as $\omega^N \mu^{N^\sigma}$, where the critical exponent $\sigma$
in the stretched exponential term does not depend on $s$, and takes the value
1/2 for each fractal from the family. The constants $\omega$ and $\mu$ depend
on both $p$ and $s$, and, in addition, $\mu$ depends on the parity of MR
generator. Besides, we have calculated numerically the stiffness dependence of
the polymer persistence length and various thermodynamic quantities (such as
free and internal energy, specific heat and entropy), for a large set of
members of MR family. We have found that semi-flexible compact polymers, on MR
lattices, can exist only in the liquid-like (disordered) phase, whereas the
crystal (ordered) phase has not appeared. Finally, the behavior of examined
system at zero temperature has been discussed.
