  In this article we report on a surprising relation between the transfer
operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators
on the space of period functions for the modular group $\PSL (2,\mathbb{Z})$.
For this we study special eigenfunctions of the transfer operators with
eigenvalues $\mp 1$, which are also solutions of the Lewis equations for the
groups $\Gamma_{0}(n)$ and which are determined by eigenfunctions of the
transfer operator for the modular group $\PSL (2,\mathbb{Z})$. In the language
of the Atkin-Lehner theory of old and new forms one should hence call them old
eigenfunctions or old solutions of Lewis equation. It turns out that the sum of
the components of these old solutions for the group $\Gamma_{0}(n)$ determine
for any $n$ a solution of the Lewis equation for the modular group and hence
also an eigenfunction of the transfer operator for this group.
