  We study positive transfer operators $R$ in the setting of general measure
spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated
path-space probability spaces $\left(\Omega,\mathbb{P}\right)$. When the
transfer operator $R$ is compatible with an endomorphism in
$\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the
Hilbert spaces $L^{2}\left(\Omega,\mathbb{P}\right)$ where the path-space
$\Omega$ may then be taken to be a solenoid. Our multiresolutions include both
orthogonality relations and self-similarity algorithms for standard wavelets
and for generalized wavelet-resolutions. Applications are given to topological
dynamics, ergodic theory, and spectral theory, in general; to iterated function
systems (IFSs), and to Markov chains in particular.
