Dynamical properties of endomorphisms, multiresolutions, similarity-and orthogonality relations

Abstract

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 $\left(Ω,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(Ω,\mathbb{P}\right)$ where the path-space $Ω$ 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.