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

TL;DR

This paper investigates transfer operators and their associated multiresolutions in measure spaces, linking them to wavelet theory, dynamical systems, and stochastic processes, with applications to IFSs and Markov chains.

Contribution

It introduces a framework connecting transfer operators with multiresolutions and wavelet structures, extending to general measure spaces and dynamical systems.

Findings

Constructed path-space probability measures for transfer operators.

Established multiresolution structures including orthogonality and self-similarity.

Applied results to IFSs, Markov chains, and ergodic theory.

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 $\left(\Omega,\mathbb{P}\right)$. 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.