Bezuglyi S., Jorgensen P.E.T. Transfer Operators, Endomorphisms, and Measurable Partitions

Springer, 2018. — 167 p. — (Lecture Notes in Mathematics 2217). — ISBN: 978-3-319-92416-8.The subject of this book stands at the crossroads of ergodic theory and measurable dynamics. With an emphasis on irreversible systems, the text presents a framework of multi-resolutions tailored for the study of endomorphisms, beginning with a systematic look at the latter. This entails a whole new set of tools, often quite different from those used for the “easier” and well-documented case of automorphisms. Among them is the construction of a family of positive operators (transfer operators), arising naturally as a dual picture to that of endomorphisms. The setting (close to one initiated by S. Karlin in the context of stochastic processes) is motivated by a number of recent applications, including wavelets, multi-resolution analyses, dissipative dynamical systems, and quantum theory.
The automorphism-endomorphism relationship has parallels in operator theory, where the distinction is between unitary operators in Hilbert space and more general classes of operators such as contractions. There is also a non-commutative version: While the study of automorphisms of von Neumann algebras dates back to von Neumann, the systematic study of their endomorphisms is more recent; together with the results in the main text, the book includes a review of recent related research papers, some by the co-authors and their collaborators.Introduction and Examples
Endomorphisms and Measurable Partitions
Positive, and Transfer, Operators on Measurable Spaces: General Properties
Transfer Operators on Measure Spaces
Transfer Operators on L1 and L2
Actions of Transfer Operators on the Set of Borel Probability Measures
Wold’s Theorem and Automorphic Factors of Endomorphisms
Operators on the Universal Hilbert Space Generated by Transfer Operators
Transfer Operators with a Riesz Property
Transfer Operators on the Space of Densities
Piecewise Monotone Maps and the Gauss Endomorphism
Iterated Function Systems and Transfer Operators
Examples