Transfer operators, endomorphisms, and measurable partitions
type: Item
instance of: scholarly article
title: Transfer operators, endomorphisms, and measurable partitions
zbMATH Open document ID: 1416.37002
DOI: 10.1007/978-3-319-92417-5
author: S. Bezuglyi
author: Palle Jorgensen
published in: Lecture Notes in Mathematics
publication date: 2018-06-29T00:00:00Z
review text: This monograph is a good modern source on the theory of transfer operators. The book is addressed to researchers of the dynamical systems theory, as well as to mathematicians from related fields. \par Chapter 1 introduces the main notions of the book, transfer operators and endomorphisms, and explains the motivation of the investigation of them. The definition of transfer operators is as follows. Let $\sigma$ be a surjective endomorphism of a standard Borel space ${(X,\mathcal{B})}.$ The linear positive operator $R$ is called transfer operator if it satisfies the pull-out property \[ R((f\circ\sigma)\cdot g)=f\cdot R(g) \] for all Borel functions ${f,g:X\to\mathbb{R}}.$ Occasionally the transfer operator is denoted as a pair ${(R,\sigma)}.$ Several classes of transfer operators are presented in the chapter (generated by finite-to-one endomorphisms; Frobenius-Perron transfer operators; Ruelle transfer operators; generated by a system of conditional measures). \par In the Chapter 2 the authors recall the basics of measurable dynamics. Here are discussed exact and ergodic endomorphisms, measurable partitions, conditional measures, disintegrations of measures and the construction of natural extensions of endomorphisms (solenoids). \par Using the point of view of transfer operators as pairs ${(R,\sigma)}$, in Chapter 3, the authors consider their general properties. In particular, they discuss the structure of semigroup with identity under the product \[ (R_1,\sigma_1)\cdot(R_2,\sigma_2)=(R_1R_2,\sigma_1\sigma_2). \] Also isomorphisms, factor maps and multiplicative properties are investigated in detail. \par Chapter 4 is devoted to properties of positive transfer operators acting on the set $M(X)$ of Borel measures defined by the formula \[ (\mu R)(A)=\int_XR(\chi_A)d\mu. \] In particular, conditions under which $\mu R$ is a finite Borel measure and $\mu R$ is absolutely continuous (or equivalent) with respect to $\mu$ are studied. It is showed that for nonsingular~$\si…
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 28-02
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 6897817
zbMATH Keywords: endomorphisms
zbMATH Keywords: measure spaces
zbMATH Keywords: transfer operators
zbMATH Keywords: invariant measures
zbMATH Keywords: measurable partitions
zbMATH Keywords: harmonic functions
reviewed by: Ivan Podvigin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-92417-5
OpenAlex ID: W2809475211
description: scientific article; zbMATH DE number 6897817