Transfer operators, endomorphisms, and measurable partitions
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~$\sigma$ and $\mu$ such that $R(1)$ is $\mu$-integrable \[ \mu R\mid_{\sigma^{-1}(\mathcal{B})}\ \sim\ \mu \mid_{\sigma^{-1}(\mathcal{B})}. \] Also the ergodic decomposition of transfer operators and relation with polymorphisms are discussed. \par In Chapter 5 the authors continue to study the properties of transfer operators on measures as well as on $L^1$ and on the Hilbert space $L^2.$ In particular, it is showed that $R$ sends equivalent measures to equivalent ones; the adjoint operator $R^*$ is well described as well as the explicit realization of $R$ in $L^2.$ Some additional relations between $R$ and $\sigma$ are considered. \par In the spirit of Chapter 4 for normalized transfer operators $R$ the authors investigate in Chapter 6 the properties of two maps \[ t_R:\lambda\mapsto\lambda R, \ \ \ s_\sigma:\lambda\mapsto\lambda\circ\sigma^{-1}, \] defined on the set ${M_1(X)}$ of probability measures. The relation between the sets of fixed points and compositions ${t_Rs_\sigma}$ and ${s_\sigma t_R}$ are discussed. \par Chapter 7 deals with a geometric variant of Wold's theorem on isometries in a Hilbert space. The authors apply this result to measure-preserving noninvertible dynamical systems. \par Chapter 8 presents the concept of the universal Hilbert space on ${(X,\mathcal{B})}$ which is space of equivalent pairs ${(f,\lambda)}$ with ${f\in L^2(X,\lambda)}.$ On the universal Hilbert space the authors consider some operators generated by normalized transfer operators and study their properties. \par Chapter 9 is devoted to study the transfer operators satisfying the Riesz property, i.e., for which there exists a uniquely determined family of Borel measures~${\{\mu_x\}_{x\in X}}$ such that \[ R(f)(x)=\int_Xf(y)\,d\mu_x(y). \] It is showed that for normalized transfer operators with Riesz property one has ${\mu_x\circ\sigma^{-1}=\delta_x}.$ \par The next Chapter 10 contains the properties of transfer operators on the space of densities, i.e., operators of the form \[ R(f)(x)=\frac{(fd\mu)\circ\sigma^{-1}}{d\mu}(x),\ \ f\in L^1(X,\mu). \] \par In Chapters 11 and 12 the authors apply the obtained results to some examples from ergodic theory, namely to transfer operators induced by iterated function systems, and particularly, related to the Gauss map and a doubling endomorphism. \par The last chapter of the book is devoted to several examples of transfer operators related to measurable partitions. In particular, Bratteli diagrams and Vershik map are considered.
scientific article; zbMATH DE number 6897817