Sergey Bezuglyi

Introduction and Examples

We present a unified study a class of positive operators called (generalized) transfer operators, and of their applications to the study of endomorphisms, measurable partitions, and Markov processes, as they arise in diverse settings. We begin with the setting of dynamics in standard Borel, and measure, spaces.

Transfer Operators on the Space of Densities

This chapter is focused on the study of an important class of transfer operators. As usual, we fix a non-invertible non-singular dynamical system ((X, {mathcal B}, mu, sigma )). Without loss of generality, we can assume that μ is a finite (even probability) measure because μ can be replaced by any measure equivalent to μ.

Piecewise Monotone Maps and the Gauss Endomorphism

The purpose of the next two chapters is to outline applications of our results to a family of examples of dynamics of endomorphisms, and their associated transfer operators.

Positive, and Transfer, Operators on Measurable Spaces: General Properties

The notions of positive operators and transfer operators are central objects in this book. We will discuss various properties of these operators and their specific realization in the subsequent chapters. Here we first focus on the most general properties and basic definitions related to these operators.

Operators on the Universal Hilbert Space Generated by Transfer Operators

Starting with a fixed transfer operator (R, σ) on ((X, {mathcal B})), we show below that there is then a naturally induced universal Hilbert space (mathcal H(X)) with the property that (R, σ) yields naturally a corresponding isometry in (mathcal H(X)), i.e., an isometry with respect to the inner product from (mathcal H(X)). With this, we then obtain a rich spectral theory for the transfer operators, for example a setting which may be considered to be an infinite-dimensional Perron-Frobenius theory. Our main results are Theorems 8.12, 8.17, and 8.18.

Transfer Operators on Measure Spaces

Our starting point is a fixed pair (R, σ) on ((X, {mathcal B})) making up a transfer operator. In the next two chapters we turn to a systematic study of specific and important sets of measures on ((X, {mathcal B})) and actions of (R, σ) on these sets of measures. These classes of measures in turn lead to a structure theory for our given transfer operator (R, σ). Our corresponding structure results are Theorems 4.14, 5.13, 5.12, 5.9, and 5.20.

Actions of Transfer Operators on the Set of Borel Probability Measures

Let (R, σ) be a transfer operator defined on the space of Borel functions (mathcal F(X, {mathcal B})). The main theme of this chapter is the study of a dual action of R on the set of probability measures (M_1 = M_1(X, {mathcal B})). As a matter of fact, a big part of our results in this chapter remains true for any sigma-finite measure on ((X, {mathcal B})), but we prefer to work with probability measures. The justification of this approach is contained in the results of Chap. 5 where we showed that the replacement of a measure by a probability measure does not affect the properties of R described in terms of measures. Our main assumption for this chapter is that the transfer operators R are normalized, that is R(1) = 1. In other chapters, we also used this assumption to prove some results.

Iterated Function Systems and Transfer Operators

In this chapter, we will discuss an application of general results about transfer operators, that were proved in previous chapters, to a family of examples based on the notion of iterated function system (IFS).

Wold’s Theorem and Automorphic Factors of Endomorphisms

In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isometry operator of a Hilbert space in a unitary part and a unilateral shift.

Endomorphisms and Measurable Partitions

In this chapter, we collect definitions and some basic facts about the underlying spaces, endomorphisms, measurable partitions, etc., which are used throughout the book. Though these notions are known in ergodic theory, we discuss them for the reader’s convenience.