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2018 | Buch

Introduction and Examples Kapitel lesen Erstes Kapitel lesen

Transfer Operators, Endomorphisms, and Measurable Partitions

verfasst von: Dr. Sergey Bezuglyi, Prof. Palle E. T. Jorgensen

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction and Examples
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 2. Endomorphisms and Measurable Partitions
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 3. Positive, and Transfer, Operators on Measurable Spaces: General Properties
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 4. Transfer Operators on Measure Spaces
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 5. Transfer Operators on L1 and L2
Abstract
Given a transfer operator (R, σ), it is of interest to find the measures μ such that both R and σ induce operators in the corresponding L p spaces, i.e., in \(L^p(X, {\mathcal B}, \mu )\). We turn to this below, but our main concern are the cases p = 1, p = 2, and p = ∞. When R is realized as an operator in \(L^2(X, {\mathcal B}, \mu )\), for a suitable choice of μ, then it is natural to ask for the adjoint operator R ∗ where “adjoint” is defined with respect to the L 2(μ)-inner product.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 6. Actions of Transfer Operators on the Set of Borel Probability Measures
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 7. Wold’s Theorem and Automorphic Factors of Endomorphisms
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 8. Operators on the Universal Hilbert Space Generated by Transfer Operators
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 9. Transfer Operators with a Riesz Property
Abstract
A well known theorem (Riesz) in analysis states that every positive linear functional L on continuous functions is represented by a Borel measure. More precisely, let X be a locally compact Hausdorff space and C c (X) the space of continuous functions with compact support. Then the well-known Riesz representation theorem from analysis says that, for every positive linear functional L, there exists a unique regular Borel measure μ on X such that
$$\displaystyle L(f) = \int _X f\; d\mu. $$
We are interested in a special case of functionals L x defined on a function space by the formula L x (f) = f(x). For Borel functions \(\mathcal F(X, {\mathcal B})\) over a standard Borel space \((X, {\mathcal B})\), the Riesz theorem is not directly applicable. We introduce in this chapter a class of transfer operators R that have the following property.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 10. Transfer Operators on the Space of Densities
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 11. Piecewise Monotone Maps and the Gauss Endomorphism
Abstract
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.
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 12. Iterated Function Systems and Transfer Operators
Abstract
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).
Sergey Bezuglyi, Palle E. T. Jorgensen
Chapter 13. Examples
Abstract
In this chapter, we discuss in detail several examples of transfer operators that are mentioned in Introduction.
Sergey Bezuglyi, Palle E. T. Jorgensen
Backmatter
Metadaten
Titel
Transfer Operators, Endomorphisms, and Measurable Partitions
verfasst von
Dr. Sergey Bezuglyi
Prof. Palle E. T. Jorgensen
Copyright-Jahr
2018
Electronic ISBN
978-3-319-92417-5
Print ISBN
978-3-319-92416-8
DOI
https://doi.org/10.1007/978-3-319-92417-5