[Paper Review] Infinite-dimensional transfer operators, endomorphisms, and measurable partitions

Sergey Bezuglyi, Palle E. T. Jørgensen|arXiv (Cornell University)|Feb 9, 2017

Mathematical Dynamics and Fractals70 references3 citations

TL;DR

This paper establishes a duality between endomorphisms of standard Borel measure spaces and positive transfer operators, extending the Perron-Frobenius theorem to infinite-dimensional settings. It introduces a universal Hilbert space framework and characterizes harmonic functions, invariant measures, and conditional expectations via transfer operators, with key results on ergodic decomposition and Radon-Nikodym derivatives for nonsingular endomorphisms.

ABSTRACT

We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel measure spaces. Hence the corresponding linear structures are infinite-dimensional. Nonetheless, we prove a number of analogues of the more familiar finite-dimensional settings, for example, the Perron-Frobenius theorem for positive matrices, and the corresponding Markov chains. Tools from the theory of operators in Hilbert space of special significance to us will be the use of a certain universal Hilbert space, as well as classes of operators in it, directly related to the central theme of duality for transfer operators. From ergodic theory, we address such questions as measurable cross sections, partitions, and Rohlin analysis of endomorphisms of measure spaces. While there are classical theorems dealing with analogous questions for automorphisms of measure spaces, a systematic study of endomorphisms is of more recent vintage;-- in its infancy. In order to make the exposition accessible to students and to researchers in neighboring areas, we have included a number of explicit examples and applications.

Motivation & Objective

• To develop a unified framework for transfer operators and endomorphisms in infinite-dimensional measure spaces.
• To generalize the Perron-Frobenius theorem for positive matrices to infinite-dimensional settings, particularly for Markov chains and stochastic processes.
• To characterize harmonic functions and invariant measures using transfer operators and measurable partitions.
• To establish a duality between endomorphisms σ and transfer operators R via Radon-Nikodym derivatives and conditional measures.
• To apply the theory to dynamical systems, including piecewise monotone maps, Gauss maps, and iterated function systems.

Proposed method

• Uses standard Borel and measure spaces to define endomorphisms σ and transfer operators R acting on measurable functions.
• Defines transfer operators R via integration over conditional measures μ_C, with R(f)(x) = ∫_{C_x} f(y) dμ_{C_x}(y).
• Applies the universal Hilbert space H(X) to represent transfer operators and study their spectral properties.
• Employs ergodic decomposition and Wold’s theorem to analyze isometric and automorphic factors of endomorphisms.
• Uses the Jacobian Jσ(x) = ∑_{y∈σ⁻¹(x)} 1/J(y) to define Rσ(h)(x) = ∑_{y∈σ⁻¹(x)} h(y)/Jσ(y), linking nonsingularity to transfer operator structure.
• Applies Rohlin’s theory of measurable partitions and cross-sections to construct invariant measures and analyze endomorphism dynamics.

Experimental results

Research questions

• RQ1 How can the Perron-Frobenius theorem be extended from finite to infinite-dimensional settings using transfer operators?
• RQ2 What is the precise duality between endomorphisms σ and transfer operators R in standard Borel measure spaces?
• RQ3 How do harmonic functions for transfer operators relate to measurable partitions and σ-invariant sets?
• RQ4 What is the role of the Radon-Nikodym derivative ωσ in characterizing invariant measures under nonsingular endomorphisms?
• RQ5 How do transfer operators on L¹ and L² spaces reflect the ergodic and spectral properties of endomorphisms?

Key findings

• A measurable function h is harmonic with respect to R if and only if it is constant on the atoms of a σ-invariant measurable partition ξ.
• The Radon-Nikodym derivative of the pushforward measure μR with respect to μ is given by W = ν_C(C), where ν_C is the conditional measure on the partition element C.
• For a bounded-to-one nonsingular endomorphism σ, the transfer operator Rσ is defined by Rσ(h)(x) = ∑_{y∈σ⁻¹(x)} h(y)/Jσ(y), with Jσ the Jacobian.
• The operator R satisfies R(fg) = fR(g) for f measurable in the base space and g in the product space, showing a multiplicative compatibility with the dynamics.
• The universal Hilbert space H(X) supports a representation of transfer operators and enables spectral analysis of endomorphisms via isometric decompositions.
• The ergodic decomposition of a transfer operator corresponds to the decomposition of the underlying endomorphism into its automorphic and completely non-singular parts.