[Paper Review] Intermittency and Regularized Fredholm Determinants

Hans Henrik Rugh|ArXiv.org|Oct 7, 1996

Advanced Topics in Algebra2 references4 citations

TL;DR

This paper studies intermittent chaotic dynamics in real-analytic interval maps with a neutral fixed point at zero. It introduces a regularized Fredholm determinant that analytically continues across the continuous spectrum [0,1], enabling a one-to-one correspondence between the zeros of the determinant and the discrete eigenvalues of the Perron-Frobenius operator, thereby providing a spectral tool for analyzing intermittency.

ABSTRACT

We consider real-analytic maps of the interval $I=[0,1]$ which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated Perron-Frobenius operator ${\cal M}$ has a decomposition $Sp ({\cal M}) = σ_c \cup σ_p$ where $σ_c=[0,1]$ is the continuous spectrum of ${\cal M}$ and $σ_p$ is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant $d(λ)$ which has a holomorphic extension to $λ\in C-σ_c$ and can be analytically continued from each side of $σ_c$ to an open neighborhood of $σ_c-{0,1}$ (on different Riemann sheets). In $C-σ_c$ the zero-set of $d(λ)$ is in one-to-one correspondence with the point spectrum of ${\cal M}$. Through the conformal transformation $λ(z) = 1/(4z) (1+z)^2$ the function $d \circ λ(z)$ extends to a holomorphic function in a domain which contains the unit disc.

Motivation & Objective

• To understand the spectral structure of the Perron-Frobenius operator for maps with a neutral fixed point at zero.
• To address the challenge of continuous spectrum obscuring discrete eigenvalues in intermittent systems.
• To develop a regularized Fredholm determinant that isolates and characterizes the pure point spectrum.
• To enable analytic continuation of the determinant across the continuous spectrum for spectral analysis.
• To establish a one-to-one correspondence between the zeros of the determinant and the point spectrum of the operator.

Proposed method

• Define the Perron-Frobenius operator M on a suitable function space for real-analytic, expanding maps with a neutral fixed point at 0.
• Decompose the spectrum of M into continuous spectrum σ_c = [0,1] and pure point spectrum σ_p with no accumulation points outside 0 and 1.
• Construct a regularized Fredholm determinant d(λ) that extends holomorphically to C − σ_c.
• Perform analytic continuation of d(λ) from both sides of σ_c to a neighborhood of σ_c − {0,1} on different Riemann sheets.
• Apply the conformal transformation λ(z) = (1+z)²/(4z) to map the determinant to a holomorphic function on a domain containing the unit disk.
• Use the transformation to relate the zero set of d(λ) to the point spectrum of M via the spectral correspondence.

Experimental results

Research questions

• RQ1 How does the spectrum of the Perron-Frobenius operator decompose for maps with a neutral fixed point?
• RQ2 Can a regularized Fredholm determinant be constructed that captures the discrete spectrum despite the presence of a continuous spectrum?
• RQ3 To what extent can the Fredholm determinant be analytically continued across the continuous spectrum?
• RQ4 What is the role of conformal mapping in transforming the spectral problem into a more tractable form?
• RQ5 Is there a one-to-one correspondence between the zeros of the regularized determinant and the eigenvalues of the Perron-Frobenius operator?

Key findings

• The spectrum of the Perron-Frobenius operator M decomposes into a continuous spectrum σ_c = [0,1] and a pure point spectrum σ_p with no accumulation points outside 0 and 1.
• The regularized Fredholm determinant d(λ) admits a holomorphic extension to the complex plane minus the interval [0,1].
• d(λ) can be analytically continued from both sides of [0,1] to a neighborhood of [0,1] − {0,1} on distinct Riemann sheets.
• The zero set of d(λ) in C − [0,1] is in one-to-one correspondence with the point spectrum of M.
• After the conformal transformation λ(z) = (1+z)²/(4z), the function d(λ(z)) extends to a holomorphic function on a domain containing the open unit disk.
• The construction provides a spectral tool for analyzing intermittency in non-uniformly hyperbolic systems via determinant-based methods.