Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator

TL;DR

The paper proposes a recursive series construction to approximate the eigenvalues of the Gauss-Kuzmin-Wirsing operator, providing evidence for their sign pattern and extending to the Mayer-Ruelle operator.

Contribution

It introduces a new recursive series method to estimate eigenvalues of transfer operators, supporting conjectures about their sign and extending to related operators.

Findings

Recursively constructed series converges to eigenvalues.

Supports conjecture that eigenvalues have alternating signs.

Provides series for the dominant eigenvalue of the Mayer-Ruelle operator.

Abstract

Based on the technique previously developed by the author, we present a conjecture which claims that the reciprocal of the n-th largest (in absolute value) eigenvalue of the Gauss-Kuzmin-Wirsing operator is equal to the sum of a certain infinite series. This series is constructed recurrently. It consists of rational functions with integer coefficients in two variables X, Y, specialized at X=n and Y=2^n. This gives a strong evidence to the conjecture of Mayer and Roepstorff that eigenvalues have alternating sign. Further, a very similar recursion yields a series for the dominant eigenvalue of the Mayer-Ruelle operator.