Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas
Number Theory
2018-01-23 v7 Dynamical Systems
Spectral Theory
Abstract
Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove an asymptotic formula for the eigenvalues of L. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.
Keywords
Cite
@article{arxiv.1210.4083,
title = {Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas},
author = {Giedrius Alkauskas},
journal= {arXiv preprint arXiv:1210.4083},
year = {2018}
}
Comments
49 pages