Dynamical zeta functions

TL;DR

This paper provides a detailed exposition of the Milnor-Thurston kneading determinant approach to dynamical zeta functions, connecting classical theory with recent developments and renewed interest in the field.

Contribution

It offers a comprehensive explanation of the kneading determinant method for dynamical zeta functions, integrating insights from Baladi, Ruelle, and recent research.

Findings

Clarifies the connection between kneading determinants and Ruelle transfer operators

Reviews recent advances and applications of the kneading approach

Provides accessible notes for researchers interested in dynamical zeta functions

Abstract

These are notes from a course given in Orsay in 2002 explaining carefully the Milnor-Thurston kneading determinant approach to dynamical zeta functions as interpreted by Baladi and Ruelle (Invent. Math. 1996). We make them available in view of the recent renewed interest in this approach. (see arXiv:1501.00294, The Milnor-Thurston determinant and the Ruelle transfer operator, HH Rugh, Comm. Math. Phys. 342 (2016) 603-614, and arXiv:1407.5313, Kneading with weights, HH Rugh, Lei Tan, J. Fractal Geom. 2 (2015) 339-375)