The Milnor-Thurston determinant and the Ruelle transfer operator

TL;DR

This paper reveals that the Milnor-Thurston determinant and the Fredholm determinant of the dual Ruelle transfer operator are identical, explaining their shared zeros and linking topological entropy with spectral properties of the transfer operator.

Contribution

It demonstrates that on an appropriate function space, the dual Ruelle transfer operator's regularized determinant equals the Milnor-Thurston determinant, clarifying their relationship.

Findings

The zeros of the Milnor-Thurston determinant match the peripheral zeros of the transfer operator's Fredholm determinant.

The dual Ruelle transfer operator's regularized determinant is identical to the Milnor-Thurston determinant.

This connection explains the common zeros related to topological entropy.

Abstract

The topological entropy $h_{\rm top}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that $\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. The purpose of this note is to show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle.