Kneading with weights

TL;DR

This paper extends Milnor-Thurston's kneading theory to weighted, piecewise continuous, monotone interval maps, establishing new identities and analyzing the spectral properties of the associated kneading determinant.

Contribution

It introduces a weighted kneading determinant and proves its properties, connecting it to the pressure and semi-conjugacy to a family of Cantor PL maps.

Findings

The weighted kneading determinant ${ m f D}(t)$ is non-zero for |t|<1/ρ₁.

${ m f D}(t)$ has a zero at t=1/ρ₁.

The map is semi-conjugate to an analytic family of Cantor PL maps.

Abstract

We generalise Milnor-Thurston's kneading theory to the setting of piecewise continuous and monotone interval maps with weights. We define a weighted kneading determinant ${\cal D}(t)$ and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure $\log \rho_1$ of the weighted system, playing the role of entropy, we prove that ${\cal D}(t)$ is non-zero when $|t|<1/\rho_1$ and has a zero at $1/\rho_1$. Furthermore, our map is semi-conjugate to an analytic family $h_t, 0 < t < 1/\rho_1$ of Cantor PL maps converging to an interval PL map $h_{1/\rho_1}$ with equal pressure