# Kneading with weights

### Hans Henrik Rugh

Université Paris-Sud, Orsay, France### Lei Tan

Université d'Angers, Angers, France

## Abstract

We generalize Milnor–Thurston's kneading theory to the setting of piecewise continuous and monotone interval maps with a weight associated to each branch. 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 every map in an analytic family $\myS_t, 0<t<1/\rho_1$ of piecewise linear maps with slopes proportional to the prescribed weights and defined on a Cantor set. When the original map extends to a continuous map $f$, the family $\myS_t$ converges as $t\rightarrow 1/\rho_1$ to a continuous piecewise linear interval map $\tilde{f}$. Furthermore, $f$ is semi-conjugate to $\tilde{f}$ and the two maps have the same pressure.

## Cite this article

Hans Henrik Rugh, Lei Tan, Kneading with weights. J. Fractal Geom. 2 (2015), no. 4 pp. 339–375

DOI 10.4171/JFG/24