# 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 $D(t)$ and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure $gρ_{1}$ of the weighted system, playing the role of entropy, we prove that $D(t)$ is non-zero when $∣t∣<1/ρ_{1}$ and has a zero at $1/ρ_{1}$. Furthermore, our map is semi-conjugate to every map in an analytic family $S_{t},0<t<1/ρ_{1}$ 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 $S_{t}$ converges as $t→1/ρ_{1}$ to a continuous piecewise linear interval map $f~ $. Furthermore, $f$ is semi-conjugate to $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