Kneading with weights

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 $\mathcal{D}(t)$ 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 $\mathcal{D}(t)$ 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 ${\mathcal{S}}_t,0<t<1/{\rho }_1$ 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 ${\mathcal{S}}_t$ converges as $t\to 1/{\rho }_1$ 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.