Kneading with weights

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