Kneading with weights

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 $\mathcal{D}(t)$ 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 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

25 pages, 3 figures