Journal of Fractal Geometry


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Volume 2, Issue 4, 2015, pp. 339–375
DOI: 10.4171/JFG/24

Published online: 2015-08-31

Kneading with weights

Hans Henrik Rugh[1] and Lei Tan

(1) Université Paris-Sud, Orsay, France

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

Keywords: Milnor–Thurston kneading theory, piecewise linear interval maps, dynamical zeta functions, kneading determinant, pressure, entropy, semi-conjugacies.

Rugh Hans Henrik, Tan Lei: Kneading with weights. J. Fractal Geom. 2 (2015), 339-375. doi: 10.4171/JFG/24