Kneading with weights

  • Hans Henrik Rugh

    Université Paris-Sud, Orsay, France
  • Lei Tan

    Université d'Angers, Angers, 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 and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure of the weighted system, playing the role of entropy, we prove that is non-zero when and has a zero at . Furthermore, our map is semi-conjugate to every map in an analytic family 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 , the family converges as to a continuous piecewise linear interval map . Furthermore, is semi-conjugate to 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