JournalsjfgVol. 2 , No. 4pp. 339–375

Kneading with weights

  • Hans Henrik Rugh

    Université Paris-Sud, Orsay, France
  • Lei Tan

    Université d'Angers, Angers, France
Kneading with weights cover
Download PDF

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 D(t){\cal D}(t) and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure logρ1\log \rho_1 of the weighted system, playing the role of entropy, we prove that D(t){\cal D}(t) is non-zero when t<1/ρ1|t|<1/\rho_1 and has a zero at 1/ρ11/\rho_1. Furthermore, our map is semi-conjugate to every map in an analytic family \mySt,0<t<1/ρ1\myS_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 ff, the family \mySt\myS_t converges as t1/ρ1t\rightarrow 1/\rho_1 to a continuous piecewise linear interval map f~\tilde{f}. Furthermore, ff is semi-conjugate to f~\tilde{f} 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