Milnor–Thurston kneading theory

Kneading theory provides an effective calculus for describing the qualitative behavior of the iterates of a piecewise monotone mapping f of a closed interval I of the real line into itself. Some quantitative invariants of this discrete dynamical system, such as the lap numbers of the iterates and the Artin–Mazur zeta function of f are expressed in terms of certain matrices and formal power series.

The basic invariant of f is its kneading matrix, a rectangular matrix with coefficients in the ring ${\displaystyle \mathbb {Z} [[t]]}$ of integer formal power series. A closely related kneading determinant is a formal power series

${\displaystyle D(t)=1+D_{1}t+D_{2}t^{2}+\cdots \,}$

with odd integer coefficients. In the simplest case when the map is unimodal, with a maximum at c, each coefficient ${\displaystyle D_{k}}$ is either ${\displaystyle +1}$ or ${\displaystyle -1}$, according to whether the ${\displaystyle (k+1)}$th iterate ${\displaystyle f^{k+1}}$ has local maximum or local minimum at c.

• Sharkovsky theorem
• Topological entropy

• Milnor, John W.; Thurston, William (1988), "On iterated maps of the interval", Dynamical systems (College Park, MD, 1986–87), Lecture Notes in Mathematics, vol. 1342, Berlin: Springer, pp. 465–563, doi:10.1007/BFb0082847, MR 0970571
• Preston, Chris (1989), "What you need to know to knead", Advances in Mathematics, 78 (2): 192–252, doi:10.1016/0001-8708(89)90033-9, MR 1029100