Kneading Theory for Iteration of Monotonous Functions on the Real Line

TL;DR

This paper extends kneading theory to families of monotonic functions on the real line, providing tools to characterize maps, define kneading determinants, and relate them to entropy and measures.

Contribution

It introduces a generalized kneading theory framework for monotonic functions, unifying classical results and enabling new analysis methods.

Findings

Characterization of monotonic maps via kneading sequences

Definition of kneading determinants and their relation to entropy

Construction of linearizing measures for these functions

Abstract

We construct a version of kneading theory for families of monotonous functions on the real line. The generality of the setup covers two classical results from Milnor-Thurston's kneading theory: the first one is to dynamically characterise an $l$-modal map by its kneading sequence, the second one is to define the concept of kneading determinant, relate it to topological entropy and use this to construct a certain type of special "linearazing measure".