Dynamics of the Takagi function and the shadowing property

TL;DR

This paper investigates the dynamics of the Takagi function, demonstrating orbit convergence, analyzing a family of scaled Takagi maps, and exploring the shadowing property, including conditions under which it fails.

Contribution

It proves orbit convergence for almost all points, introduces scaled Takagi maps, and characterizes when these maps lack the shadowing property.

Findings

Orbits of the Takagi function converge to 2/3 for almost every point.

The Takagi function possesses the shadowing property.

Certain scaled Takagi maps do not have the shadowing property for specific parameters.

Abstract

The Takagi function $T:[0,1]\to \mathbb{R}$ is a classical example of a continuous nowhere differentiable function. In this paper, we study the discrete dynamical system generated by the Takagi function. First, we prove that for almost every point $x\in [0,1]$, the orbit $(T^n(x))_n$ converges to $2/3$. We introduce the family of Takagi maps, given by $\textbf{T}_\gamma=\gamma \cdot T$, where $\gamma>0$ is a parameter. We also study the shadowing property for this family of maps. We show that the Takagi function has the shadowing property. Additionally, we provide two distinct techniques that allow us to find values of the parameter $\gamma$ for which $\textbf{T}_\gamma$ fails to have the shadowing property. Finally, we pose some open questions.