Dynamics of the Takagi function and the shadowing property

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 $\textbf{T}_γ=γ\cdot T$, where $γ>0$ 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 $γ$ for which $\textbf{T}_γ$ fails to have the shadowing property. Finally, we pose some open questions.

Figures (6)

• Figure 1: The graphs of $G_3$, $G_4$ and $T$.
• Figure 2: The orbit $(T^n(1/6))_n$ calculated by a computer, where $T\approx G_{13}$.
• Figure 3: The graph of the function $G_2=g_1+g_2$
• Figure 4: The graph of the $T$ and $T^2$ around $z\in \text{Fix}\,(T)$
• Figure 5: The quasi-tangent point $x=13/24\approx0.541666667$ and $\textbf{T}_{\gamma}$, where $\gamma=\gamma(x)=13/15$.

Theorems & Definitions (50)

• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4