Let g: ℝ →ℝ be a differentiable bounded function, and let h: ℝ^2 →ℝ^2 be given by h(t, x)=(λt, λ

Step 1: **Understand the function $h$ and its action on points.**u000AThe function $h(t, x)$ transfo

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Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable bounded function, and let $h: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by $$ h(t, x)=\left(\lambda t, \lambda^{2-s}(x-g(t))\right) $$ where $\lambda>1$ and $0<s<2$. Show that graph $f$ is a repeller for $h$, where $f$ is the function $$ f(t)=\sum_{k=0}^{\infty} \lambda^{(s-2) k} g\left(\lambda^k t\right) . $$ Thus functions of Weierstrass type (see (11.7)) can occur as invariant sets in dynamical systems.

   Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable bounded function, and let $h: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by
$$
h(t, x)=\left(\lambda t, \lambda^{2-s}(x-g(t))\right)
$$
where $\lambda>1$ and $0<s<2$. Show that graph $f$ is a repeller for $h$, where $f$ is the function
$$
f(t)=\sum_{k=0}^{\infty} \lambda^{(s-2) k} g\left(\lambda^k t\right) .
$$

Thus functions of Weierstrass type (see (11.7)) can occur as invariant sets in dynamical systems.

Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer 2nd Edition

Chapter 13, Problem 9

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Step 1

** The function $h(t, x)$ transforms a point $(t, x)$ in $\mathbb{R}^2$ to another point $(\lambda t, \lambda^{2-s}(x - g(t)))$. Here, $\lambda > 1$ and $0 < s < 2$. The first component $\lambda t$ scales $t$ by a factor of $\lambda$, which stretches the $t$-axis. Show more…

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Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable bounded function, and let $h: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by $$ h(t, x)=\left(\lambda t, \lambda^{2-s}(x-g(t))\right) $$ where $\lambda>1$ and $0<s<2$. Show that graph $f$ is a repeller for $h$, where $f$ is the function $$ f(t)=\sum_{k=0}^{\infty} \lambda^{(s-2) k} g\left(\lambda^k t\right) . $$ Thus functions of Weierstrass type (see (11.7)) can occur as invariant sets in dynamical systems.

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