I have in mind a path $\gamma:[0,\infty)\rightarrow\mathbb{R}^2$ with some conditions, and I'm wondering if I can parameterize it nicely. I will denote $\gamma(t)$ by $\gamma_t$. Also, to avoid complex analysis, I will let $i=\bigl( \begin{smallmatrix}0 & -1\\ 1 & 0\end{smallmatrix}\bigr)$ be the $\pi/2$ rotation matrix. Here are the conditions on $\gamma$:
- $\gamma_0=(1,0)$.
- $\gamma$ is infinitely differentiable with derivatives given by $\gamma^{n+1}=\mathbf{u}(i\gamma^n-\gamma^n)$ where $\mathbf{u}(\cdot)$ returns unit vector.
I strongly believe that $\lim_{t\rightarrow\infty}\gamma_t=\mathbf{0}$ and possibly there exists some $T$ for which $\gamma_t=\mathbf{0}$ whenever $t\geq T$. In this case, $2$ is violated for $t\geq T$. Anyway, I am most interested in parameterizing such a path if it unique.
Edit: On working with $2$ I have an easier condition:
$$\gamma^{n+1}=\frac{1}{2}(\mathbf{i-1})\gamma^n$$
where $\mathbf{i-1}=\bigl(\begin{smallmatrix}0 & -1\\ 1 & 0\end{smallmatrix}\bigr)-\bigl( \begin{smallmatrix}1 & 0\\ 0 & 1\end{smallmatrix}\bigr)=\bigl( \begin{smallmatrix}-1 & -1\\ 1 & -1\end{smallmatrix}\bigr)$ which has determinant $2$.