Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be said about iterates of $f$, specifically:

• Is there a closed-form expression (in function of $n,x$) for the value $f^{\circ n}(x)$ of the $n$-th iterate of $f$ when $n\in \mathbb{N}$ (or indeed $n\in \mathbb{Z}$, letting $f^{\circ(-1)}(x) = \sqrt{x(2-x)}$ stand for the reciprocal of $f$)?

• Does there exist a $1$-parameter group $(f^{\circ\tau})_{\tau\in\mathbb{R}}$ of increasing auto-homeomorphisms of $[0,1]$ (meaning $f^{\circ\varsigma} \circ f^{\circ\tau} = f^{\circ(\varsigma+\tau)}$ for $\varsigma,\tau\in\mathbb{R}$) such that:

  • $f^{\circ n}$ coincides with the $n$-iterate of $f$ for $n\in\mathbb{N}$ (of course $1$ is enough here),

  • $1-f^{\circ\tau}(1-x) = f^{\circ(-\tau)}(x)$ for all $\tau,x$ (this holds for integer $\tau$),

  • for $x$ fixed, $f^{\circ\tau}(x)$ is a continuous function of $\tau$, and

  • for any $\tau$ we have $f^{\circ\tau}(x) = 2\,(\frac{x}{2})^{2^\tau} + o(x^{2^r})$ when $x\to 0$

  ? (Note: I don't know if these conditions are enough to make such a family $(f^{\circ\tau})_{\tau\in\mathbb{R}}$ unique even if it exists.)

Ideally I'm hoping for an answer to both questions at once in the form of a closed-form expression of $f^{\circ\tau}(x)$, but I'd already be happy with an answer to either part.

Edit: as it's been suggested to me elsewhere, it seems that we can define $f^{\circ\tau}(x)$ by $\lim_{n\to+\infty} f^{\circ(-n)}(2\cdot(\frac{1}{2}f^{\circ n}(x))^{2^\tau})$: numerically this appears to work very well, but it's not clear to me why this limit even exists.