Strength of attraction of fixed points Consider a smooth map $f: \mathbb{R} \rightarrow \mathbb{R}$ with an attracting fixed point $F$. Then, we have


*

*if $f'(F) \ne 0$, $F$ is a "simple" attracting fixed point,

*if $f'(F) = 0$, $F$ is a super-attracting fixed point,

*if $f^{(k)}(F) = 0$ for $1 \le k \le n$, we can say $F$ is super-attracting of order $n$,

*if $f^{(k)}(F) = 0$ for all $k \ge 1$, we could call $F$ a mega-attracting fixed point (I don't think that's a "standard" piece of terminology, but I like it!).


Examples:


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*$f(x) = \frac{\sin(x)}{2}$ has a simple attracting fixed point at $0$,

*$f(x) = x^2$ has a super-attracting fixed point at $0$,

*$f(x) = x^k$, $k > 1$, has a super-attracting fixed point of order $k-1$ at $0$,

*$f(x) = \begin{cases} e^{-\frac{1}{x^2}}, & \mbox{if } x \ne 0 \\ 0, & \mbox{if } x = 0\end{cases}$ has a mega-attracting fixed point at $0$, yet is not a constant function.


What are the upper and lower bounds on the "rate" of attraction for each class, and what's the proof? I suppose the absolute maximum "strength" of attraction of a mega-attracting fixed point, and thus of all fixed points, is the strength of the mega-attractor of a constant function, but what about the minimums, and what about the other classes?
Also, is there a generalization of these concepts to continuous flows, and to more complicated attracting sets?
 A: As ioannis galidakis noted, the "rate of attraction" has a concrete meaning in numerical analysis, which is the asymptotics of the difference $|x_n-F|$ where $x_n$ is a sequence such that $x_n\to F$ and $x_n=f(x_{n-1})$ for $n>1$. In practical terms it helps to focus on the logarithm of difference, $$D_n=|\log |x_n-F||$$ which roughly corresponds to the number of digits of $x_n$ that are the same as the digits of $F$. (This is of obvious interest when $F$ is a root of an equation we are trying to solve.) 


*

*Simple attracting fixed point has $|x_n-F| \approx |f'(F)||x_{n-1}-F|$. Hence, $D_n$ grows linearly with $n$. 

*Super-attracting fixed point of order $k$ has $|x_n-F| \approx |c_{k+1}| |x_{n-1}-F|^{k+1}$ where $c_{k+1}$ is the first nonzero term of the Taylor expansion of $f(x)-F$. Hence, $D_{n}$ grows exponentially,  $D_n\approx (k+1)^n$.

*Mega-attracting: according to the above, $D_n$ grows faster than every exponential, i.e., has superexponential growth. 


There is no upper bound on the growth of $D_n$ in the mega-attracting case. Indeed, given a rapidly convergent monotone sequence $x_n\to 0$, we can construct a $C^\infty$ function such that $f(x_n)=x_{n+1}$ for all $n$. (Use disjoint smooth bumps of height $x_{n+1}$ supported near each $x_n$. For the derivative of order $k$ to have limit zero at the origin, we need $(x_n-x_{n+1})^k x_{n+1} \to 0$ as $n\to\infty$. This is satisfied for every $k$ provided $x_n$ goes to zero rapidly enough.)
