Sheldon's answer on MSE is nice.
Thank you Sheldon.
I made an intresting observation relating things to complex dynamics.
The main thing is the mysterious looking change \( x \) -> \( x - 2^t / x. \)
My observation can be considered positive or negative , intresting or dissapointing , it depends on taste I guess and the hope for nontrivial analogues.
But the idea of having some function \( g(x) \) for which every real iterate \( g^{[t]}(x) \) " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ).
- Maybe variants of this exist in calculus textbooks / papers but its very " dynamical " in nature -
Anyway here it is :
\( f_x(t) = x - 2^t / x \)
\( f_x(f_x^{[-1]}(t) + 1) \)
=> \( x - 2^T / x \) with \( T = f_x^{[-1]}(t) + 1 \)
=> \( x - 2/x * 2^{f_x^{-1} (t)} \)
=> \( x - 2/x * Solve(q,x - q/x = t) \)
Solve .. => \( q = x(x-t) \)
Thus :
\( x - 2/x *x(x-t) = x - 2(x-t) = x - 2x + t = -x + t \)
Which is trivial.
Reminds me of this quote :
" Young man, in mathematics you don't understand things. You just get used to them. "
John von Neumann.
Btw I considered doing the things (steps above) in reverse : showing \( x \) -> \( x - t/x \) is valid from the validity of \( x -> -x + t. \)
regards
tommy1729
" the master "
Thank you Sheldon.
I made an intresting observation relating things to complex dynamics.
The main thing is the mysterious looking change \( x \) -> \( x - 2^t / x. \)
My observation can be considered positive or negative , intresting or dissapointing , it depends on taste I guess and the hope for nontrivial analogues.
But the idea of having some function \( g(x) \) for which every real iterate \( g^{[t]}(x) \) " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ).
- Maybe variants of this exist in calculus textbooks / papers but its very " dynamical " in nature -
Anyway here it is :
\( f_x(t) = x - 2^t / x \)
\( f_x(f_x^{[-1]}(t) + 1) \)
=> \( x - 2^T / x \) with \( T = f_x^{[-1]}(t) + 1 \)
=> \( x - 2/x * 2^{f_x^{-1} (t)} \)
=> \( x - 2/x * Solve(q,x - q/x = t) \)
Solve .. => \( q = x(x-t) \)
Thus :
\( x - 2/x *x(x-t) = x - 2(x-t) = x - 2x + t = -x + t \)
Which is trivial.
Reminds me of this quote :
" Young man, in mathematics you don't understand things. You just get used to them. "
John von Neumann.
Btw I considered doing the things (steps above) in reverse : showing \( x \) -> \( x - t/x \) is valid from the validity of \( x -> -x + t. \)
regards
tommy1729
" the master "