maybe i sound naive.
but i think that is naive.
in other words , intuitively it may seem simple :
f(x) = f(g(x))
we set q(g(x)) = q(x)+1 and p(x) is 1 periodic and t(x) is the inverse of p(x).
then f(x) = p(q(x))
now to find g(x) from f(x) we ' believe ' that reversing the above will help :
f(x) = p(q(x))
t(f(x)) = q(x) (*)
t(f(x)) + 1 = q(x)+1
from (*) we know that inv q(x) = invf(p(x)). lets call that F(x).
then
from q(g(x)) = q(x)+1 = t(f(x)) + 1 we get
g(x) = F(t(f(x)) + 1)
( assuming the correct branches are taken )
this formula looks powerfull ... lets try
exp(x) = exp(g(x))
g(x) = log( sin (2pi * (1/(2pi) arcsin(exp(x)) + 1) ) )
we simplify
g(x) = log( sin( arcsin(exp(x)) + 2pi) ) )
this doesnt look good ... we simplify further
g(x) = log(sin(arcsin(exp(x)))))
now note that we cannot work with the branches of log , since we need to know an invariant of exp to know the branches of log.
if we do not know that exp(x + 2pi i) = exp(x) we do not know that log(x) + 2pi i is another branch of log(x) and visa versa !
we could find that by investigating log and exp , but that is not a general method.
the riemann surfaces of a function and its inverse reveals branches and invariants , but not easily in closed form rather some numerical values.
so how do we reduce g(x) = log(sin(arcsin(exp(x))))) ?
sure g(x) = log(sin(arcsin(exp(x))))) implies that g(x) = x.
but g(x) = x is a uselessly trivial result.
my example included functions we understand well , but of course more complicated ones are not so easy to analyse.
i am thus tempted to conclude that no decent theory of invariants exists without investigating the riemann surfaces.
more precisely , i assume that we must find a function mapping one branch to the next , therefore requiring to know the structure of the riemann surface ( shape and cuts ) and expressing that function as a taylor series , which we compute by taking nth derivatives of the analytic continuations that take us to the next branch value of the imput.
although that may work in many cases , since we need to understand the structure of the surface alot , it still might not be - or lead to - a complete theory of invariants.
i conclude that , unless i am missing something important , that solving f(x) = f(g(x)) for g(x) is non-trivial.
however my knowledge is limited , maybe someone can "open my eyes" with some theorems or theories.
since functions like tetration and the alike are more exotic than any standard function i therefore find it justified to consider finding invariants , even if they dont have closed forms.
regards
tommy1729
but i think that is naive.
in other words , intuitively it may seem simple :
f(x) = f(g(x))
we set q(g(x)) = q(x)+1 and p(x) is 1 periodic and t(x) is the inverse of p(x).
then f(x) = p(q(x))
now to find g(x) from f(x) we ' believe ' that reversing the above will help :
f(x) = p(q(x))
t(f(x)) = q(x) (*)
t(f(x)) + 1 = q(x)+1
from (*) we know that inv q(x) = invf(p(x)). lets call that F(x).
then
from q(g(x)) = q(x)+1 = t(f(x)) + 1 we get
g(x) = F(t(f(x)) + 1)
( assuming the correct branches are taken )
this formula looks powerfull ... lets try
exp(x) = exp(g(x))
g(x) = log( sin (2pi * (1/(2pi) arcsin(exp(x)) + 1) ) )
we simplify
g(x) = log( sin( arcsin(exp(x)) + 2pi) ) )
this doesnt look good ... we simplify further
g(x) = log(sin(arcsin(exp(x)))))
now note that we cannot work with the branches of log , since we need to know an invariant of exp to know the branches of log.
if we do not know that exp(x + 2pi i) = exp(x) we do not know that log(x) + 2pi i is another branch of log(x) and visa versa !
we could find that by investigating log and exp , but that is not a general method.
the riemann surfaces of a function and its inverse reveals branches and invariants , but not easily in closed form rather some numerical values.
so how do we reduce g(x) = log(sin(arcsin(exp(x))))) ?
sure g(x) = log(sin(arcsin(exp(x))))) implies that g(x) = x.
but g(x) = x is a uselessly trivial result.
my example included functions we understand well , but of course more complicated ones are not so easy to analyse.
i am thus tempted to conclude that no decent theory of invariants exists without investigating the riemann surfaces.
more precisely , i assume that we must find a function mapping one branch to the next , therefore requiring to know the structure of the riemann surface ( shape and cuts ) and expressing that function as a taylor series , which we compute by taking nth derivatives of the analytic continuations that take us to the next branch value of the imput.
although that may work in many cases , since we need to understand the structure of the surface alot , it still might not be - or lead to - a complete theory of invariants.
i conclude that , unless i am missing something important , that solving f(x) = f(g(x)) for g(x) is non-trivial.
however my knowledge is limited , maybe someone can "open my eyes" with some theorems or theories.
since functions like tetration and the alike are more exotic than any standard function i therefore find it justified to consider finding invariants , even if they dont have closed forms.
regards
tommy1729