03/12/2008, 09:20 PM
Just a short note, why this thread is called Bummer:
From a "proper" analytic iteration \( f^{\circ t} \) one would expect that it is analytic everywhere in the domain of the definition.
But we know that the regular iteration at a fixed point is the only analytic iteration that does not introduce a singularity at that fixed point (no oscillating first or higher order derivative when approachin the fixed point).
Conclusio:
There is no analytic iteration \( \exp_b^{\circ t} \), \( 0<b<\eta \), that is analytic at both fixed points for most \( t \), though for integer \( t \) it is.
As every tetration \( {^x}b \) (that satisfies \( {^{x+1}}b=b^{^xb} \), \( {^1b}=b \) and analyticity) can be written as \( {^x}b=\exp_b^{\circ x}(1) \) for some analytic iteration \( \exp_b^{\circ x} \), this statement can be called a bummer as there is no "proper" analytic iteration.
From a "proper" analytic iteration \( f^{\circ t} \) one would expect that it is analytic everywhere in the domain of the definition.
But we know that the regular iteration at a fixed point is the only analytic iteration that does not introduce a singularity at that fixed point (no oscillating first or higher order derivative when approachin the fixed point).
Conclusio:
There is no analytic iteration \( \exp_b^{\circ t} \), \( 0<b<\eta \), that is analytic at both fixed points for most \( t \), though for integer \( t \) it is.
As every tetration \( {^x}b \) (that satisfies \( {^{x+1}}b=b^{^xb} \), \( {^1b}=b \) and analyticity) can be written as \( {^x}b=\exp_b^{\circ x}(1) \) for some analytic iteration \( \exp_b^{\circ x} \), this statement can be called a bummer as there is no "proper" analytic iteration.