slog_b(sexp_b(z)) How does it look like ?

[tommy1729]

\( \text{slog}(\text{sexp}(z))=z+\theta_n(z)+n\text{period} \), where theta is a 1-cyclic function, which goes to zero as \( \Im(z) \) goes to infinity, and theta is uniquely determined by n.
- Sheldon

Lets say z is close to a real > 0 , then for most z : slog(sexp(z)) = z.

This suggests that \( theta_n(z))-n\text{period} \) is almost Always flat and not differentiable.

This theta seems different from the periodic base change theta , and from the periodic sexp theta (the one that changes one C^oo solution of tetration into another one ).

In other words - and no offense - but that is a very very general answer ; a family of functions without much details or properties and not like any seen before ( I think ).

Also Im still unsure about alot of things. I blame myself mainly but my confusions are not resolved yet.

Lets say n :

theta depends only on n.

But what does n depend on ? On z I guess , but how ??
This is probably equivalent to asking when to switch branches ?
Somehow I guess that relates to the case when interpolation of iterations exp^[a/b](x_0) for a complex x_0 and integer a,b leads to intersecting points. ( we interpolate fractions by density to get app. continu line of reals ).
And switching branches seems related.

Second :

Why L as ' invariant ' ? Why not 2pi i ?

afterall iterations of exp(z) + 2pi i seems to suggest branches that differ by 2 pi i !? Like exp(u+2pi i) = sexp(slog(u+2pi i)+1) = v.
Seeming giving a branch difference of 2pi i.
Said otherwise log(sexp(a)) = sexp(a-1) or sexp(a-1) + 2pi i.
Hence a branch difference of 2pi i ?


Then again a branch difference of 2pi i is the log , not the sexp.
All quite confusing id say.

Or does " period " mean 2pi i here ? ( for base e)

Is it just me, or is tetration tricky ?

(Sometimes I believe I like tricky stuff, hence my intrest in tetration and number theory.)

And what the ideas z1 - z2 = L or z1 - z2 = 2pi i ?

But some good news too. I think your conjecture from 4/24 is correct and provable. Or partially provable.

( btw I seem to remember mick mentioning such a conjecture on MSE , i believe from you. ( I know you have MSE account ) But I forgot some of his arguments about it , did you discuss it with him ? or did we write anything uselfull about it ? I probably should have written things down :/ )

regards

tommy1729