08/11/2009, 03:42 PM

(08/06/2009, 08:56 PM)Gottfried Wrote: (1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.

(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.

No, I dont follow your line of thoughts. Somehow you assume that one of both is the perfect iteration while the other is the bad one with sinoidal deviation, dont you?

Imho both have the same right to live (though only the lower fixed point is usable for a tetrational). Each of them is holomorphic at its fixed point while non-holomorphic at the other fixed point.

I guess with matrix power iteration along the points p in the interval [2,4] we can smoothly deform the tet[p=2] into tet[p=4], where tet[p] is always a superfunction/iteration that though will neither be holomorphic at any of both fixed points.

Remember that matrix power iteration coincides with regular iteration when applied to fixed points.

The oscillating behaviour is not contained in the "bad" tetrational but born through taking the difference. *Both* contribute to the oscillation.

The behaviour must be oscillating as we know that f^{-1}(g(z)) - z is 1-periodic.