11/12/2009, 03:23 PM

actually I think that the regular tetration at the lower fixed point is not continuable to bases .

Actually I think the Shell-Thron region is the domain of holomorphy, every point on the boundary is singular. This became clear to me some days ago:

1. The regular iteration depends on the derivative of at the fixed point.

2. If is the fixed point then the derivation is

3. The fixed point is given by , or .

4. The problematic value of the derivation at the fixed point is . In this case is not analytic at . I guess this implies that is not analytic at .

5. is exactly at those in the image of the unit circle under of the inverse of , i.e. under which is just the boundary of the Shell-Thron region.

Actually I think the Shell-Thron region is the domain of holomorphy, every point on the boundary is singular. This became clear to me some days ago:

1. The regular iteration depends on the derivative of at the fixed point.

2. If is the fixed point then the derivation is

3. The fixed point is given by , or .

4. The problematic value of the derivation at the fixed point is . In this case is not analytic at . I guess this implies that is not analytic at .

5. is exactly at those in the image of the unit circle under of the inverse of , i.e. under which is just the boundary of the Shell-Thron region.