09/29/2008, 01:03 AM

As I now see, even if the uniqueness criterion is valid, the precondition:

for each

can not be true for any for (and probably for each )!

Michał Misiurewicz [1] showed 1981 that the Julia set of is the whole (that is not directly visible from the fractal). The Julia set is the boundary of the set of all such that . That means that the set and its complement is dense in . In every neighborhood of any complex number there is a complex number such that and also a such that !

And that implies that , which contains an open non-empty set, always contains points such that .

[1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X

for each

can not be true for any for (and probably for each )!

Michał Misiurewicz [1] showed 1981 that the Julia set of is the whole (that is not directly visible from the fractal). The Julia set is the boundary of the set of all such that . That means that the set and its complement is dense in . In every neighborhood of any complex number there is a complex number such that and also a such that !

And that implies that , which contains an open non-empty set, always contains points such that .

[1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X