02/11/2009, 03:14 PM
Thank you Gottfried for the good illustrations.
The first thing I like to mention is that they help to understand *regular* iteration.
If we have a fixed point at 0 and dont chose the right half iterate then the derivatives will oscillate towards 0. So we would demand that all derivations of the half iterate exists in 0, or moreover that the half iterate is analytic in 0.
And indeed this is a uniqueness criterion (though we have to take special care of the case \( f'(0)=1 \) where an analytic iterate may not exist: \( e^x-1 \) is our most famous example)! It determines the regular iteration.
Next thing: Only monotony of the (first) derivative does not suffice. Little changes keep the derivative monotonous. I am not sure about the other derivatives. I will look whether I can more precisely support this thesis.
Third, minimizing the rectangles. The first thing is that the area of all rectangles is infinite. The second thing is that it depends on which part of your initial function you start the rectangles. So I see not much hope about this approach. Though the idea came also to my mind when I drawed those pictures that you now provide.
The first thing I like to mention is that they help to understand *regular* iteration.
If we have a fixed point at 0 and dont chose the right half iterate then the derivatives will oscillate towards 0. So we would demand that all derivations of the half iterate exists in 0, or moreover that the half iterate is analytic in 0.
And indeed this is a uniqueness criterion (though we have to take special care of the case \( f'(0)=1 \) where an analytic iterate may not exist: \( e^x-1 \) is our most famous example)! It determines the regular iteration.
Next thing: Only monotony of the (first) derivative does not suffice. Little changes keep the derivative monotonous. I am not sure about the other derivatives. I will look whether I can more precisely support this thesis.
Third, minimizing the rectangles. The first thing is that the area of all rectangles is infinite. The second thing is that it depends on which part of your initial function you start the rectangles. So I see not much hope about this approach. Though the idea came also to my mind when I drawed those pictures that you now provide.