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just another uniqueness musing
#1
Hi -

I'm not really happy to start a new thread here on a subject, which we had discussed several times and in some depth. But I am not able to follow the discussion of Henryk and Dmitri, that's too elaborate maths currently.
So my *basic* fiddlings/reinventing the wheels(?) may be inappropriate in the recent uniqueness-threads - but well, one moderator may move this post to an appropriate place...
Ok, so much as a foreword.

(I'm reproducing a msg to sci.math here, which I sent today)

I'm still trying to learn more about this uniqueness thing. This time I've considered to look at the problem via the function graphs.

For instance, for the function f(x) = exp(x) the function f°0.5(x) should have a curve somehow between the f°0(x) = x and the f°1(x) = exp(x) lines/curves.

I thought about creating the trajectory for f°0.5(x) depending on one initial setting, say f°0.5(1) = 1 .
The trajectories of iterates of f°(h+0.5)(x) give then more or less smooth curves (when interpolated). From the graphs it is obvious, that there is one initial setting which gives the smoothest result.

Images for various initial settings (-1,-0.7,-0.37,-0.2,0)
To have better graphs I used f(x) = exp(1/2 x)

See images below.

The best "smoothness", on one hand, seems simply that, where the derivatives are monotonuous.
But the "smoothness" can also be expressed by the sum (or integral?) of the area of the rectangles which occur, when the trajectories f°h(x) and f°h(f°0.5(x)) are drawn and the vertical lines at x=h and x=h+1 and the horizontal lines at the according y-values are taken as borders of rectangles (thin green and blue lines)

The worst initial settings give rectangles all of area zero, so I assume, the best guess (smoothest graph) gives the maximal sum of areas, so the "unique best definition of f°0.5(x)" is then that, which gives the highest area/integral (or something related to this)

Seems to be a useful/meaningful uniqueness-criterion?

From this would then also occur the idea to use the maximizing criterion on some integral-formula for that criterion (and possibly for the general case).

Gottfried


[Image: attachment.php?aid=439]

   

   

   


   


Attached Files Image(s)
   
Gottfried Helms, Kassel
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#2
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 where an analytic iterate may not exist: 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.
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#3
[attachment=441]

your post is very intresting Gottfried.

in fact it shows the following important result :

if f(x) ( being f(f(x)) = exp(x) ) exists and is unique on a finite (real) interval I ,

then f(x) exists and is unique for all real x.

---

i think this can be extended to the property of analytic but im not sure. more specific f(x) analytic on interval I -> real-analytic.

---

does your data match robbins ?

do you reach the same x for f(x) = 0 ?

can you say more about the areas of the rectangles in a real interval based upon the estimate ?

---

Gottfried's post leads to a way to extend f(x) to a given (real) domain.

i was already aware of this concept , but its nice to see a graph.

in particular i prefer these 'rectangles' to estimate the neighbourhood of x = 0 given f(x) in an interval (a,b) with a,b > 1.

this way we exploit the higher precision of my method for x > 0 and ' transfer ' that to the neighbourhood of x = 0.


nice work Gottfried.


high regards

tommy1729
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#4
Yes, I see - the area is either infinite or zero. One may think of taking the area in some finite boundaries - but then we may be able to construct a function, which produces a lower area in that region (compensated by a relatively bigger area outside that region) So I see, this is no conclusive solution, yet.

The only idea that remains is, to consider the length of the guessed curve of f°0.5(x) in an interval of f°0.5(x) .. f°2.5(x). (but also... I've never dealt with line-integrals). This length should be minimal if the amplitude of the overlaid mod(1)-function is minimal. But this is just crude guess - should be checked on paper first...

Hmmm.

[update]
Just added two more pictures.

The first gives mirrored graphs, which actually means the functional inverses f°-1(x) are also shown.


   


Then we can compute distances for each function and its inverse (the distances are measured along the antidiagonal direction). The "smoothest" version of f°0.5(x) has a "smooth" graph for the distances; if all derivatives are monotonuous only of that selected curve then... we have a special case...
The red line shows the distances of the "smoothest", the orange line some other, not optimal, estimate for f°0.5(x)
[/update]

Gottfried

   

[update2]
Here is the "distances"-view where the "mirror"-graphs is just rotated by 45 deg. Also a third interpolation was included. Perhaps the optimal view of the matter
[/update2]

   
Gottfried Helms, Kassel
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#5
tommy1729 Wrote:in fact it shows the following important result :

if f(x) ( being f(f(x)) = exp(x) ) exists and is unique on a finite (real) interval I ,

then f(x) exists and is unique for all real x.

that interval I has to be large enough though.

it has to 'connect' two rectangles.

however perhaps someone can get rid of that restriction and arrive at " any finite interval " ?

regards

tommy1729
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#6
Gottfried Wrote:The only idea that remains is, to consider the length of the guessed curve of f°0.5(x) in an interval of f°0.5(x) .. f°2.5(x). (but also... I've never dealt with line-integrals). This length should be minimal if the amplitude of the overlaid mod(1)-function is minimal. But this is just crude guess - should be checked on paper first...

Sounds not bad, but here again the length is infinite, and if you chose f(a) for the shortest length in one interval [a,F(a)] - i.e. a straight line - then in the interval [F(a),F(F(a))] the length is longer then for a better choose in the first interval.
So over which and how many intervals do you want to measure the length without being arbitrary?
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