Convergence of iterations and selfsimilarity

The greek physicist and mathematician Archimedes said: “give me some place I can stand and I will move the earth”. It has been early recognized how important constants are in a ever changing world. The role of invariants under transformations plays not only in mathematics but only as a symmetry consideration a very important role in modern physics. I was early interested in mathematical constants which are the result of an iteration and independent on the initial value.

During my time at highshool I looked for example at the convergence of the iteration f(x), f(-f(x)), f(-(-f((x)))..

For f(x) = e^x this iteration converges to g where e^g =1/g and g = 0.5671432904...This constant is well known.

Similar we can define for a given value a by the relation ba^b = 1 the iteration series a, a**(-x), a**(a**(-x)), which converges to b.

Here some examples:

It is interesting that for a = 4 the iteration converges to 1/2. Above some value the iteration oscillates between two numbers.

We can see this iteration as a solution for the problem to find x such that xa^x = 1

The convergence is shown in the figure above. A bifurcation is observed, at a = e^e = 15.1543...

Above the bifurcation point there are 2 solutions b₁ and b₂ which solve now the problem

xa^y = ya^x = 1. Also it follows that x^x = y^y. This of course does not imply x=y!

The solution is x=b₁, y=b_2. How fast converges the iteration as a approaches the bifurcation limit?

A magnification is shown with two different number ot total iterations (5000 and 200000). The iteration converges at the bifurcation point to the value 1/e. It shows that as the value a converges e^e the convergence to 1/e reduces very much. The iteration oscillates above this value between two values which approach slowly 0 and 1.

It was strange that at the same time around 1975 or so I found selfsimilar structures very fascinating. How iterations and selfsimilar structures are connected I could not imagine at that time!

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