11/03/2020, 05:00 PM
(This post was last modified: 11/04/2020, 02:36 PM by Gottfried.

*Edit Reason: tiny inconsistency between "key" in image and used order of periodic points*)
From my considerations on n-periodic points I came to the question, what would the Kneser-method produce for the orbit over such periodic points, with fractional steps of some small order.

I think, the Kneser-method for base \( \small b=\sqrt 2 \) gives pretty smooth orbit/curves for iteration with fractional heights, so I used that base.

I can find n-periodic points of any order, at least (arbitrarily near) approximates (using my method discussed in the previous thread).

Assume for instance the set of 6-periodic points (call it for instance "6-cycle") having the following approximate values:

such that \( \small{p_{k+1}=b^{p_k}} \) then, in a plot generated with Excel, we see the six points, and connected by some cubic-spline-interpolation, which is of course only by accident near to the closed curve generated by the fractional iterations using the Kneser- (or any other interpolation-) method.

See this picture first. I "normalized" the y-(imaginary-) axis by \( \small \Im{(z)} / \tau \) where \( \small \tau= \pi / \ln b \) *(for better comparability when this picture are made with another base)* :

Note: the "key" for the cycles, mentioned in the legend of the plot, gives the periodic points in the inverse order (p1,p6,p5,p4,p3,p2) because the software uses logarithmizing instead of exponentiation)

The Kneser-iterations produce as well a smooth curve do far, but much different from the cubic spline; the light-blue segment even makes a big journey out-of-bounds of the plot here.

The next segments of the orbit begin to evolve to "overcycle": the pink segment continues the light blue segment and even generates some spiral around the origin. The gold and the brown segments even don't continue the attaching segments smoothely.

Here is a bigger view: it may give a bit better imagination, but is still far from exposing the full chaos:

Problem: the source in Fatou.gp is difficult to edit. I'd like to be able to compute in higher precision (while not sending Pari/GP in the nirvana of neverending loops). Also it seems as if the orbits of -say- tet(z,1..2) is different or sometimes different from b^tet(z,0..1).

Questions:

- can someone check this curves in general by recomputing data with the Kneser-method?

- can someone check whether the Kneser-calculation can be made consistent at the switching points, where the segments don't follow smoothely?

- the chaotic ranges of the orbits are surely caused by the branch-cuts of the logarithm or by some other well known property alike. On the other hand, we've *infinitely many* n-periodic points, somehow randomly shuttered in the shown segment. For the Schröder-function this should be catastrophic: these are points, which cannot be iterated towards the fixpoints. What does this mean for the "moving towards the fixpoint"-part in the mechanism?

I think, the Kneser-method for base \( \small b=\sqrt 2 \) gives pretty smooth orbit/curves for iteration with fractional heights, so I used that base.

I can find n-periodic points of any order, at least (arbitrarily near) approximates (using my method discussed in the previous thread).

Assume for instance the set of 6-periodic points (call it for instance "6-cycle") having the following approximate values:

Code:

` p1=6.574747674571700333009302782482639405458 + 0.9684029672628275112988471898707035489652*I`

p2=9.218591772304179221754557226368239753533 + 3.215625927832110139905357739034367499059*I

p3=10.75598408660258595556412864882513957578 + 21.91051323491973949320807501979124161907*I

p4=10.70582570926180333437881525870964054377 + 40.18332039944084859802348678029895039952*I

p5=8.546967145179356295028740245483622451155 + 39.96464873483437244009418575109150854376*I

p6=5.464835241819467227108868806348013192730 + 18.55139938049736513812617532327500721009*I

such that \( \small{p_{k+1}=b^{p_k}} \) then, in a plot generated with Excel, we see the six points, and connected by some cubic-spline-interpolation, which is of course only by accident near to the closed curve generated by the fractional iterations using the Kneser- (or any other interpolation-) method.

See this picture first. I "normalized" the y-(imaginary-) axis by \( \small \Im{(z)} / \tau \) where \( \small \tau= \pi / \ln b \) *(for better comparability when this picture are made with another base)* :

Note: the "key" for the cycles, mentioned in the legend of the plot, gives the periodic points in the inverse order (p1,p6,p5,p4,p3,p2) because the software uses logarithmizing instead of exponentiation)

The Kneser-iterations produce as well a smooth curve do far, but much different from the cubic spline; the light-blue segment even makes a big journey out-of-bounds of the plot here.

The next segments of the orbit begin to evolve to "overcycle": the pink segment continues the light blue segment and even generates some spiral around the origin. The gold and the brown segments even don't continue the attaching segments smoothely.

Here is a bigger view: it may give a bit better imagination, but is still far from exposing the full chaos:

Problem: the source in Fatou.gp is difficult to edit. I'd like to be able to compute in higher precision (while not sending Pari/GP in the nirvana of neverending loops). Also it seems as if the orbits of -say- tet(z,1..2) is different or sometimes different from b^tet(z,0..1).

Questions:

- can someone check this curves in general by recomputing data with the Kneser-method?

- can someone check whether the Kneser-calculation can be made consistent at the switching points, where the segments don't follow smoothely?

- the chaotic ranges of the orbits are surely caused by the branch-cuts of the logarithm or by some other well known property alike. On the other hand, we've *infinitely many* n-periodic points, somehow randomly shuttered in the shown segment. For the Schröder-function this should be catastrophic: these are points, which cannot be iterated towards the fixpoints. What does this mean for the "moving towards the fixpoint"-part in the mechanism?

Gottfried Helms, Kassel