Hi Sheldon -

likely a typo in the formula:

I'm still thinking about your post and probably need some more time. (I experimented with some plots so far). However, since you introduce the continued fractions here - that made me thinking, whether periods of quadratic roots are different (and the curves look different) from other irrational periods since the continued fraction is periodic; especially whether the golden ratio for the period-parameter gives a distinctive curve. Then we have the difference between algebraic irrationals and transcendents; for the latter the liouville-numbers might be a special class in regard of the approximation to a dense line. That numbers have the property to have irrationality measure of infinity and arbitrarily "close to" rational numbers - much "closer" than any algebraic irrational number. Thus it might be possible, that they are some bridge between the rational periods with their distinct accumulation points and the seemingly smooth curve of the golden-ratio period on the other extreme.

What I'm looking at at the moment is your problem of the analycity of the superfunction -I just have to get familiar with the understanding of that general problem first. The only thing what I know beforehand is, that I can only create a formal powerseries for the superfunction if u is an irrational root of the complex unity - the standard approach to the diagonalization of the associated Bell-matrix does not work for repeated eigenvalues (which occur, if u is a rational such root). But I think I need one or two days to chew all this to more clarity. (Here in Hassia/Germany I have a nice four-day "short holiday" with a free (christian) day today and my half-term allows me to stay away from university tomorrow... so I'll invite the sun for that very welcome break :-) )

Also, there was a nice graph accidentally, when I forgot to remove the connecting lines of the excel-plot of some trajectories - just an artistic impression:

(The header is not correct/was not finished when the data were added to the graph)

Gottfried

likely a typo in the formula:

(06/22/2011, 03:23 PM)sheldonison Wrote: case, Then the period "works". For all integers, we can define the superfunction for x=(m mod period), starting from initial value L+z:For the parameter of the superfunction I think you want to write m (or for exp_b the iterator/height x ?)

. And for

I'm still thinking about your post and probably need some more time. (I experimented with some plots so far). However, since you introduce the continued fractions here - that made me thinking, whether periods of quadratic roots are different (and the curves look different) from other irrational periods since the continued fraction is periodic; especially whether the golden ratio for the period-parameter gives a distinctive curve. Then we have the difference between algebraic irrationals and transcendents; for the latter the liouville-numbers might be a special class in regard of the approximation to a dense line. That numbers have the property to have irrationality measure of infinity and arbitrarily "close to" rational numbers - much "closer" than any algebraic irrational number. Thus it might be possible, that they are some bridge between the rational periods with their distinct accumulation points and the seemingly smooth curve of the golden-ratio period on the other extreme.

What I'm looking at at the moment is your problem of the analycity of the superfunction -I just have to get familiar with the understanding of that general problem first. The only thing what I know beforehand is, that I can only create a formal powerseries for the superfunction if u is an irrational root of the complex unity - the standard approach to the diagonalization of the associated Bell-matrix does not work for repeated eigenvalues (which occur, if u is a rational such root). But I think I need one or two days to chew all this to more clarity. (Here in Hassia/Germany I have a nice four-day "short holiday" with a free (christian) day today and my half-term allows me to stay away from university tomorrow... so I'll invite the sun for that very welcome break :-) )

Also, there was a nice graph accidentally, when I forgot to remove the connecting lines of the excel-plot of some trajectories - just an artistic impression:

(The header is not correct/was not finished when the data were added to the graph)

Gottfried

Gottfried Helms, Kassel