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location Austin, TX
age 52
visits member for 2 years, 1 month
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I'm interested in tetration, and complex dynamics and iteration theory. I'm seeking to learn more about math, so I can prove some of the things I've posted on some other forums.


Nov
24
comment Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?
Many of the points on the boundary aren't rational and aren't in an attracting basin. For example the point at the tip, $z \approx -0.2281554936539618 + 1.115142508039937i$, which is one of the solutions of $x^3 + 2x^2 + 2x + 2 = 0$, which repeats after a preperiod of 2. There are also chaotic irrational points on the boundary which are not solutions of algebraic equations. But, if you limit yourself to rational points, than I think there are only a few examples complex points that aren't in an atracting basin, like $z=i$.
Nov
19
revised Is there an inverse operation of tetration for values between 0 and .3?
added 42 characters in body
Nov
19
answered Is there an inverse operation of tetration for values between 0 and .3?
Nov
18
comment If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
The Op's question involves $g(z+\theta(z))$ where $\theta(z)$ is real periodic. So the real strip would naturally be the real period of $\theta(z)$. And in that strip, if $\theta(z)$ is entire, then $\theta(z)$ would take on all values. btw, just as there are formal equations for the Schroeder functions, that lead to periodic $f^z$ functions, so one can also do the reverse! From the $g(z)$ complex periodic function, I can generate an $f(z)$ with any real iteration. I like ... I choose the period of $\theta(z)$. btw. (not sure how to prove it, but I've done the Taylor series calculations).
Nov
18
comment If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
You may want to add the complex-dynamics tag, since the known solution involves complex dynamics.
Nov
18
comment If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
Thanks Mick. Even though it does seem obviously true, I'm not sure how to prove there is no solution where all three functions are entire. Though I've been thinking about related problems for awhile. The thought process is that periodic functions must go to a constant in some direction in the complex plane, and yet $\theta(z)$ is taking on all values in every 1-cyclic strip... perhaps one can show that $g(z+\theta(z))$ must also take on all values in every strip?
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
grammar; in the last picture description
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
modest improvements
Nov
18
comment If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
In this solution, $\theta(z)$ is analytic, with singularities at $|\Im(z)|\approx 14.0788i$, $g(z)$ is entire, and $h(z)$ is analytic over the right half of the complex plane, and near the real axis, except where noted. The Op's question seems to suggest that all three functions, $\theta(z),\;g(z),\;h(z)$ be entire. I'm pretty sure that's impossible since the amplitude of an entire $\theta(z)$ grows arbitrarily large as $|\Im(z)|$ increases.
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added fourier equations
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added 47 characters in body
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added 35 characters in body
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added 4 characters in body
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
solution uses complex dynamics
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added graph of g(z)
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added $\theta(z)$ graph
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
pictures, more detail, changed s(z) to f(z), and used theta(z) for the real periodic function
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
pictures, more detail, changed s(z) to f(z), and used theta(z) for the real periodic function
Nov
18
comment If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
Turns out this example, iterating $z \mapsto z+z^2-0.01$, has a very small $\theta(z)$ sinusoidal amplitude, on the order of $5\cdot 10^{-40}$. In fact, $g(z) \approx h(z)$. By definition, $\theta(z)=h^{-1}(g(z))-z$.
Nov
18
revised If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?
added 91 characters in body