04/20/2023, 04:09 PM
(This post was last modified: 04/21/2023, 08:39 AM by Shanghai46.)
I'm working on my iteration formula, and I made a few hypothesis based on intuition and tests. These seem really useful in order to find the final interval restrictions of it.
If you have an answer "it's true", could you link to a demonstration, or to a property that demonstrates that, and if it's false, could you provide a counter example, as well as, if possible, put some restrictions to make it work, then giving the demonstration? Thank you.
ONE : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\). Let's consider the real interval \(I\) which is the biggest monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\), and that all positive iterations of \(f(x_0)\) are inside the interval \(I\).
is \(\forall n\in\mathbb{N}, |f^{\circ n+1}(x_0)-\tau| < |f^{\circ n}(x_0)-\tau|\) true? (kinda answered by Tommy but without evidence..... So......)
TWO : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\) so that \(f'(\tau)<0\). Let's consider the real interval \(I\) which is a monotomic interval (not the biggest) so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\).
If \(x_0\) and \(f(x_0)\in I\), are ALL positive iterations of \(f(x_0)\) inside \(I\)?
THREE : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\) so that \(f'(\tau)>0\). Let's consider the real interval \(I\) which is THE BIGGEST monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\).
are ALL positive iterations of \(f(x_0)\) inside \(I\)?
FOUR : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\). Let's consider the real interval \(I\) which is the biggest monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\), and that all positive iterations are inside the interval \(I\).
is \(\forall r\in\mathbb{R}, arg(f^{\circ r}(x_0)-\tau)=arg(f'(\tau)^r) \) true? (\(arg\) is the argument in the complex plane)
Here they are. Thank you.
If you have an answer "it's true", could you link to a demonstration, or to a property that demonstrates that, and if it's false, could you provide a counter example, as well as, if possible, put some restrictions to make it work, then giving the demonstration? Thank you.
ONE : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\). Let's consider the real interval \(I\) which is the biggest monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\), and that all positive iterations of \(f(x_0)\) are inside the interval \(I\).
is \(\forall n\in\mathbb{N}, |f^{\circ n+1}(x_0)-\tau| < |f^{\circ n}(x_0)-\tau|\) true? (kinda answered by Tommy but without evidence..... So......)
TWO : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\) so that \(f'(\tau)<0\). Let's consider the real interval \(I\) which is a monotomic interval (not the biggest) so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\).
If \(x_0\) and \(f(x_0)\in I\), are ALL positive iterations of \(f(x_0)\) inside \(I\)?
THREE : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\) so that \(f'(\tau)>0\). Let's consider the real interval \(I\) which is THE BIGGEST monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\).
are ALL positive iterations of \(f(x_0)\) inside \(I\)?
FOUR : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\). Let's consider the real interval \(I\) which is the biggest monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\), and that all positive iterations are inside the interval \(I\).
is \(\forall r\in\mathbb{R}, arg(f^{\circ r}(x_0)-\tau)=arg(f'(\tau)^r) \) true? (\(arg\) is the argument in the complex plane)
Here they are. Thank you.
