4 hypothesis about iterated functions

[tommy1729]

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......)

Also wrong, I found a counter example

Wrong ?
Here you left out some conditions compared to the original I think.
Anyway remember there are no cyclic points by definition since all converge to the fixpoint.

Notice that if the function has at its fixpoint the derivative -1 or so then more likely there are cyclic points.

Im curious for your counterexample.


regards

tommy1729