Questions tagged [iterated-function-system]

This tag is used both for questions about iterated function systems in fractal geometry (finite families of contractions $f: X \to X$ on a complete metric space $(X,d)$ that are used to construct fractals) and questions about iterated function systems in probability theory (a random process associated to a finite family of maps $f_i:E \to E$ on a topological space $E$ and corresponding probabilities $p_i(x)$ for each $x \in E$).

70 questions
Filter by
Sorted by
Tagged with
29 views

$\mathbb K_3$, attractor of a pair of contractions

Reading this, I wanted to do the classic demonstration again by myself but there are points that bother me. Let $$C_0=[0,1], C_1=[0,\frac13]\cup[\frac23,1]...$$We have the classical definition of the ...
• 3,223
27 views

• 21.4k
48 views

Notational differences b/w Iterated functions & exponentiation.

https://en.wikipedia.org/wiki/Exponentiation#Iterated_functions https://en.wikipedia.org/wiki/Function_composition#Functional_powers https://calculus.subwiki.org/wiki/Higher_derivative Why does ...
1 vote
147 views

$f(f(f(x))) = x$ with $f(x) \ne x$ and $f(f(x)) \ne x$

I recently came to wonder if there are function that, when applied, iteratively, become a fix point, but only after a certain amount of iteration. Formally, let's define the following: $f_1(x) = g(x)$,...
• 13
184 views

• 9,245
392 views

Are there known points on the boundary of the Mandelbrot set which iterate forever?

Where $f(z)=z^2+c$ is the Mandelbrot iteration function, are there any known complex numbers $z$ such that iterating $z\to f(z)$ to infinity retains $z$ on the boundary (i.e. it does not explode to ...
• 1,597
1 vote
71 views

Are there "half step", or other subdivisions, of the Mandelbrot iteration function?

The Mandelbrot Set is generated by iterating $f(z)=z^2+c$. Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions? ...
• 1,597
1 vote
50 views

iterations of $f(x)=\dfrac{ax^4+bx^3+cx^2+dx+e}{a'x^3+b'x^2+c'x+d'}$

About iterated functions I read here that for example for $f(x)=Cx+D$ we can calculate quite simply that: $f^{[n]}(x)=C^nx+\dfrac{1-C^n}{1-C}D$ Likewise, I would like to obtain an explicit expression ...
• 175
77 views

Construct a function $f$ that $f(f(z))=-z(1-z)$ without using any information linked to $f(f(2))=2$

Consider the analytic function $g(z)=-z(1-z)$ which is a generator of Logistic Sequence with multiplier $-1$, having 2 global fixed point, $g(0)=0$ and $g(2)=2$. From classical dynamics, whenever a ...
62 views

Limit of an iterated function system

Consider the function $h(x) = 2x(1-x).$ My goal is to find $$\lim_{n \to \infty} h^n \left ( \frac{1}{4} \right ),$$ where $h^n (x_0)$ is the $n^{\text{th}}$ iterate of the function $h(x)$ at the ...
316 views

86 views

• 1,313
48 views

Number of zeroes of iterated function $f(x)=x^2-3/2$

This problem has been bothering me for a long time, and I come back to it every couple of months or so but can never seem to make any progress: Let $f(x)=x^2-3/2$. Consider the function f^{\...
147 views

IFS of compact set

In complete metric space $X$, there exists a unique non-empty compact set of $X$ called attractor for any iterated function system (IFS). But, is there an IFS for any nonempty compact subset of $X$?
• 71
Is $\lim\limits_{n\to\infty}(A_{n+1}-A_{n})$ finite?
The following question is related to my question here about behavior of this sequence $a_{n}=(1-\frac12)^{ \left( \frac12-\frac13 \right)^{...^{ \left( \frac{1}{n}-\frac{1}{n+1} \right)}}}$. Now I ...