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 $x \in E$).

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$\mathbb K_3$, 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 ...
Stéphane Jaouen's user avatar
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Symmetry of King's dream fractal (orbit of a 2-dimensional iterated function system is a parallelogram?)

For fixed real numbers $a,b,c,d$ define the map $f : \mathbb{R}^2 \to \mathbb{R}^2$ by $$f : (x, y) \longmapsto (\phi(ax) + b \cdot \phi(ay),\hspace{1em} \phi(cx) + d \cdot \...
Watson's user avatar
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How can i solve this optimization problem effectively?

Recently, i met an optimization problem $$ \arg \min_{\mathbf{x}}\Vert \mathbf {Kx} - \mathbf{y} \Vert^2_2+\frac{\eta \Vert \mathbf{Dx} -\mathbf d \Vert_2^2 }{\Vert \mathbf{Dx} \Vert_2^2} $$ from ...
Leung Joe's user avatar
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Fixed point convergence of vector scheme

I need to prove the fixed point convergence of a vector scheme which is as follows $$\mathbf{A}\mathbf{X} = b \max\{\mathbf{J}(\mathbf{X}+\mathbf{a}),0\}$$ where the max-operator operates elementwise; ...
coenhobelix's user avatar
3 votes
1 answer
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Loops in a binary iterated function system

Take a function $f(x)$ which operates on numbers written in binary. It is a three step operation: Split the number into digits in even- and odd-numbered places. (For example, the number $\underline{1}...
Elliott Price's user avatar
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For a convergent iterated function, is the distance between the nth iteration and the fixed point always decreasing as n increases?

Suppose we have a function $f$ that has a fixed point $\tau$. Let's also consider a real number $x_0$, that, when we infinitely apply the function $f$ to it, it converges towards $\tau$. Also, $x_0$ ...
Pierre Carlier's user avatar
2 votes
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62 views

How to find the limit to infinity of the iterative function used in the Mandelbrot set with a fixed C.

I'm trying to do my math IA on the effect of changing complex constant c in the mandelbrot set on the series's convergence, but I don't know how to algebraically solve the limits for iterative ...
Ram Tewari's user avatar
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Does this fractal mapping have a name?

I was playing with a mapping $f:[0,1] \to \text{Sierpinski's Triangle}$, and I'd like to know if there's a name for this kind of thing. [It's a lot like playing the chaos game in reverse order.] The ...
Nick C's user avatar
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Do the self-similarities of a fractal function govern its convergence properties?

Let the graph of a function take a fractal form, such as the following representation of the Collatz conjecture topologically conjugated to the interval $[\frac12,1)\to[\frac12,1)$ In this example, ...
it's a hire car baby's user avatar
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Generalization of the sliding ladder problem

Suppose we have a ladder of length $1$, and it's sliding down the $y-$axis. We know that the curve enveloped by it is an astroid: However, what if we iterate this process? We call this astroid curve $...
Hanging Pawns's user avatar
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Can a self-similar shape not be a polytope yet have a non-fractal boundary?

Looking at the Wikipedia page https://en.wikipedia.org/wiki/Rep-tile all examples of rep-tiles that are not polygons have fractal boundaries. In general, if the Hutchinson attractor of an IFS of ...
Numeral's user avatar
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Iterated function with $a_{n} = {F^{2n}(1)} $ and $b_{n} = {F^{2n+1}(1) }$. How to show that $a_n$ is increasing and $b_n$ is decreasing?

Consider function $ F(x) = \sqrt{1+\frac{1}{x}}$. Let $a_{n} = {F^{2n}(1)} $ and $b_{n} = {F^{2n+1}(1) }$ . How can I show that $ a_{n} $ is increasing and $b_{n} $ is decreasing and what is their ...
NobodyKnows's user avatar
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Derivative of an iterative vector function?

I am trying to solve the following problem I have some vector-matrix product of the form $\textbf{y} = \textbf{w} \cdot \textbf{X}(\textbf{x})$. Here, $\textbf{x}=[x_1, x_2, ... , x_N]$ and the matrix ...
APMATH24's user avatar
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For any composition sequence $s$ of maps $h(X)=X/2, \ f(X)=(3X + 1)/2$, there exists an integer $X$ such that its Collatz sequence contains $s$

Let $h(X) = X/2$, and $f(X) = (3X + 1)/2$. Then clearly every iteration $g^i(X), X \in \Bbb{Z}$ the Collatz mapping $$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \ X = 1\pmod 2 \end{...
Daniel Donnelly's user avatar
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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 ...
ThinkMachine_'s user avatar
1 vote
2 answers
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$ 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: $f_1(x) = g(x)$,...
mimre25's user avatar
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Fourier series of iterated sin / Diagonalization of an infinite matrix of Bessel functions

We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$ We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \...
al4085's user avatar
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10 votes
2 answers
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If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, is it true that $f(x) = x + 1/2$?

If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, where $f: \Bbb R \rightarrow \Bbb R$ is real-analytic, bijective, monotonically increasing, is it true that $f(x) = x + 1/2$? I have tried to represent $f(x)$ ...
Newone's user avatar
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Chaos game and fractal polygons

Let $V=\{v_1,v_2,\cdots,v_n\}$ the set of $n$ vertices of a regular polygon. For $n=3$ the chaos game implies that with an arbitrary point on the plane $x_0$, and by applying the recursive relation $$...
polfosol's user avatar
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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 ...
spraff's user avatar
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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? ...
spraff's user avatar
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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 ...
PouJa's user avatar
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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 ...
Leo Warvin Peng's user avatar
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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 ...
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4 votes
1 answer
316 views

Does the truth of one imply the other? A simple Collatz generalization in terms of primes.

Let $f_i:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1: $$f_0(n) = \begin{cases} n/2, & \text{if}\ 2\mid n\\ 3n+1, & \text{...
Math777's user avatar
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Iterated function system of a compact set.

This is a reference request and so I'll be skimpy on definitions, unless asked. A well known result in fractal geometry says that given an iterated function system (IFS) $\{S_1,...,S_m\}$, there ...
tangentbundle's user avatar
1 vote
1 answer
51 views

Can any compact subset of $\mathbb{R^2}$ be written as a suitable YFS attractor?

I'm wondering. Can any compact subset of $\mathbb{R^2}$ be written as a suitable IFS attractor? Can someone explain? Thank you for visiting my question.
bosanac's user avatar
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1 answer
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What is the simplest function yielding Fibonacci numbers upon iteration?

A while ago I asked a question on trigonometric functions that are iteratively periodic. That is, after a finite number of iterations (compositions of the function with itself) the function returns to ...
Max Muller's user avatar
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1 vote
1 answer
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Are the invariant sets of all iterated function systems necessarily fractal?

An iterated function system is defined as a finite set of contraction mappings, defined over a complete metric space $X$, and iteration is defined as sequential composition of these contraction ...
user918212's user avatar
1 vote
1 answer
47 views

Is there any way to figure out if a system of two recurrence relations will converge?

Context: I'm trying to write code that generates 2D Iterated Function System (IFS) fractals based on some affine transformations. I want to generate the fractal till convergence when possible. The ...
Zantorym's user avatar
7 votes
4 answers
231 views

The real function $f$ such that $\log \cdots \log (f)$ is strictly convex on its domain for any number of $\log$'s

Does there exist a function $f: (a,b) \to \mathbb R$ ($a,b$ are allowed to be infinity) such that $\log \cdots \log (f)$ is strictly convex on its whole domain of definition for an arbitrary number (...
No One's user avatar
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Are there "well-behaved" functions which iterate infinitely without repeating while also being bounded?

In this video the presenter shows that for strictly increasing functions $f(x)=f^{-1}(x) \implies f(x)=x$ because when you iterate $f$ it either increases forever or decreases forever. We cannot solve ...
spraff's user avatar
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6 votes
2 answers
221 views

Proving the limiting behavior of functions containing iterated trigonometric functions.

I remember years ago coming across some seemingly non-trivial (ie. non-fixed point related) limits describing to the behavior of infinitely iterated trigonometric functions, but I can't for the life ...
cmpeq's user avatar
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1 answer
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clarification of definition of ifs

Could any-one tell me what they meant by dots here for both the places used? especially with probabilities?
Myshkin's user avatar
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Given an Iterated System of Functions, how to find an estimate of the fractal dimension of the image?

Given an Iterated System of Functions, say $f_1,...,f_n$, with associated transition probabilities $\sigma_{ij}$ for each $f_{i}$ to each $f_{j}$, how do we go about finding an estimate to the fractal ...
Alex's user avatar
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Carleman matrix for sum of two functions

Carleman matrices are useful to express the composition of two functions. Recall that the Carleman matrix of a function $f$ is an infinite matrix whose $i,j$ element is $\frac{1}{j!}\frac{{\rm d}^j}{{\...
Monaca's user avatar
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2 answers
69 views

Iterates of $\frac{\sqrt{2}x}{\sqrt{x^2 +1}}$ converge to $\text{sign}(x)$.

In this post, a comment states that if $f(x):= \dfrac{\sqrt{2}x}{\sqrt{x^2 +1}}$ and $F_n:=\underbrace{f\circ \dots\circ f}_{n\text{ times}}$, then the pointwise limit $\lim\limits_{n \to \infty} F_n$ ...
ABIM's user avatar
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2 votes
2 answers
104 views

How is this interesting fractal generated?

Can someone help me with the steps through which this fractal was generated. My first observation was that we are making a vertical line at half the distance but notice that reconstructions on the ...
jd_sal's user avatar
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1 vote
1 answer
164 views

Fractal construction and IFS (iterated function system): defining a specific construction with IFS

The setting: We consider the plane $\mathbb R^2$ with its canonical euclidean structure. The canonic base is written $(e_1,e_2) = ((1,0)^\top,(0,1)^\top)$. Let's consider $I = [a,b]e_1$ (with $a<b$...
MiKiDe's user avatar
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For which $k$ we have: $|x_k-\alpha|\le10^{-16}|\alpha|$ where $\alpha$ is a solution of the equation $10x-\sin x=3$?

Let iterative method which is determined by the formula $x_{n+1}=\frac{\sin (x_n)+3}{10}$ for $x_0=0.33$. For which $k$ we have: $|x_k-\alpha|\le10^{-16}|\alpha|$ where $\alpha$ is a solution of the ...
dsk62's user avatar
  • 307
5 votes
1 answer
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Solving the iterated equation $f^{\circ n}(x)=f(x)^k$

On my spare time, I'm trying to solve equations of the form $$f^{\circ n}(x)=f(x)^k,\quad n,k\in \mathbb{Z}$$ where $f^{\circ n}(x)=f\circ f\circ\dots\circ f$, $n$ times. I know $f(x)=x^{\sqrt[n-1]{k}...
NoetherianCheese's user avatar
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1 answer
280 views

nonlinear system of equations solved by iteration method does not converge

I've modeled an electrical machine by means of a magnetic equivalent circuit. I've reached to a system of equations like this $ [f(F_N)]_{N\times1}^q=[P_N]_{N\times N}^q*[F_N]_{N\times1}^q-[\phi_N]_{N\...
Scrat68's user avatar
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0 votes
2 answers
105 views

Complex dynamics for non-holomorphic functions

Are there any resources for studying the dynamics of functions that are not holomorphic? Specifically, I am trying to study (for a paper) the behaviour of the function $$N(z) := z - \frac{2\overline{...
matty_k_walrus's user avatar
0 votes
1 answer
86 views

Partial derivative of an iterated function

I'm trying to find the form of the partial derivative for the following function: $F(F(x_{1},x_{2}),x_{2})$ I tried using the rules of composition: $\frac{\partial F}{\partial x_{2}}=\frac{\partial ...
Julian's user avatar
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6 votes
0 answers
100 views

Does Khinchin's constant have an analog for nested radicals?

Edit: as multiple users have pointed out, the premise of my question assumes some canonical representation of real numbers as infinite nested radicals. There does not seem to be any such ...
Descartes Before the Horse's user avatar
2 votes
1 answer
112 views

Proving restriction is surjective

Let $g:M\to M$ and $M_g = \bigcap_{n=1}^\infty g^n(M)$. Then the restriction $f = g|_{M_g}:M_g\to M_g$ is surjective if: a)$M$ is compact and $g$ is continuous or b)#$g^{-1}(y)<\infty$ for all $y\...
MathNewbie's user avatar
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2 votes
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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^{\...
Franklin Pezzuti Dyer's user avatar
3 votes
0 answers
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 $X$?
Zilkadde's user avatar
2 votes
1 answer
183 views

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 ...
zeraoulia rafik's user avatar
1 vote
1 answer
129 views

How can multivariate iterated functions be generalized?

Alright, so we all know how single variable functions work with iteration: If f(x) = x^2 Then f^2 (x) = (x^2)^2 This works really well because in with only one independent variable there is one ...
Divide Me By Zero's user avatar