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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23 views

Is there a function $f(k,n)$ which describes a $k$-fold iteration of $\sqrt{n}$? What would be the properties of such a function?

Is there a function $f(k,n)$ that takes the square root of $n$ $k$ times? This would be the graph of $f(x,2)$: This would be the graph of $f(x,3)$:
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41 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 ...
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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: $f_1(x) = g(x)$,...
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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\, \...
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376 views

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 $f(x)$ ...
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28 views

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 $$...
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56 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 ...
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28 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? ...
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41 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 ...
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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 ...
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20 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 ...
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295 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{...
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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 ...
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1answer
41 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.
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Is there a special term for "loop connector"?

Let: $$ y = f^{k}(n) $$ $$ n, k \in \Bbb{N} $$ $$ A, B, C, D $$ $$ A \cap B = A \cap C = B \cap C = \emptyset $$ $$ A \cap D = D$$ And: $$ \forall k, n : $$ $$ f^{4k-3}(4n-3) \in A $$ $$ f^{4k-2}(4n-2)...
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1answer
63 views

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 ...
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1answer
55 views

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 ...
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1answer
41 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 ...
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186 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 (...
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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 ...
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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 ...
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38 views

clarification of definition of ifs

Could any-one tell me what they meant by dots here for both the places used? especially with probabilities?
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26 views

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 ...
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55 views

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}{{\...
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63 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 $\lim\limits_{n \to \infty} F_n$ ...
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95 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 ...
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91 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 $a<b$...
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30 views

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 ...
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59 views

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}...
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1answer
95 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\...
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54 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{...
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1answer
63 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 ...
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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 ...
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1answer
39 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\...
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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^{\...
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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 $X$?
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175 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 ...
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1answer
55 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 ...
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1answer
127 views

Five-fold symmetric Sierpinski gasket pentagon using Iterated Functional System

Ran across an old article from Computer Graphics World by Michael F. Barnsley and Alan D. Sloan entitled Chaotic Compression on Iterated Functional Systems and modified their Table 1 $W$ matrix by ...
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Real world applications of partial metric

What are the real-world applications of the partial metric notion introduced by Steve G Matthews? I am writing a paper connecting partial metric with iterated function systems and trying to find the ...
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435 views

Matlab code for iterated function system [closed]

$f_1(x)=1.2x(1-x), f_2(x)=2.8x(1-x)$ on $\mathbb R$, Now from a starting point $x_0=0.25$,an the iterated function system $x_{n+1}=f_{\omega_n}(x_n), n=0,1,2,\dots$ where $\omega_n$ are i.i.d ...
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170 views

Do iterative formula only solve for one positive root?

For all the questions I have done using an iterative formula, the quadratic equation only ever solves to one root, I am not sure is this always the case, or is there a method to find multiple roots? ...
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1answer
101 views

Iterations of function over rationals

Let $F = \mathbb{Q}\setminus\{-1,0,1\}$ and $f$ the function defined over F by $f(x) = \dfrac{x^2-1}{x}$. Show that : $$ \bigcap_{n\ge 1} f^n(F)= \emptyset$$ I don't have many ideas for this problem....
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142 views

Asymptotic behavior of $a_{n+1}=\frac{a_n^2+1}{2}$

Define a sequence as follows: $$a_0=0$$ $$a_{n+1}=\frac{a_n^2+1}{2}$$ I would like to know the asymptotic behavior of $a_n$. I already know (by roughly approximating $a_n$ with a differential equation)...
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90 views

Iterated function in differential equations

I am currently working on a problem that involves differential equations that contains iterated functions. The problem can be described as that one seeks the solution for the equation $$ \dot{x} = f^...
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44 views

iteractively solve power equation

I have an equation like that: $$ x = (x-A)^{2/3}+B $$ And I want to find the value of x. The problem is that during the iteration, x can became negative and then appears complex number. But that ...
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1answer
59 views

How many mappings have an IFS?

In most of paper about iterated function systems and fractals which I read, an iterated function system is taken as finite set of contraction mappings like $\{f_1,f_2,...,f_n\}$. In some paper, the ...
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99 views

understanding $\theta,\Theta, d\theta, \mu(d\theta)$

Let $(\mathcal S, d)$ be a metric space. let $\{f_\theta: \theta \in \Theta\} $ be a family of Lipschitz functions on $\mathcal S$ and $\mu$ be a probability distribution on $\Theta$. Suppose that $\...
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2answers
253 views

Collatz $2x + 1$ conjecture?

Do we know of any Collatz theorem involving similar functions. For example what do we know about iterations of: $$ f(x) = \begin{cases} \dfrac{x + 1}{2} \text{, if } x \text{ is odd}. \\ 2x + 1, \...
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35 views

selecting a number according to probability

Could anyone just tell me what does he, mean by at the point $2$, selecting a number according to the probabilities, $p(x_0)$? does he meant to chhose the largest one out of $p_i(x_0), i=1,2,\dots,N$?,...