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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24 views

correct notation for iterative equation with variable bounds

I'm currently writing my PhD-thesis in engineering and stumbled upon a set of equations, which is quite straightforward to write in programming languages either using a ...
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1answer
35 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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4answers
158 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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61 views

The coefficients on terms of $z_{n+1} = z_n^2 + z_0$ a generalization

I was thinking about the iterative equation we use to generate mandelbrots, but expressed in polar coordinates. I wanted to see if it was possible to write the nth iteration as some sum. $z_n=z_{n-1}^...
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21 views

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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2answers
90 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 ...
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36 views

a form of central limit theorem of a markov chain

Could anyone explain to me what it means by the central limit theorem for the following Markov chain? $X_{n+1}=f_{\omega_n}(X_{n}),\quad n=0,1,2,\dots$, where $\omega_0, \omega_1\dots$ are i.i.d ...
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1answer
29 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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23 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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26 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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59 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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2answers
89 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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1answer
60 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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27 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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1answer
55 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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11 views

Find exponent of the convergence of the iterative method

Find exponent of the convergence of the iterative method in which iterative function $\phi$ such that $\phi(\alpha)=\alpha$ has in the point $\alpha$ derivatives of the order $1,...,n$ equal $0$ and $...
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1answer
52 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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2answers
42 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
40 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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70 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 ...
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1answer
33 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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36 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^{\...
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70 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 $X$?
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1answer
174 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
22 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
87 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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0answers
28 views

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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1answer
76 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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59 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
100 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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1answer
131 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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1answer
62 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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2answers
38 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
57 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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2answers
85 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
142 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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0answers
30 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 $p_i(x_0), i=1,2,\dots,N$?,...
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1answer
82 views

Is it possible to simplify this nested GCD?

Is it possible to simplify this nested GCD? $$\gcd\bigg(\gcd(m^2,\sigma(m^2)),\frac{m^2}{\gcd(m^2,\sigma(m^2))}\bigg)$$ Here, $\gcd(m^2,\sigma(m^2))>1$ and $\sigma(m^2)$ is the sum of divisors ...
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1answer
56 views

Limit involving iterated function $f_a(x)=x^2+a^2$

I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the ...
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134 views

Self-similarity dimension for IFS or attractor of IFS

We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set ...
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1answer
38 views

convert iterated function to continuous function

Let's say we have a function: $$v(t_n) = a\cdot(t-t_{n-1}) + v(t_{n-1}) \cdot d^{t-t_{n-1}}$$ Where $v$ = veolcity $t$ = time $a$ = acceleration $d$ = friction or damping What is basically does is ...
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1answer
52 views

Number of solutions for composite function

If $f(x)=4x(1-x)$ The function is defined over $\Bbb{R}$. Find the number of real solutions of $$f\circ f\circ f(x)=\frac x3.$$
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104 views

Behavior of 2-norm of k-th power of matrix

I got this problem from Greenbaum's book of iterative methods. In page 14 he mentions that the 2-norm of matrix $A^k$ is asymptotically behaves like $v \left( \begin{array} { c } { k } \\ { j - 1 } \...
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1answer
52 views

understanding iterated function system by Markov chain [closed]

The Iterated Function System at node $i$ is a discrete time Markov chain on the state space ${\cal S}_i=\mathbb{R}^d$. The chain is specified by an integer $m$ and a collection of maps $f_j^{(i)}: ...
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1answer
299 views

logistic map with lamba greater than 4

I was doing some recreational math about the logistic map. (If you're not familiar with what the logistic map is, here are some links you can check out) https://www.youtube.com/watch?v=ETrYE4MdoLQ ...