# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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\... 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 $$N(z) := z - \frac{2\overline{... 1answer 40 views ### Partial derivative of an iterated function I'm trying to find the form of the partial derivative for the following function: F(F(x_{1},x_{2}),x_{2}) I tried using the rules of composition: \frac{\partial F}{\partial x_{2}}=\frac{\partial ... 0answers 70 views ### Does Khinchin's constant have an analog for nested radicals? Edit: as multiple users have pointed out, the premise of my question assumes some canonical representation of real numbers as infinite nested radicals. There does not seem to be any such ... 1answer 33 views ### Proving restriction is surjective Let g:M\to M and M_g = \bigcap_{n=1}^\infty g^n(M). Then the restriction f = g|_{M_g}:M_g\to M_g is surjective if: a)M is compact and g is continuous or b)#g^{-1}(y)<\infty for all y\... 0answers 36 views ### Number of zeroes of iterated function f(x)=x^2-3/2 This problem has been bothering me for a long time, and I come back to it every couple of months or so but can never seem to make any progress: Let f(x)=x^2-3/2. Consider the function$$f^{\... 0answers 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$X$? 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 ... 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 ... 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 ... 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 ... 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 ... 0answers 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? ... 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.... 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)... 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 $$\dot{x} = f^... 2answers 38 views ### iteractively solve power equation I have an equation like that:$$ x = (x-A)^{2/3}+B $$And I want to find the value of x. The problem is that during the iteration, x can became negative and then appears complex number. But that ... 1answer 57 views ### How many mappings have an IFS? In most of paper about iterated function systems and fractals which I read, an iterated function system is taken as finite set of contraction mappings like \{f_1,f_2,...,f_n\}. In some paper, the ... 2answers 85 views ### understanding \theta,\Theta, d\theta, \mu(d\theta) Let (\mathcal S, d) be a metric space. let \{f_\theta: \theta \in \Theta\} be a family of Lipschitz functions on \mathcal S and \mu be a probability distribution on \Theta. Suppose that \... 2answers 142 views ### Collatz 2x + 1 conjecture? Do we know of any Collatz theorem involving similar functions. For example what do we know about iterations of:$$ f(x) = \begin{cases} \dfrac{x + 1}{2} \text{, if } x \text{ is odd}. \\ 2x + 1, \... 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$p_i(x_0), i=1,2,\dots,N$?,... 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 ... 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 ... 0answers 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 ... 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 ... 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 $$f\circ f\circ f(x)=\frac x3.$$ 0answers 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 } \...
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)}: ...