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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Iterating the map $f(n)=1+D(\sigma(n)-1)$
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
The Leibniz rule implies ...
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IFS and Hausdorff dimension
Let $(X,d)$ be a complete metric space, $\{ T_1, \dots, T_m \}$ an iterated function system of similarities defined on the set of compact nonempty subsets of $X$ and $F$ the corresponding fixed point (...
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Open set condition (IFS)
Let $(X,d)$ be a complete metric space, $\{ T_1, \dots, T_m \}$ an iterated function system of similarities defined on the set of compact nonempty subsets of $\mathbb{R}^n$ and let $F$ be the ...
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Hausdorff-Dimension and Ahflors-Regularity
I have come across the following result in fractal geometry
Given a non-empty bounded subset $A\subseteq \mathbb{R^n}$ and a Borel-regular measure $\mu$ on $\mathbb{R^n}$ with $0 < \mu(A) \leq\mu(\...
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When does a Moran equation $\sum r_i^n=1$ correspond to an iterated function system?
Background
Let $\psi_1, \psi_2, ... \psi_n :\mathbb{R}^n\to\mathbb{R}^n$ be similarity mappings defining an iterated function system $\psi$. To each $\psi_i$ we assign the similarity coefficient $r_i \...
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Constructing a non-similar right triangle from the altitude and half-hypotenuse of a right triangle
Inside right triangle with side lengths $(a,b,c)$ and angles $(A,B,90^\circ)$,
we can construct a (usually) non-similar right-triangle with side lengths $\left(\frac{a^2-b^2}{2c}, \frac{2ab}{2c}, \...
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Question regarding the "Hausdorff dimension formula" in Falconer's book "Fractal Geometry"
I am currently working through the proof of the following theorem (Thm 9.3 p.130 Falconer)
Let $\{S_1,...,S_m\}$ be an IFS with ratios $0 < c_i < 1$ for which the open set condition holds, i.e. ...
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Why does a cube show up when running an IFS of this hypertetrahedron?
I was messing around with iterated function systems and ran the chaos game, except instead of using a triangle it uses a hypertetrahedron, and the original function that computes where to place each ...
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Proof for the existence of a compact attractor
Let $(X,d)$ be a complete metric space and let $f_1,\ldots,f_n$ be contractions with Lipschitz constants $q_i$. Then a unique non-empty compact set exists such that $K=\bigcup_{i=1}^n f_i(K)$.
Now the ...
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functional iteration and convolution [closed]
Question
$$ H(u,t)= u^{-1} (u X-1+e^{-uX }) $$
$$ H_T(u)=sup H(u,t) $$
$$H_T(u e^{-a v} ) <H_T(u) $$
$$ H_T(u)< u c^{-2} +2c_1 A(c_1 u) + H_T(u e^{-a v})$$
the author iterates the above equation ...
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Is an infinite composition of bijections always a bijection?
Main Question
Suppose I have a sequence of real valued functions $f_1:X_0\rightarrow X_1,...,f_n:X_{n-1} \rightarrow X_n,...,$ and I then, with $\circ$ denoting function composition, define
$$g_n : ...
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Does the sequence of cosines converge for all complex numbers?
Now asked on MO here.
Define $f_1(z) = \cos(z)$, $f_{n+1}= \cos(f_n (z)) $, The question is Does $\lim\limits_{n \to \infty}f_n(z)$ exist for all $z \in \mathbb{C}$? And if the answer is no what is ...
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Prof. Knuth lecture about $ \pi $ and random maps
In this video, Prof. Knuth talks about an interesting combinatorial problem:
suppose you have a random map $ f\colon \{ 1, 2, 3,\ldots, n \} \rightarrow \{ 1, 2, 3,\ldots, n \}$. If you consider the ...
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Question on iterated fixed points of $x^2-2$
Let $P(x)=x^2-2$. Let $P_n(x)$ denote the $n^{th}$ iteration of P. I was asked to prove that the equation $P_n(x)=x$ has all distinct real roots.
My attempt:
I tried using induction, but I'm not sure ...
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Hutchinson's Notation in Fractals and Self-Similarity
I have been examining https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf for my thesis but couldn't find an explanation for two notations in the paper. ...
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Reflecting a fractal
Assume you have a certain IFS (iterated function system - a finite set of contractions) given by affine transformations.
As reflections are affine transformations, any reflection of an IFS fractal ...
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Dividing a polyhedron into two similar copies of itself
The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
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Iterated function system with a fractional number of contractions
Iterated function systems can be used to generate fractals. One starts off with a simple geometric figure and applies the IFS infinitely many times to obtain a fractal. For example, in the case of the ...
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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]...$$We have the classical definition of the ...
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How can i solve this optimization problem effectively?
Recently, i met an optimization problem
$$ \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} $$
from ...
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Is the Lipschitz constant of the Hutchinson operator the maximum of the Lipschitz constant of its components?
Let $(X,d)$ be a metric space, define $F(X)$ as the set of all non-empty compact subsets of $X$, for $A,B \in F(X)$ we define
$$d(A,B) = \sup_{a \in A} \, \inf_{b \in B} d(a,b)$$
We now define the ...
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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}...
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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$ ...
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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 ...
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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 ...
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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, ...
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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 $...
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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 ...
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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 ...
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Moran's theorem Open set condition Koch Curve
I am currently studying some measure theoretic fractal geometry, and I am trying to learn how to use Moran theorem. The statement I currently have is:
Moran's theorem:
if $F_1,...,F_N : \mathbb{R}^d \...
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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 ...
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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{...
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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 ...
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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 :$$ \sin( p\, \...
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If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, is it true that $f(x) = x + 1/2$?
If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, where $f: \Bbb R \rightarrow \Bbb R$ is real-analytic, bijective, monotonically increasing, is it true that $f(x) = x + 1/2$?
I have tried to represent $f(x)$ ...
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Chaos game and fractal polygons
Let $V=\{v_1,v_2,\cdots,v_n\}$ the set of $n$ vertices of a regular polygon. For $n=3$ the chaos game implies that with an arbitrary point on the plane $x_0$, and by applying the recursive relation
$$...
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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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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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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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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 ...
user545426
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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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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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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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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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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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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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