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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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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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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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 ...
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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 ...
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 ...
139 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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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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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 ...