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 $x \in E$).
Questions tagged [iterated-function-system]
93 questions
59
votes
4 answers
Does this Fractal Have a Name?
I was curious whether this fractal(?) is named/famous, or is it just another fractal?
I was playing with the idea of randomness with constraints and the fractal was generated as follows:
Draw a point at the center of a square.
Randomly choose any…
Silver
- 1,556
47
votes
0 answers
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 angled triangles
$1:\sqrt{2}$ parallelograms
The…
Kepler's Triangle
- 3,599
22
votes
5 answers
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
$$g_n : X_0 \rightarrow X_n\;\text{by}\;g_n(x_0) = (f_n…
Aidan O'Keeffe
- 486
- 3
- 11
11
votes
2 answers
If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, is it true that $f(x) = x + 1/2$?
If $f(f(x)) = x+1, f(x+1) = f(x) + 1$, where $f: \Bbb R \rightarrow \Bbb R$ is real-analytic, bijective, monotonically increasing, is it true that $f(x) = x + 1/2$?
I have tried to represent $f(x)$ as power series in a neighborhood of arbitrary…
Newone
- 153
8
votes
1 answer
Asymptotic behavior of $a_{n+1}=\frac{a_n^2+1}{2}$
Define a sequence as follows:
$$a_0=0$$
$$a_{n+1}=\frac{a_n^2+1}{2}$$
I would like to know the asymptotic behavior of $a_n$. I already know (by roughly approximating $a_n$ with a differential equation) that
$$a_n\sim 1-\frac{2}{n}$$
as $n\to\infty$.…
Franklin Pezzuti Dyer
- 40,930
- 9
- 80
- 174
7
votes
2 answers
Prof. Knuth lecture about $ \pi $ and random maps
In this video, Prof. Knuth talks about an interesting combinatorial problem:
suppose you have a random map $ f\colon \{ 1, 2, 3,\ldots, n \} \rightarrow \{ 1, 2, 3,\ldots, n \}$. If you consider the values $( 1, f(1), f(f(1)), \ldots \}$, what is…
user967210
- 1,464
- 5
- 15
7
votes
4 answers
The real function $f$ such that $\log \cdots \log (f)$ is strictly convex on its domain for any number of $\log$'s
Does there exist a function $f: (a,b) \to \mathbb R$ ($a,b$ are allowed to be infinity) such that $\log \cdots \log (f)$ is strictly convex on its whole domain of definition for an arbitrary number (though finitely many) of $\log$'s?
If such…
No One
- 8,465
6
votes
1 answer
Fourier series of iterated sin / Diagonalization of an infinite matrix of Bessel functions
We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$
We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \sin(x) ) = \sum_{k=1, \,odd}^{\infty} 2 \, J_k(p) \, \sin( k…
al4085
- 309
6
votes
1 answer
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:
$$f_0(n) =
\begin{cases}
n/2, & \text{if}\ 2\mid n\\
3n+1, & \text{otherwise} \\
\end{cases}$$
Noting that $2$ and $3$ are…
Math777
- 738
6
votes
2 answers
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 of me remember how to construct the proof.
Can…
cmpeq
- 63
6
votes
0 answers
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 representation.
Khinchin's constant is the peculiar number $K$…
5
votes
1 answer
Self similar set which does not fulfill the open set condition
Informally, a set is considered self similar if it consists of smaller copies of itself. If this set fulfills the so called open set condition, one can easily calculate the Hausdorff Dimension (see for example Wikipedia).
It is easy to find some…
Keba
- 2,602
- 14
- 30
5
votes
0 answers
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 the sufficient conditions that $z$ should satisfy…
pie
- 8,483
5
votes
1 answer
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 $f(x)=x^{\sqrt[n-1]{k}}$ is a solution, but I cannot prove if there's…
NoetherianCheese
- 506
4
votes
1 answer
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 \to\mathbb{R}^d $ such that $|F_i(x) - F_i(y)| =…
Samael Manasseh
- 778