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Questions tagged [iterated-function-system]

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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 $x \in E$).

93 questions
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1 vote
0 answers
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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 ...
Augusto Santi's user avatar
  • 3,054
2 votes
2 answers
422 views

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 (...
Mths's user avatar
  • 53
1 vote
1 answer
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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 ...
Mths's user avatar
  • 53
2 votes
1 answer
84 views

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(\...
Olimani's user avatar
  • 105
2 votes
0 answers
162 views

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 \...
Kepler's Triangle's user avatar
  • 3,745
1 vote
0 answers
55 views

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}, \...
R B's user avatar
  • 21
1 vote
1 answer
211 views

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. ...
Olimani's user avatar
  • 105
0 votes
0 answers
53 views

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 ...
Hart Traveller's user avatar
  • 13
1 vote
1 answer
104 views

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 ...
Dave the Sid's user avatar
  • 189
0 votes
1 answer
96 views

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 ...
brando.dawson's user avatar
  • 33
22 votes
5 answers
3k views

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 : ...
Aidan O'Keeffe's user avatar
  • 521
5 votes
0 answers
195 views

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 ...
pie's user avatar
  • 9,329
7 votes
2 answers
305 views

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 ...
user967210's user avatar
  • 1,518
1 vote
0 answers
145 views

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 ...
Dailin Li's user avatar
  • 29
0 votes
0 answers
64 views

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. ...
Dave the Sid's user avatar
  • 189

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