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 $x \in E$).
93 questions
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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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Equal Lyapunov exponents
I'm working with 2D iterative map for which i calculate first and second Lyapunov exponents. For some range of parameter these exponents are equal each other. I wonder, does it imply something? I ...
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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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