Questions tagged [fractals]
Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
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Causal fermion systems fromm fractal geometry
Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...
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Diffeomorphisms preserving "nice" smooth functions
Let $\mathbb{R}^2\supset D=\{(x,y)\in\mathbb{R}^2|x^2+y^2<1\}$ be the open unit disc, and $U\subset\mathbb{R}^2$ be the interior of Koch's snowflake, as constructed in Falconer's book Fractal ...
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Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$
Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space?
A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder ...
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Rössler attractor, Convergence of box counting to estimate the fractal dimension
At the moment I want to estimate the fractal dimension of the Rössler attractor. I have written a program, which is counting the boxes hitted N(ɛ) (with ɛ := side length of boxes) by a trajectory on ...
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Fractal homological algebra
The usual definition of a chain complex requires for the indices to be integer numbers. However, taking inspiration from the theory of Hausdorff dimension, one can think of 'fractal' chain complexes (...
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Hausdorff dimension of $X\times X$
I am thinking of the following question:
Let $X\subseteq \mathbb R$. Is it true that
$$
\mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)?
$$
My thoughts:
We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...
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Complexity of the Mandelbrot set on rationals
Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals.
A point $c$ is contained within the Mandelbrot set $M$ if the following ...
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Relationship between the Hurst exponent and the alpha parameter
I have a question about the relationship between the Hurst exponent $H$ and the $\alpha$ parameter in the autocorrelation function when long memory is present. As we know in this case the decay of the ...
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Literature on the total variation of fractal graphs/fractal Brownian motion?
I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study ...
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♢ ⧫ ⬠: the fourth kind of Penrose tiling?
It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There ...
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Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
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C^1 fractals in statistical mechanics
It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions....
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Quantitative estimates on space filling curves
To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
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Sierpinsky pyramid L-System
The Wikipedia page for L-Systems describes many of them, including a couple rules that converge toward the Sierpinsky triangle. That particular fractal also has a 3D version, which basically uses ...
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Is the limit of classical Laver tables connected anywhere?
Let $A_{n}=(\{1,\dots,2^{n}\},*_{n})$ be the $n$-th classical Laver table. Then $*_{n}$ is the unique operation on $\{1,\dots,2^{n}\}$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and
$x*_{n}1=x+1\...
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Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
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What is a precise definition of a twisted fibration of one fern over another?
I am curious if there is a notion of a "twisted fibration" of fractals. Since there are many classes of fractals, I'll ask specifically about L-systems.
How can we precisely define the twisted ...
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Is this a quasi-crystal and/or a fractal?
I'm not too familiar with quasi-crystals, but I was recently playing around with a particular discrete function and I got the following neat pattern:
This is just a small segment, but as far as I ...
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Box dimension of the set of Pisot numbers?
A Pisot number is an algebraic integer bigger than $1$ with all of its Galois conjugates having modulus less than $1$. The set of Pisot numbers is known to be countably infinite and is not dense in $(...
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Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
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Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
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Is the area of the Mandelbrot provably computable?
Recall the Mandelbrot set $M$ is the set of points $c$ in the complex plane such that the sequence $z_0 = 0, z_{n+1} = z_n^2 + c$ is bounded. It is well-known that $M$ is a compact set of positive ...
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A question about Julia set for quadratic family
Let $P_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P_c)$ is an injective map. Is there some reference about it? Any comments and reference ...
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Why are the Julia sets so simple? (quadratic family)
I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.
My question is specifically ...
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A one dimensional fractal like set with the same line width within a bounded area? [closed]
Say that we have a line $\left(0,0.5\right)$. I want a process that can split that line and half and move that half a bit, and then take half of that half and moved it and so on, so that by the end ...
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Can we define the Mandelbrot set in terms of the generating function of the Catalan numbers?
For each complex number $c$, define $P_{0}(c)=0$ and $P_{n+1}(c) = (P_{n}(c))^{2} + c$ . The Mandelbrot set is the set of complex numbers c for which $|P_{n}(c)|$ stays bounded as $n\rightarrow \...
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Did Gaston Julia ever get to see a computer-generated image of his eponymous set?
I learned from Wikipedia that Gaston Julia died in 1978. Is it known if he ever got to see a computer-generated image of the set named after him?
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Random Cantor sets on the unit interval
Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For ...
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Usable Change-of-Variables Formula for Hausdorff Measure
Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
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Integral over the Cantor's set Hausdorff dimension
As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...
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Unexpected occurrences of the Sierpinski triangle
The probably most well-known occurrence of the Sierpinski Triangle is as the odd entries of the Pascal triangle
Some month ago however, there was an article about mathematical models of sandpiles ...
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Product Fractals
Here is a theorem found in the Falconer's book on fractal geometry:
Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
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Critical set of a self similar structure
A self similar structure is a triple $L=(K, S, \{F_i\})$ where $K$ is a compact metric space, $S$ is a finite set and for every $i\in S$ the functions $F_i:K\rightarrow K $ are injective and ...
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Fundamental system of neighborhoods. Self similar set
I have been reading the text Analysis on Fractals of Jun Kigami. There is a theorem about the fundamental system of neighborhoods of a point in a self similar set. It is stated as follows
Let $\...
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Does the Mandelbrot set have infinite VC dimension?
Define a binary classifier for points in the complex plane, whose parameter $\theta$ is an isometry of $\mathbb{C}$, and which classifies $z \in \mathbb{C}$ based on whether or not $\theta(z)$ is in ...
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A point set of power series with coefficients in {-1, 1}. Connected or not?
Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...
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Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?
Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...
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Quantification of the extent of periodicity in a time series using fractal analyses
I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...
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What results exist for functions with regionally fluctuant fractal dimension?
I'm interested in functions that have a varying fractal dimension at different scales and/or regions. Has this been investigated in detail? I'd be interested in results and references in this area of ...
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Self-Similar Graphs
Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs.
Many definitions of fractal dimensions (...
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Lightning strike fractal formula [closed]
I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or ...
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Are there any exact results for Hausdorff Measure?
The computation of the Hausdorff measure is extremely difficult due to the infimum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...
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Fractal in discrete time series/discrete time sequence
Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series?
I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
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What is the Hausdorff dimension of this fractal?
Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= \sum_{i=h}^\...
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Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension
Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...
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Fractal dimension of scaling limits of discrete structures
Let $S$ be the set of positive integers whose base-three expansion contains only the digits 0 and 2. The discrete set $S$ in a sense has (negative) fractal dimension $(\log 1/2)/(\log 3)$, since if ...
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On the boundary of the twindragon
Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a non-empty ...
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Precise density estimates for Cantor sets
Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...
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Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?
For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision (...
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What is "graph-directed iterated function"?
Im translating an article about Rauzy fractal and I ran into this sentence:
...