fractals

Is the limit of classical Laver tables connected anywhere?

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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 $x*_{n}1=x+1\mod 2^{n}$ whenever $x,y,z\in A_{n}$. Let $L_{n}:\{0,\dots,2^{n}-1\}\rightarrow\{0,\dots,2^{n}-1\}$ be the mapping that reverses the ordering of the digits in the binary expansion of the number in $\{0,\dots,2^{n}-1\}$. More explicitly, $$L_{n}(\sum_{k=0}^{n-1}2^{k}a_{k})=\sum_{k=0}^{n-1}2^{n-1-k}a_{k}$$ whenever $a_{0},\dots,a_{n-1}\in\{0,1\}$. Let $L_{n}^{\sharp}:\{1,\dots,2^{n}\}\rightarrow\{1,\dots,2^{n}\}$ be the mapping … Read more

Fourier coeffients of Cantor measure

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For 0<θ<12, denote by μθ the uniform Cantor measure with dissection ratio θ. It is not hard to show that the Fourier–Stieltjes transform of μθ is ˆμθ(ξ)=∞∏k=1cos(πθkξ) (up to scaling and constant multiple). It is well known that ˆμθ(ξ)→0 as ξ→∞ if and only if θ is a Pisot number. Now we are concentrated on … Read more

Can this number be interpreted as a fractal dimension?

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Under Goldbach’s conjecture, let’s denote for a large enough integer n by r0(n) the quantity inf and by k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n)). Let’s now consider the density of primes around n at two different scales: first at large scale, hence a global density, \delta_{0}(n):=\frac{\pi(2n)}{2n} and then at small scale, hence a local density, \delta_{1}(n):=\frac{k_{0}(n)}{2r_{0}(n)}. My idea is that, … Read more

Algebraicity of the “outer” boundary of the Mandelbrot set

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Let M be the Mandelbrot set and let λ∈M,μ∈C be algebraic numbers. Let tλ,μ be defined as tλ,μ=sup Question: Is t_{\lambda,\mu} an algebraic number? (If it’s any easier – I’m also interested in the case when \lambda and \mu are both Gaussian rationals.) Answer Clearly every piecewise real algebraic curve has this property. In general, … Read more

Usable Change-of-Variables Formula for Hausdorff Measure

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Let Hs be the s-dimensional Hausdorff measure, let D be a nonsingular matrix. Consider the change of measure formula: ∫Af(Dx)dHs(x)=∫DAf(y)dD∗Hs(y) , where D∗Hs(M)=Hs(D−1M) is the pushforward of the Hausdorff measure. Is it possible to find such a function a(x) that ∫DAf(y)dD∗Hs(y)=∫DAf(y)a(y)dHs(y) ? I’m very interested in a usable general change of variables formula; does that exist? Answer … Read more

Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

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Call a Borel set A⊆[0,1] good if 0<dim(A)≤¯dimM(A)<2dim(A), where dim(A) is the Hausdorff dimension of A and ¯dimM(A) is the upper Minkowski dimension of A. (Note that the second inequality in the string of three inequalities holds automatically.) Does a compact set X⊆[0,1] of positive Hausdorff dimension necessarily contain a subset which is a good … Read more

algebraic structure of fractals

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I am wondering whether there exists an algebraic structure(group or modules etc) possess some kind of self-similarity, (sub-group or sub-module have an identical structure with itself) and “irregularity”(undefined) at the same time? if it exists can we find a functor between the category of fractals and this category of algebraic structure in order to study … Read more

Approximating fractal curves

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Is there a known algorithm for approximating a fractal curve, say as specified by some iterative procedure e.g. a Koch snowflake, in terms of f−1(0) for some “simple” function f? Specifically, consider the set F={(x,y)|f(x,y)=0} where f(x,y)=∑Nn,m=−Ntnmeinx+imy and f is real. I would like a procedure to determine the parameters tnm such that the set … Read more

Self-similarity of a dendrite fractal

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The Julia set of the map z↦z2+i is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three generators, but maybe there are more. Perhaps I am after an “iterated function system” which will generate the dendrite … Read more