Questions tagged [iterated-function-system]

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17 questions
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Union of attractors (Fractals)

Is union of attractors of different IFSs (iterated function systems) attractors of another IFS? Is union of fractal sets always fractal? I wonder the answers for intersection and difference. I could ...
24 views

Do iterative formula only solve for one positive root?

For all the questions I have done using an iterative formula, the quadratic equation only ever solves to one root, I am not sure is this always the case, or is there a method to find multiple roots? ...
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Iterations of function over rationals

Let $F = \mathbb{Q}\setminus\{-1,0,1\}$ and $f$ the function defined over F by $f(x) = \dfrac{x^2-1}{x}$. Show that : $$\bigcap_{n\ge 1} f^n(F)= \emptyset$$ I don't have many ideas for this problem....
113 views

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)...
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selecting a number according to probability

Could anyone just tell me what does he, mean by at the point $2$, selecting a number according to the probabilities, $p(x_0)$? does he meant to chhose the largest one out of $p_i(x_0), i=1,2,\dots,N$?,...
60 views

Is it possible to simplify this nested GCD?

Is it possible to simplify this nested GCD? $$\gcd\bigg(\gcd(m^2,\sigma(m^2)),\frac{m^2}{\gcd(m^2,\sigma(m^2))}\bigg)$$ Here, $\gcd(m^2,\sigma(m^2))>1$ and $\sigma(m^2)$ is the sum of divisors ...
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Limit involving iterated function $f_a(x)=x^2+a^2$

I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the ...
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Self-similarity dimension for IFS or attractor of IFS

We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set ...
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convert iterated function to continuous function

Let's say we have a function: $$v(t_n) = a\cdot(t-t_{n-1}) + v(t_{n-1}) \cdot d^{t-t_{n-1}}$$ Where $v$ = veolcity $t$ = time $a$ = acceleration $d$ = friction or damping What is basically does is ...
If $f(x)=4x(1-x)$ The function is defined over $\Bbb{R}$. Find the number of real solutions of $$f\circ f\circ f(x)=\frac x3.$$