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This is a generalization of the question Are there mini-mandelbrots inside the julia set?

@Hagen raises an issue I was afraid of, which is that even the mini-Mandelbrots in the Mandelbrot set are not exact copies. I don't know enough to give a rigorous answer to this question, but I'm thinking of whatever (implicit) standard is used for identifying mini's within the Mandelbrot itself. Though it would certainly be interesting if one can find an exact duplicate of the Mandelbrot set somewhere inside a "nice" fractal without effectively constituting the entirety of that fractal.

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    $\begingroup$ Depends a bit on what you accept as mini-Mandelbrot (you know even those "copies" of $M$ in $M$ itself are not exactly copies ...) $\endgroup$ Apr 6, 2014 at 22:49
  • $\begingroup$ @HagenvonEitzen Are they copies under a scale and transform? What I mean is, if you see a "mini-Mandelbrot" after zooming in 100x at (1, 1.5) would it be exactly the same after taking $\frac z{100}-(1+1.5i)$? $\endgroup$
    – kleineg
    Aug 4, 2014 at 17:20
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    $\begingroup$ This poster noticed mini-Mandelbrots showing up in the bifurcation locus of a root-finding map! math.stackexchange.com/questions/2069219/… $\endgroup$
    – Vectornaut
    Apr 2, 2019 at 1:48

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Iterating the cubic polynomial, $z \mapsto c(z^3-3z)$, starting with z=1, one gets a cubic mandelbrot, whose "mini-mandelbrots" are conjugate to the quadratic mini-mandelbrots. These images are from Wolf Jung's Mandelbrot program, under cubic polynomials with semi-conjugate to quadratic.

cubic mandelbrot, iterating as above zoom upper middle zoomed in on mini-mandelbrot in cubic mandelbrot

Here is one more example, iterating $z \mapsto c(z^5-5z)$, starting with z=1, which also has mini-mandelbrots. quintic mandelbrot, iterating c(z^5-5z)

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Images of the boundary of the Mandelbrot set, $\partial{M_2}$, under quasiconformal maps with arbitrarily small quasiconformal distortion appear in the bifurcation locus of any holomorphic family of rational maps:

$$f:X \times \mathbb{C}^*\to\mathbb{C}^*$$

when $X$ is connected and the degree of $f_t$ is at least $2$ for all $t$ in $X$, and the bifurcation locus is not simply empty.

See McMullen, Theorem 4.3 and Corollary 4.4 in particular.

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There are distorted mini-copies of the Mandelbrot fractal in the Mandelbar fractal.

The Mandelbar fractal is self-similar to itself and to the Mandelbrot fractal, which is generated with a similar iterating function, except the Mandelbar iterating function squares the complex conjugate.!

See here: http://en.wikipedia.org/wiki/Tricorn_(mathematics)

It appears that this result generalizes to the Multibrot-n fractals, as well.

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