There is a relatively famous fractal, called the Burning Ship fractal, which is usually presented as an analogue of some sort to the Mandelbrot set. The Mandelbrot set has two definitions. Both define a parameterized function f[c](z) = z^2 + c (over the complex numbers). One definition is that M is the set of all c such that the Julia set J(f[c]) is connected. The other is that it is the set of all c such that the sequence of iterates of f[c] starting with 0 does not converge to infinity. It is a theorem, and not a super easy theorem, that both definitions produce the same set. That is, that J(f[c]) is connected precisely when the 0-origin iteration sequence of f[c] converges to infinity. Call these the "connectedness definition" and the "convergence definition". The Burning Ship fractal is defined by substituting a different parameterized function, g[c](z), which doesn't have a simple formula because it isn't even analytic -- that is, it takes z apart into its real and imaginary parts, does some stuff to them, and then puts them back together. You can look up the rule on Wikipedia. The Burning Ship fractal is defined using convergence only, with no attention paid, apparently, to the question of whether the connectedness definition yields the same fractal -- that is, whether the theorem linking connectedness and convergence is even true of g, as it is of f. I have a strong intuition that it's not. Has anybody investigated what the connectedness-analog fractal generated by g looks like?