I've read the claim of a fractal image application to be able to show infinite levels of zoom for Julia sets for the classic iteration formula $z_{i+1}:=z_i^2+c$. The application has a realtime interface, so I guess the images can be computed within some upper time limit. They don't provide this feature for Mandelbrot sets, and clearly state that those are limited to double precision floating point calculations.
Now I wonder what technique may be used to provide 'unlimited' zoom.
I guess there is some kind of self similarity, so that a small detail of a Julia set can be replaced by a several magnitudes larger fractal set (with some other, Julia like algorithm) that resembles the same image. Of course, the will be an infinite loop of images at some point, as the parameter space created by the zooming user interface must be finite.
Are the known properties of the Julia set to explain this?
The claim that any application can generate all Julia sets at any level of detail is certainly exaggerated, as it's been proven that there are non-computable Julia sets. See for, the work of Braverman and Yompolsky in this paper on the arxiv.
Nonetheless, most Julia sets are computable and all display loose a type of self-similarity. Specifically, the Julia set of a function $f$ is invariant under the inverse image of $f$. For a quadratic Julia set for a function of the form $f_c(z)=z^2+c$, this means that the Julia set $J$ satisfies $$J = -\sqrt{J-c} \, \bigcup \, \sqrt{J-c}.$$ Here's an illustration for $f(z)=z^2-1$.
Note that the Julia set is not strictly self-similar, as the functions $\pm\sqrt{z-c}$ are not similarities. They are not even contractions. They do, however, form something like an iterated function system that can be used to generated $J$. Suppose, for example, that we want to zoom in to the region outlined in red in the top of the figure below.
The lower left show of the figure shows a simple zoom. The lower right shows a zoom with detail generated by applying the function $$F(z)=\sqrt{1-\sqrt{z+1}}$$ to the entire image.