Can the Mandelbrot set be designed through inequalities?

Question

Many years ago I found an inequality that directly controlled whether a point $\displaystyle c$ belongs or does not belong to the Mandelbrot set. Roughly, it was something like this: If $\displaystyle g(c) \leq 1/4$ then $\displaystyle c$ belongs to $\displaystyle M$ (ie to the Mandelbrot set), where $\displaystyle g(c)$ is a definite function. This inequality was efficient on any scale and the only limitation was the numerical accuracy of the computer. The same function $ \displaystyle g (x)$ was easy to adapt to the design of any set defined by the relation $\displaystyle f_c (z)=z^n + c$. The set $\displaystyle M$ contains discs and cardioids, some of which are distorted. The same function $\displaystyle g$ was functional in all these cases. These are included in the good news.

However, the $\displaystyle g$ function was not complete, as it often malfunctioned at seemingly random points $\displaystyle c$, but it was possible to fix the problem with additional interventions. I had started to improve the $\displaystyle g$ function by gradually removing these interventions, but then the computer's hard drive broke down. I had not printed the method, and since then I have not dealt with this issue. I've even forgotten the basics about the Mandelbrot set. However, I recently found in the margins of a book some of my illegible notes that were the starting point for the construction of the $\displaystyle g$ function. I am unable to verify the accuracy of these notes, but I remember that on the basis of these it was possible to check whether a point $\displaystyle c$ belongs to the large cardioid of $\displaystyle M$ with an initial form of the above inequality.

$\displaystyle r_m=\frac{\sqrt{3 \pm 2 \sqrt{3-8x_m}}}{4}$

$\displaystyle \left (= \sqrt{x_m^2 + y_m^2} \right)$

$\displaystyle y_m= \pm \frac{\sqrt{3-16x_m^2 \pm 2 \sqrt{3-8x_m}}}{4}$

I'm not happy that I can only provide this information, but the purpose is to motivate research.

Note : inequality can not be used to color the points that do not belong to $\displaystyle M$, so they must be colored through the standard method of repetition. This is due to the fact that a significant part of the necessary information is removed in return for the speed. Of course this is important because all the beauty of the Mandelbrot ensemble is in its atmosphere.

Secondary note: At many intersections of $\displaystyle M$ it is obvious that if we zoom in on this area an island will appear. It is possible to find out how large the zoom must be for nz the island to appear by checking a random point near the intersection, but i do not remember the relevant relationship.

Answer 1

You should state your question more clearly. Yes, Mandelbrot set can be described by an inequality, namely $M=\{ z:u(z)\leq 0\}$ where $u(c)=v_c(c),$ and $$v_c(z)=\limsup_{n\to\infty}2^{-n}\log|(f^n_c)|,$$ This function is relatively easy to compute approximately for any $c$ since the convergence to the limit is fast. It does not follow that it is easy or possible to compute its zero set. Indeed, it is known proved that $M$ is not semi-algebraic, and thus "not computable" in any strictly defined sense,

L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 1–46.

There is also the book: Complexity and Real Computation By Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale.

M. Braverman and M. Yampolsky, Computability of Julia sets. Algorithms and Computation in Mathematics, 23. Springer-Verlag, Berlin, 2009.