The Mandelbrot set (the black region below) is defined as the set of all points c in the complex plane such that iterating f꜀(z) = z^2 + c starting from z = 0 doesn't diverge to infinity.

Certain values of c have a periodic cycle, a set of points such that, if you repeatedly apply f꜀, starting at one of the points, you will cycle through all of them and return to the point you started at. For example, f₀ has a 1-cycle {0}, since f₀(0) = 0, and f₋₁ has the 2 cycle {0,-1}, since f₋₁(0) = -1, and f₋₁(-1) = 0. A periodic cycle is attractive if repeatedly applying f꜀ to values close to one of the values of the cycle results in the the values of successive iterations approaching the values in the cycle.

An interior component of the Mandelbrot set in which all the maps f꜀ have an attractive periodic cycle is called hyperbolic. For example, all the maps f꜀ for c in the main cardioid have an attractive 1-cycle. The density of hyperbolicity conjecture states that all interior regions of the Mandelbrot set are hyperbolic, and they are dense in the Mandelbrot set.

Resolves YES if density of hyperbolicity is proven, and NO if it is disproven.

Note that it is implied by /JosephNoonan/is-the-mandelbrot-set-locally-conne