Inverted Lambda

Inverted Lambda begins with the Lambda fractal’s logistic recurrence and adds a rational correction. The first term, c·z(1−z), is the standard complex logistic map. The second, 0.18/(z² + c), is reciprocal and quadratic; when its denominator approaches zero, it can hurl the orbit to very large values.

That fraction changes the rules. The plain logistic map is a polynomial, smooth everywhere; Inverted Lambda is a rational map with movable poles. An orbit can head toward infinity through unbounded growth, or by wandering too close to the divisor. The parent’s round lobes remain, but jagged seams, knot-like tangles, and sudden scale changes cut through them.

The 0.18 is fixed, not another parameter. It keeps the polynomial and rational contributions both active within a moderate number of iterations.

Inverted Lambda is a hybrid construction joining two well-studied complex-dynamics families. Its logistic base comes from the lambda fractal tradition studied by Mandelbrot (1980). Adding a reciprocal quadratic term to a polynomial map relates to singular perturbation, which Curt McMullen explored in the late 1980s for rational maps zⁿ + λ/zⁿ.

This particular pairing—a logistic map and a reciprocal quadratic with a fixed constant numerator—appears to be a novel fractal-rendering formulation. It does not trace to one historical paper; it draws on ideas developed across decades of complex-dynamics research. “Inverted Lambda” describes the construction rather than a term from the research literature.

The parameter plane keeps the logistic family’s rounded lobes, but sharp seams, dense knots, and patches where color suddenly deepens or brightens interrupt them. Near a pole, an orbit makes a large jump, and the escape-time coloring records that break.

It can resemble a smooth surface cracked from within. Far from the poles, the parent Lambda structure is still legible; the rational correction draws angular, crystalline detail across it.