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Critical point

A critical point of a function is a place where its slope is zero in every direction: a peak, a pit or a saddle. Counted with signs, plus for peaks and pits and minus for saddles, the critical points of a function on a closed surface always add up to its Euler characteristic.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

The dark lines are one point's orbit

Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.

dynamics · Period-doubling A random surface and the region above height 1. A smooth Gaussian random field on a 64 by 64 torus and its excursion set above 1: 14 pieces, 0 holes, Euler characteristic 14; expected 13.8.

Pieces minus holes on a random surface

Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.

topology · Euler characteristic

Named alongside it

The objects these essays reach for when they reach for this one.

AttractorBifurcationChaosEuler characteristicFeigenbaum constantInvariant densityLogistic mapMorse theoryPeriod-doublingRandom fieldSaddle pointTorus

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