Concept

Contraction

A map that brings every pair of points closer together by at least a fixed factor smaller than one. Such a map has exactly one fixed point, and applying it over and over from any start converges to that point, with the error shrinking by the factor at every step.

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

A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other.

A point that pulls, and a point that pushes

Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.

dynamics · Fixed points

Sperner's corridor to the same fixed point, on a gentle map and a twisted one. Two 32 by 32 labelled grids; the corridor crosses 44 triangles reading 100 corners on the gentle map and 148 triangles reading 204 corners on the twisted one.

What it costs to find the point that stays put

Brouwer's theorem promises a point a map leaves where it is, and Sperner's corridor walks to it. On a gentle map the walk is short; twist the map about its fixed point and the corridor follows every turn, so its cost grows like the square of the grid. Halving the square by winding number finds the same point in a number of readings proportional to the grid's side whatever the map does — and no method that only reads the map can do fundamentally better.

topology · Fixed points

Named alongside it

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

Fixed point Brouwer Cobweb Convergence Convergence rate Derivative Iteration Linearisation Logistic map Nonconstructive Orbit Stability

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