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Concept

Invariant set

A set that a map or flow sends into itself, so that a point starting inside never leaves. The interesting ones are much smaller than the space they sit in, and the dynamics restricted to them is what a system's long-run behaviour is.

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

Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3.

Stretch, fold, and what is left

A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

dynamics · Strange attractor The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

dynamics · Strange attractor

Named alongside it

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

Cantor setChaosSelf-similarityStrange attractorFractal dimensionHorseshoeIterationSurfaceSymbolic dynamicsVolume

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