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Concept

Conjugacy

A change of coordinate carrying one map onto another, so that the two are the same map differently labelled. It preserves everything defined from orbits and neighbourhoods — cycles, entropy, chaos — and destroys everything defined from distances or derivatives.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order.

The orbit written as a word

Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.

dynamics · Symbolic dynamics The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · Iteration Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold.

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

dynamics · Iteration The same loop, seen from two places. An annulus with two marked points and a loop based at the first. Two paths join the points, differing by a full turn round the hole, and each carries the loop to a loop based at the second point.

The group a space has at a point

The loops of a space form a group once a starting point is fixed, and the fixing looks like an arbitrary choice that ought to be removable. It is removable, but only up to conjugation, and the residue is exactly what makes a non-commutative fundamental group harder to state than to compute.

topology · Homotopy

Named alongside it

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

IterationPeriodic orbitSymbolic dynamicsArcsine distributionBase pointBinary expansionChange of variableDense orbitDoubling mapEquivalence relationErgodicityFundamental group

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