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Concept

Time average

The average of a quantity along one trajectory of a system, taken over longer and longer stretches of time. Ergodic theory asks when it equals the space average over all states, which is when one long run can stand in for the whole system.

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

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 Where one Hénon orbit spends a million steps. A density map of 1000000 steps of one Hénon orbit counted into 420 × 140 cells: 2542 cells visited, half the time spent in 608 of them.

Where the time goes on an attractor

A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.

dynamics · Strange attractor

Named alongside it

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

Arcsine distributionConjugacyErgodic theoremErgodicityFractalHenon mapInvariant densityInvariant measureIterationLorenz systemMeasurePeriodic orbit

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