Crystallography & Symmetry
patterns derived from their groups
Concept

Unique ergodicity — where it appears

The property that every long window of a sequence gives the same limiting frequency for a patch, wherever the window was cut. It is what makes the frequency of a patch a number belonging to the chain rather than to the sample, and for a primitive substitution the number is an eigenvector entry.

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

How fast a window's answer settles: 1/L on the chain, 1/√L on a shuffle. The largest error a window of each length makes about a block's frequency, over every position the window can take, on logarithmic axes. The upper line is a shuffle of the chain's own letters — same frequencies, no order — and its slope is close to −½, which is the random walk a sequence with no structure produces. The lower line is the Fibonacci chain itself and its slope is close to −1. The frequency of a block in the chain is therefore something a finite window measures rather than approaches: to know it to a part in a thousand needs a window of a thousand tiles, not a million.

The average is the same wherever it is taken

A measurement is made on a window somewhere, and the question is whether the answer belongs to the chain or to the window. For the Fibonacci chain the error falls as one over the window's length; for a shuffle of the same letters it falls as one over the square root, and the two exponents are fitted rather than asserted.

aperiodic · Complexity How often each block of 5 occurs. Every block of length 5 in the fibonacci chain, with its frequency from the Perron eigenvector of the block substitution and again from a count over a chain of 46368 letters. The two share nothing: one is a linear algebra problem over a matrix of integers, the other a loop over a string. The eigenvalue of the block matrix is the inflation factor of the letter matrix, which is a second check and a stronger one — a chain inflates at one rate whatever length of window is being counted.

How often each patch occurs

That a patch has a frequency at all is the ergodic theorem. What the frequency is turns out to be an eigenvector: the substitution acts on blocks as well as on letters, the block matrix has a Perron vector, and its entries are the frequencies exactly. For the Fibonacci chain those entries take three values at every length, and the three values are the three gaps of a rotation.

aperiodic · Complexity

Named alongside it

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

Patch frequencyBounded discrepancyContinued fractionFactor complexityThe Fibonacci chainInflation factorPerron eigenvectorRandom walkSturmian sequenceSturmian wordSubstitutionThree distance theorem

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