Hypercubic billiard word: Difference between revisions
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* A ''d''-ary billiard scale has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying this bound need not be billiard scales. | * A ''d''-ary billiard scale has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying this bound need not be billiard scales. | ||
* An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well. | * An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well. | ||
* If a ternary billiard scale has a well-formed | * If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4. | ||
== Determining whether a scale word is a billiard scale == | == Determining whether a scale word is a billiard scale == | ||