Hypercubic billiard word: Difference between revisions
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Formally, let | |||
* ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub>, ..., ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many X<sub>''i''</sub> steps); | |||
* ''n'' = ''a''<sub>1</sub> + ... + ''a''<sub>''r''</sub> be the length of ''w''; | |||
* ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''<sub>1</sub> = 0 at a point (0, α<sub>1</sub>, α<sub>2</sub>, ... α<sub>''r''-1</sub>) where α<sub>''i''</sub> and α<sub>''j''</sub>/α<sub>''i''</sub> for ''i'' ≠ ''j'' are irrational. | |||
We say that ''w'' is a '''billiard scale''' if any line in generic position of the form (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + ''v''<sub>0</sub> has intersections with coordinate level planes ''x''<sub>''i''</sub> = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line. | |||
[[Category:Theory]][[Category:Billiard scales]] | |||
Revision as of 00:55, 2 June 2022
Formally, let
- w be a scale word with signature a1X1, ..., arXr (i.e. w is a scale word with ai-many Xi steps);
- n = a1 + ... + ar be the length of w;
- L be a line of the form L(t) = (a1, ..., ar)t + v0, where v0 is a constant vector in Rr. We say that L is in generic position if L intersects the hyperplane x1 = 0 at a point (0, α1, α2, ... αr-1) where αi and αj/αi for i ≠ j are irrational.
We say that w is a billiard scale if any line in generic position of the form (a1, ..., ar)t + v0 has intersections with coordinate level planes xi = k ∈ Z that spell out the scale as you move in the positive t direction along that line.