Hypercubic billiard word: Difference between revisions

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== Formal definition ==

Formally, let

* ''w'' be a scale word with signature ''a''1X1, ..., ''a''''r''X''r'' (i.e. ''w'' is a scale word with ''a''''i''-many X''i'' steps);

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* ''L'' be a line of the form ''L''(''t'') = (''a''1, ..., ''a''''r'')''t'' + '''v'''0, where '''v'''0 is a constant vector in '''R'''''r''. We say that ''L'' is ''in generic position'' if ''L'' intersects the hyperplane ''x''1 = 0 at a point (0, α1, α2, ... α''r''-1) where α''i'' and α''j''/α''i'' for ''i'' ≠ ''j'' are irrational.

We say that ''w'' is a '''billiard scale''' if any line in generic position of the form (''a''1, ..., ''a''''r'')''t'' + ''v''0 has intersections with coordinate level planes ''x''''i'' = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.

We say that ''w'' is a '''rank-'''r '''billiard scale''' if any line in generic position of the form (''a''1, ..., ''a''''r'')''t'' + ''v''0 has intersections with coordinate level planes ''x''''i'' = ''k'' ∈ '''Z''' that spell out the scale as you move in the positive ''t'' direction along that line.

== Properties ==

* [[Mos]]ses are rank-2 billiard scales

* A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes

[[Category:Theory]]

[[Category:Billiard scales]]

[[Category:Scale]]

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+== Formal definition ==
 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);
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 * ''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.
+We say that ''w'' is a '''rank-'''r '''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.
+== Properties ==
+* [[Mos]]ses are rank-2 billiard scales
+* A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes
 [[Category:Theory]]
 [[Category:Billiard scales]]
 [[Category:Scale]]