Elsevier

Discrete Mathematics

Volume 309, Issue 8, 28 April 2009, Pages 2053-2066
Discrete Mathematics

Directional complexity of the hypercubic billiard

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Abstract

We consider a minimal rotation on the torus Td of direction ω. A natural cellular decomposition of the torus is associated to this map. We consider an infinite orbit for this map. We compute the complexity of the associated word. Under some hypothesis on the direction, we obtain an exact formula which shows that the order of magnitude is nd. This result is related to the billiard map inside a hypercube of Rd+1.

Keywords

Ergodic theory
Symbolic dynamic
Billiard
Complexity

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