Chess billiards. (English) Zbl 1522.37052
From the text: Our article has two aims. We first collect various wellknown results on circle homeomorphisms and apply them
to the chess billiard map to deduce certain interesting
results, in particular answering some questions posted in
[C. R. H. Hanusa and A. V. Mahankali, Eur. J. Comb. 95, Article ID 103341, 26 p. (2021; Zbl 1471.37034)]. The second goal is to prove new results about the chess
billiard. We prove some general results on periodic points;
then we turn to the study of the chess billiard in a polygon.
In particular, we prove results about the chess billiard map
in triangles, in the square, and in other centrally symmetric
domains. Our results on the square are undoubtedly the
most interesting of the article.
The ultimate goal of the study of chess billiards is to decide for which convex domains and directions the rotation
number is rational.
MSC:
| 37E10 | Dynamical systems involving maps of the circle |
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
| 37C27 | Periodic orbits of vector fields and flows |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
Citations:
Zbl 1471.37034References:
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