Outer billiards outside regular decagon: periodicity of almost all orbits and existence of aperiodic orbit
https://doi.org/10.22405/2226-8383-2019-20-2-406-441
Abstract
Outer billiards were introduced by B. Neumann in 1950s and became popular in 1970s due
to J. Moser; Moser considered outer, or dual, billiard as toy model of celestial mechanics. The
problem of stability of the Solar system has such a property that “it’s easy to write ???? equations
of particles motion down but hard to understand this motion intuitively”; according to this,
Moser suggested to consider Neumann’s outer billiard problem which has the same property.
One of classical examples of dynamical systems is an outer billiard outside regular ????-gon; in
particular, this billiard is connected with problems of existence of aperiodic trajectory and of
fullness of periodic points. These problems resolved only for a few number of a special cases.
In case ???? = 3, 4, 6 table is a lattice polygon; as a consequence, there are no aperiodic points,
and periodic points form a set of full measure. In 1993, S. Tabachnikov was managed to find an
aperiodic points in case of regual pentagon; it was done using renomalization scheme — method
which has a fundamental importance in research of self-similar dynamical systems.
According to R. Schwartz, cases which are next by complexity are ???? = 10, 8, 12; in these
cases, and also in case ???? = 5, it’s possible to build a renomalization scheme which, as R. Scwartz
writes, “allows one to give (at least in principle) a complete description of what is going on.”
Later, author was managed to discover self-similar sturctures and build renormalization
scheme for cases of regular octagon and dodecagon.
This article is devoted to outer billiard outside regular decagon. The existence of an aperiodic
orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of
such an outer billiard are proved to be periodic. All possible periods are explicitly listed. The
work is based on classical technology of search and research of renormalization scheme. Periodic
structures which occur in case ???? = 10 are similar to periodic structures in case ???? = 5, but has
their own features.
to J. Moser; Moser considered outer, or dual, billiard as toy model of celestial mechanics. The
problem of stability of the Solar system has such a property that “it’s easy to write ???? equations
of particles motion down but hard to understand this motion intuitively”; according to this,
Moser suggested to consider Neumann’s outer billiard problem which has the same property.
One of classical examples of dynamical systems is an outer billiard outside regular ????-gon; in
particular, this billiard is connected with problems of existence of aperiodic trajectory and of
fullness of periodic points. These problems resolved only for a few number of a special cases.
In case ???? = 3, 4, 6 table is a lattice polygon; as a consequence, there are no aperiodic points,
and periodic points form a set of full measure. In 1993, S. Tabachnikov was managed to find an
aperiodic points in case of regual pentagon; it was done using renomalization scheme — method
which has a fundamental importance in research of self-similar dynamical systems.
According to R. Schwartz, cases which are next by complexity are ???? = 10, 8, 12; in these
cases, and also in case ???? = 5, it’s possible to build a renomalization scheme which, as R. Scwartz
writes, “allows one to give (at least in principle) a complete description of what is going on.”
Later, author was managed to discover self-similar sturctures and build renormalization
scheme for cases of regular octagon and dodecagon.
This article is devoted to outer billiard outside regular decagon. The existence of an aperiodic
orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of
such an outer billiard are proved to be periodic. All possible periods are explicitly listed. The
work is based on classical technology of search and research of renormalization scheme. Periodic
structures which occur in case ???? = 10 are similar to periodic structures in case ???? = 5, but has
their own features.
References
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Review
For citations:
Rukhovich F.D. Outer billiards outside regular decagon: periodicity of almost all orbits and existence of aperiodic orbit. Chebyshevskii Sbornik. 2019;20(2):406-441. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-2-406-441
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