M.Sc Student | Pinsky Tali |
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Subject | Characterization of Periodic Orbits by Topological Considerations for a Kicked Particle |

Department | Department of Mathematics |

Supervisors | Professor Emeritus Bronislaw Wajnryb |

Professor Emeritus Shmuel Fishman (Deceased) |

We give an order relation on a certain family of
periodic orbits which exist for a certain type of homeomorphisms of the torus
called shear type homeomorphisms. Each orbit in this family is characterized by
its rotation number, which is some rational number in the unit interval. The
order relation is then induced by a natural order relation on rational numbers
and pairs of rational numbers which are Farey neighbors. The work is thus an
analog of Sharkovskii’s theorem for a specific
two-dimensional case.

The
topological tools used in order to achieve this are the Bestvina-Handel
algorithm, Markov partitions and theorems regarding isotopy stability of
periodic orbits. The results are then used for analyzing a concrete physical
system, which is a one-dimensional system (hence with a two-dimensional phase
space). The system is a quantum chaotic system called the kicked accelerated
particle, consisting of non-interacting particles subject to gravitation and
kicked by a periodic electromagnetic field. This system can be modeled as a map
on the two-torus which is of shear type, and the periodic orbits found for the
map have physical significance as they give rise to accelerator modes. The
order relation found then gives a global qualitative understanding of the
structure of existence of these periodic orbits, and so existence of
accelerator modes, depending on the parameters of the physical system.