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Periodic Data of Torus Diffeomorphisms with One Saddle Orbit

Student: Bosova Anna

Supervisor: Olga Pochinka

Faculty: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

Educational Programme: Mathematics (Bachelor)

Year of Graduation: 2019

When studying discrete dynamical systems, that is, when studying the behavior of the orbits of the map f defined on a given compact manifold, periodic data of the map play an important role. Over the past forty years, many results have emerged showing that some simple assumptions about the periodic data of the system lead to profound conclusions about its global behavior. One of the most well-known results in this direction is the effect discovered by A. Sharkovsky, that the display of a segment of an orbit of period three entails chaos, manifested in the existence of an orbit of any period. In the work of V. Grines, O. Pochinki, S. Van Strien, it was shown that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. In an article by T. Medvedev, E. Nozdrinova, O. Pochinki, this problem was solved in a general formulation, that is, the periods of source orbits are calculated for a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. The purpose of this thesis was to establish a complete list of periodic data of torus diffeomorphisms with one saddle orbit, provided that at least one nodal point of the map is fixed.

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