Open AccessClassification of rough transformations of a circle from a modern point of view
Author(s) -
Anna E. Kolobyanina,
E. V. Nozdrinova,
O. V. Pochinka
Publication year - 2018
Publication title -
žurnal srednevolžskogo matematičeskogo obŝestva
Language(s) - English
Resource type - Journals
eISSN - 2587-7496
pISSN - 2079-6900
DOI - 10.15507/2079-6900.20.201804.408-418
Subject(s) - mathematics , invariant (physics) , manifold (fluid mechanics) , pure mathematics , dynamical systems theory , completeness (order theory) , rotation number , topology (electrical circuits) , mathematical analysis , combinatorics , mechanical engineering , physics , quantum mechanics , engineering , mathematical physics
In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.