Abstract
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.
Highlights
In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classication of circle's rough transformations in canonical formulation
In the modern theory of dynamical systems such problems are understood as the complete topological classication: nding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants
In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle
Summary
Íèæíèé Íîâãîðîä (ðàíåå-Ãîðüêèé) ïî ïðàâó ìîæíî ñ÷èòàòü ìåñòîì ðîæäåíèÿ ãèïåðáîëè÷åñêîé òåîðèè. Ã. Ìàéåðîì [2] áûëî ââåäåíî ïîíÿòèå ãðóáîñòè äëÿ äèíàìè÷åñêèõ ñèñòåì ñ äèñêðåòíûì âðåìåíåì (êàñêàäîâ) íà îêðóæíîñòè. Äîêàçàòåëüñòâî ïîëíîòû ìíîæåñòâà íàéäåííûõ èíâàðèàíòîâ, òî åñòü äîêàçàòåëüñòâî òîãî, ÷òî ñîâïàäåíèå ìíîæåñòâ òîïîëîãè÷åñêèõ èíâàðèàíòîâ ÿâëÿåòñÿ íåîáõîäèìûì è äîñòàòî÷íûì óñëîâèåì òîïîëîãè÷åñêîé ýêâèâàëåíòíîñòè (ñîïðÿæåííîñòè) äâóõ äèíàìè÷åñêèõ ñèñòåì èç G;. Íàñ áóäóò èíòåðåñîâàòü êëàññû ýêâèâàëåíòíîñòè äèôôåîìîðôèçìîâ f : S1 → S1 îòíîñèòåëüíî îòíîøåíèÿ òîïîëîãè÷åñêîé ñîïðÿæåííîñòè: äèôôåîìîðôèçìû f, f ′ : S1 → S1 íàçûâàþòñÿ òîïîëîãè÷åñêè ñîïðÿæåííûìè, åñëè ñóùåñòâóåò ãîìåîìîðôèçì h : S1 → S1 òàêîé, ÷òî hf = f ′h. Ìàéåð [2]ïðèøåë ê âûâîäó, ÷òî ãðóáûå äèôôåîìîðôèçìû (îáîçíà÷èì èõ ìíîæåñòâî ÷åðåç G) èìåþò î÷åíü ïðîñòóþ äèíàìèêó, êîòîðàÿ ñ ñîâðåìåííîé òî÷êè çðåíèÿ ìîæåò áûòü îïèñàíà ñëåäóþùèì ïðåäëîæåíèåì (ñì., íàïðèìåð, [10]). Êîìïîíåíòû ñâÿçíîñòè ìíîæåñòâà Wωsi \ ωi íàçûâàþòñÿ óñòîé÷èâûìè ñåïàðàòðèñàìè ñòîêà ωi, à êîìïîíåíòû ñâÿçíîñòè ìíîæåñòâà Wαui \ αi íåóñòîé÷èâûìè ñåïàðàòðèñàìè èñòî÷íèêà αi
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