On Some Invariants of Dynamical Systems on Two-Dimensional Manifolds (Necessary and Sufficient Conditions for the Topological Equivalence of Transitive Dynamical Systems)

Aranson, Samuil; Гринес, Вячеслав Зигмундович

doi:10.1070/sm1973v019n03abeh001784

Cited by 32 publications

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“…The full topological invariant of a highly transitive flow on a closed hyperbolic orientable surface is a homotopic rotation class (to within the action of the group of covering transformations), introduced by Aranson and Grines in 1973 [27]. As the following theorem shows, the type of asymptotic direction of a trajectory influences some dynamical properties of the trajectory.…”

Section: Theorem 25 ([40]) For Every Irrational a There Is A Cherrymentioning

confidence: 99%

“…If the surface M is endowed with a geometric structure of a Riemannian 2-manifold of constant negative curvature -1, then tile full topological invariant of a highly transitive flow carl be represented as a geodesic framework [27,28,74,75]. This representation of the full topological invariant is more convenient because the set of geodesic laminations can be endowed with the structure of a topological space.…”

Section: Theorem 25 ([40]) For Every Irrational a There Is A Cherrymentioning

confidence: 99%

“…This results in a new topological invariant, namely, a homotopic rotation class, introduced by Aranson and Grines in 1973[27].Before[27,281, some homological invariants of surface flows (see See. This allows us to define a Poincar6 rotation number for flows on a torus (a closed surface of genus 9 = 1).…”

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confidence: 99%

“…Lemma 2.2 ([27]). Let it, ['2 be two different lifts of the nontrivial a~(c~)-recurrent trajectory l. Then ~(~)(ll) # ~(~)(i2).…”

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confidence: 99%

“…The situation is quite dilferent on the orientable closed surface M of genus g >_ 2.Theorem 2.4 ([27]). Nielsen's results imply that the homotopic rotation orbit is topologically invariant.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Qualitative theory of flows on surfaces (A review)

Aranson¹,

Zhuzhoma²

1998

J Math Sci

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PrefaceThe geometric study of flows on manifolds has a long history in mathematics, even though it was not considered as a special field until it appeared in the works of Poincar4, Lyapunov, and Bendixon. After that, the subject developed rapidly, and, at the present time, it is the focus of extensive research.It is impossible to touch upon all sides of the geometric theory of flows in one article. The purpose of this review is to present some aspects of the geometric theory of flows on 2-manifolds (of course, within the frame of our knowledge). Our main interest is the topological structure and the topological equivalence of surface flows, asymptotic properties of semitrajectories and the closing lemma, etc. (see the contents).However, our article was not written for specialists only; we have also attempted to provide an introduction to the subject.We would like to thank the many mathematicians who took part in valuable discussions of problems of the theory of dynamical systems, and especially D. Anosov, I. Bronshtein, V. Grines, G. Levitt, M. Malkin, Shigenory Matsumoto, I. Nikolaev, and A. Zorlch.

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Section: Theorem 25 ([40]) For Every Irrational a There Is A Cherrymentioning

confidence: 99%