Flows on 2-dimensional manifolds: an

overview

Igor Nikolaev∗ Evgeny Zhuzhoma†

CRM-2600April 1999

∗CRM, Universite de Montreal, Montreal, H3C 3J7, Canada; nikolaev@crm.umontreal.ca†Dept. of Mechanics and Mathematics,University of Nizhny Novgorod,23 Gagarin Ave, Nizhny Novgorod,603600

Russia, zhuzhoma@mailcity.com

Abstract

This is a monograph accepted to the Springer-Verlag series ”Lecture Notes in Mathe-matics”. According to the copyright agreement the authors are not advised to post theentire text via Internet. We do so, however, for the Introduction, the Memoirs and theBibliography. The volume contains 300 pages.

Resume

Le present est un livre accepte par Springer-Verlag dans serie ”Lecture Notes in Mathe-matics”. Selon l’accord du droits d’auteur, nous ne pouvons pas presenter le text integralsur Internet. Pourtant, nous le faisons pour l’Introduction, les Memoires et le Liste dereference. Le volume est 300 pages.

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To the memory of E. A. Leontovich–Andronova

and

To our parents

Foreword

Time–evolution in low–dimensional phase spaces is a fundamental chapter of Modern Dynamics.Despite more than 100 years of brilliant history, the inflow of new ideas in this area continues. Bynow one can clearly see only the lower levels of this giant ”tree” planted by Poincare, while the ”top”is hardly observable.

The objective of this survey is to give an up-dated, detailed picture in the case when dimensionof the phase space does not exceed 2; in other words, continuous flows on 2-dimensional manifoldsare considered. Moreover, we adopt a ”geometric” philosophy: the less analysis is used to obtainthe phase portrait of dynamical system, the better.

The necessity of such a volume is evoked primarily by absence of the study which covers possiblyall spectrum of emerging problems. Recent monograph of Aranson et al. [28] is of introductorynature and is addressed mostly to the students; for the people working at the edge of research thesubject requires a wider treatment. We hope, however, that this volume inherits the elementarystyle of [28] and will be helpful both to the advanced students and researchers.

Secondly, certain old results have been revised recently using statistical, algebraic and combi-natorial methods, the possibility which seemed unrealistic earlier. We tried to include all of them,paying special tribute to the classical results.

We did not hesitate to make historical comments, sometime bold, but not intentional. We bringour excuses to people whose names are omitted or unproportionally cited; we thank those whobrought to our attention their contributions.

Each chapter of the book is accompanied by the exercises and open problems, the old and thenew, which are believed to stimulate further development of the area.

Around 1886 H. Poincare was fascinated by non–trivial recurrent trajectories on the 2–dimensionaltorus, cf [199]. He called such trajectories ”non–closed Poisson stable”; because ”stable”has differentmeaning today, they are called ”recurrent” with an accent on their ”self–approaching” nature.

There are no non–trivial recurrent trajectories in the plane, but they are usual in the 3–dimensional space. In this sense, flows on surfaces occupy an intermediate position between flowsin R2 and flows in R3: the topology of surface imposes certain constraints on the behavior of flowmaking theory non–trivial, but manageable.

Qualitative methods preveal in the works of I. Bendixson (‘Sur les courbes definies par desequations differetielles’, Acta Math. 24, 1-88) and A. M. Lyapunov (‘Probleme general de la stabilitedu mouvement’, Soc. Math. Kharkov, 1892, translated in: Ann. Fac. Sci. Toulouse 9 (1907), 203-474), who applied Poincare technique of ”cross–sections” to the study of isolated singular points.Under rather general assumptions the analytic methods have been eventually abandoned in favorof purely geometric reasoning. In particular, Bendixson proved that if a singularity has a finitenumber of separatrices then they split the neighborhood of singularity into the sectors of threetypes: elliptic, parabolic, hyperbolic.

Bendixson’s work was crucial in the classification of limit sets of individual trajectory. To honorits creators, this theory is called ”Poincare–Bendixson”. Progress in the Poincare–Bendixson theoryis connected with the names of Kneser [133], Leontovich and Maier [142], [152]. On the existenceof quasiminimal sets, see [19], [36], [153], [169], [212].

Poincare distinguishes two types of non-trivial recurrent trajectories: everywhere dense in aregion and nowhere dense (exceptional). If the latter belong to minimal sets then the minimal setis called non-trivial. Such a set is locally homeomorphic to the product of the Cantor set and thesegment. Poincare constructed continuous flow on the torus which has a non-trivial minimal set;he conjectured that such a flow can be analytic.

In 1932 A. Denjoy [92] proved in negative the Poincare conjecture: he showed that it is false even

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in the class C2. Later in 1937, T. Cherry [85] found that the conjecture is true for quasiminimal sets.Namely, he constructed an analytic flow on torus with nowhere dense quasiminimal set consistingof the hyperbolic saddle and non–trivial recurrent semi–trajectories which are everywhere dense inthe quasiminimal set.

In 1964 A. Schwartz [216] generalized Denjoy’s theorem; he proved that C2-flows on compactsurfaces can have minimal sets only of three types: equilibrium point, periodic trajectory or wholesurface M (in this latter case M = T 2). The converse of the Denjoy-Schwartz’s theorem has beenobtained by D. Neumann [170] and C. Gutierrez [113].

It is noteworthy that structure of the Cherry’s example was unclear in the time of Cherry.Cherry investigated only particular cases leaving the general problem open. Due to the works ofAranson [26], de Melo, Martens, Mendes & van Strien [159] the structure of the Cherry flows isbetter understood today. In the maximal generality, however, the Cherry’s problem is still open.

Note that progress in this area is due to the advances in 1-dimensional dynamics, stimulatedrecently by the works of J. Ch. Yoccoz [252].

In whole, the 30-ties were extremely fruitful for the qualitative theory. In an endeavor tounderstand self-oscillations in the electric circuits, A. A. Andronov realized that adequate languagehere should be a ”geometric” one – those elaborated by Poincare, Lyapunov and Bendixson. In thecollaboration with Lev Pontryagin [5], he introduced the notion of ”rough” systems and studied thestructure of ”rough” systems in the plane. Andronov’s work inspired development of the hyperbolictheory of dynamical systems of S. Smale as well as stability theory of geodesic flows of D. Anosov.

Later on, M. M. Peixoto [193] generalized Andronov-Pontryagin results to the orientable closedsurfaces M . Peixoto proved density of structural stable vector fields in the space of all C1 vectorfields on M .

Working at the Andronov’s department (that time a tiny institution located in the downtownof Nizhny Novgorod), A. G. Maier and E. A. Leontovich began an active research in the geometrictheory of differential equations. Together with A. A. Andronov, E. A. Leontovich introduced a

notion of ”degrees of non–stability” for the flows [2]. Interesting results in this direction have beenobtained by Aranson [18], Sotomayor [231] and Teixeira [237].

By the end of 30-ties Maier proved series of theorems which became a framework of Poincare-Bendixson theory for orientable surfaces of genus g ≥ 1. He proved that the topological closure ofnon-trivial recurrent trajectory is a quasiminimal set where every trajectory is dense. From thispoint of view, it is easy to obtain a list of limit sets of individual trajectory. Estimates on thenumber of quasiminimal sets on orientable and non-orientable surfaces have been given by Maier[152], Aranson [19] and Markley [155].

Maier’s works rose an idea of ”spectral” decomposition of flow into the ”simpler” flows. By”simpler” one understands either flows without quasiminimal sets, or flows with exactly one quasi-minimal set. First type of flows is called ”regular”and the second is called ”chaotic”. Under ”spectraldecomposition” one understands a surgery of surface with the help of special closed curves so thaton every new component flow has the ”simplest” structure. G. Levitt proved the existence of suchdecomposition for flows with saddle equilibria [145]. For wider class of flows the decompositiontheorem was established by C. Gardiner [106].

Again to 30-ties date back two papers of A. Weil [247], [248] in which he gave a geometricinterpretation to ”rotation numbers” of Poincare. Weil’s method allowed to generalize ”rotationnumbers” to the surfaces of higher genus. According to Weil the differential equations are irrelevantin the study of asymptotic behavior of trajectories. He formulated his theorem for arbitrary curveswhose image at the universal covering tends to the absolute. For curves on the hyperbolic surfacesWeil conjectured similar statement.

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Due to unclear reasons Weil did not publish the proof of his statement, and Weil’s idea wasabandoned till 60-ties. By that time Weil’s construction got attention, independently, by D. Anosovand N. G. Markley. Regretfully, the bulk of Markley’s results remained unpublished (except [153]-[156]), the author pleased himself with thesis [155]. Anosov shifted slightly the accents in the Weil’sconjecture and showed that trajectory of the covering flow either remains in a bounded region of theuniversal covering (in this case the behavior is ‘Poincare-Bendixson’s), or goes to the absolute withan asymptotic direction provided flow has a finite number of equilibria. Later on Anosov generalizedthis result (see [7], and an overview in [13]). He also formulated series of questions on existenceof the asymptotic directions for special curves, leaves of foliations, laminations, etc. Anosov askedabout deviation of curves from the geodesics with same asymptotic directions. We call such sort ofproblems ”Anosov-Weil”.

Back in 50-ties, the geometric theory of flows has been largely influenced by ergodic ideas dueto Birkhoff, Kolmogorov and Rokhlin. In this direction, there were obtained several importanthomology and cohomology invariants. The most familiar one is Schwartzman’s asymptotic cycle[218]. Modification of the Schwartzman’s cycle gives Katok’s fundamental class [129], [131] whichis a ”local” invariant of the topological and smooth equivalence of the surface flows. Let us signala close idea of Zorich’s cycles [256]. According to the Birkhoff’s ergodic theorem, classical rotationnumber of Poincare may be also represented as an asymptotic cycle.

Despite the fact that flows on compact surfaces are statistically ”weak”(L. S. Young [254] showedthat the metric as well as topological entropy vanish in the case of surface flows), almost all surfacesmay carry smooth ergodic (Blokhin [66]) and mixing (Kochergin [136]) flows. Recently, Sinai andKhanin [227] showed that ”typical” transitive component of the area–preserving flow on torus ismixing.

In the last decades a lot of activity was connected with the operator algebra methods. C∗-algebras are believed to describe completely ”dynamics” of the system, the main problem here ishow to ”read-off” pertinent information. In many cases it can be done effectively as it was shownin the works of E. G. Effros and C. L. Shen on the irrational rotation algebras, see e.g. [98]. Thisapproach gives a deep insight to the old problems, in particular to the problem of description ofasymptotic properties of trajectories.

It is clear that methods used for the ”regular” and ”chaotic” flows are principally different.For ”regular” flows the invariants of combinatorial nature are required, while for ”chaotic” flowsthe asymptotic ones are important. Standard examples are Peixoto graphs and Poincare rotationnumbers, respectively. For flows in the plane and on the sphere, Leontovich and Maier introduced ascheme – complete topological invariant which boils down to description of the special trajectoriesand their mutual position; cf [142],[143]. For Morse–Smale flows the complete invariant is graph withthe rotation system [176]. Rotation system is the cyclic order of the adjacent edges given in everyvertex of the graph. Two invariants (scheme and graph) are brought together in the orbital complexproposed by T. O’Brien and D. Neumann [172]. A combinatorial classification of non–wanderingflows was obtained in [181].

It is unclear to whom the credit of classifying minimal flows on torus should be assigned. Forsuch flows complete invariant is given by the Poincare rotation number, defined modulo actions ofthe modular group. Using Markley’s classification of the Denjoy’s homeomorphisms of the circle,Aranson, Medvedev & Zhuzhoma classified Denjoy and Cherry flows on torus and sphere [44], [38],[39], [40].

For ”chaotic” flows on orientable, closed, hyperbolic surfaces Aranson & Grines [30] introduceda homotopy rotation class, which represents asymptotic direction of non-trivial recurrent semi-trajectories. For the irrational flows this invariant is complete, again up to the action of deck

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transformations. With the help of the same invariant Aranson & Grines classified non–trivialminimal sets on surfaces [31].

Apparently under the influence of Thurston’s works, G. Levitt proposed an invariant of ”chaotic”flows – so-called geodesic framework. He obtained a Whitehead classification of flows on the hy-perbolic surfaces generalizing the Aranson & Grines result [30]. Note that in different terms, the”geodesic frameworks” were constructed in 1978 again by Aranson & Grines [31]. They establishedremarkable fact that flows with non-trivial minimal set are equivalent to flows with minimal setconsisting of geodesics.

Geodesic laminations can be endowed with a topological structure. When this is done, onecan study ”perturbations” of the geodesic frameworks [42]. In particular, Aranson, Medvedev &Zhuzhoma proved continuity of the geodesic frameworks with regard to perturbations of irrationalflows. This result is analogue to continuity of Poincare rotation numbers relatively perturbationsof torus flow.

In 40-ties W. Kaplan [125]–[128] initiated ”global” study of flows on the non–compact surfaces.Kaplan obtained a complete topological invariant for the fixed–point–free flows in the plane; heshowed that the trajectories of such flows are the level curves of harmonic functions relative acomplex structure on the plane. (In this context, Kaplan formulated five problems, four of which areopen sofar.) Synthesis of Kaplan’s invariant and Leontovich–Maier scheme gives Markus’ invariant[157], which is a complete invariant of flows with fixed points in the plane.

Caused by the study of Hamiltonian flows in higher dimensions, there is a growing interesttowards flows on non–compact surfaces of finite and infinite genus. Hector–Beniere proved theexistence of minimal flow on a non–compact surface of positive genus [62]; in the series of examplesdue to Neumann [171], Inaba [124], Beniere–Meigniez [63] and Aranson–Zhuzhoma [50] it is shownthat to large extent flows on non–compact surfaces are akin to flows in the higher dimensionalspaces.

Acknowledgments. It is our pleasure to thank people who helped us in many ways. Among those areD. V. Anosov, G. Belitskii, G. Elliott, G. Forni, V. Z. Grines, J. Guckenheimer, G. Hector, T. Inaba,A. Katok, A. Ya. Kopanskii (who wrote §4.4 of this survey), R. Langevin, L. Lerman, G. Levitt,M. I. Malkin, N. G. Markley, S. Matsumoto, V. Medvedev, C. Possani, I. Putnam, P. Schweitzer,L. Shilnikov, Ya. Sinai, Ya. Umanskii and A. Zorich.

Our teachers, friends and often co-authors S. Kh. Aranson and I. U. Bronstein did much for us,both in science and in life.

Professor Dana Schlomiuk (Montreal University) arranged a financial support for the project.We are grateful to Igor Lutsenko (CRM and Princeton) for the permanent technical aid.

We thank a referee (IMPA) for useful remarks which helped us to correct some mistakes.

The second author was partially supported by RFBR grant 96-01-00236 and by the INTAS-RFBR grant 95-418.

Montreal, October 1998

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Second century of the geometric theory of flows started with a sad new: In 1997 in Nizhny Novgorodpassed away E. A. Leontovich–Andronova. The whole era is over. She was born 7 years beforePoincare’s death, when he already wrote his fundamental works on the dynamical systems. EvgenyaAleksandrovna was the wife of A. A. Andronov. She used to occupy one room with A. G. Maieras they jointly prepared Russian translation of Poincare’s major works. She was the author ofnumerous papers and several monographs on the geometric theory of flows — see, for example, [1],[2], [3], [4].

One of the authors of present survey was Leontovich’s student and visited E. A. when she wasalready retired. Up to the very death she kept a serenity, interested in developments and sorrowed onrecent disarray of science in Russia. Her favorite reading at that time varied from the perestroika’sherald ”Novy Mir” to the classics of Leo Tolstoy. As a detail, E. A. liked recalling P. Novikov’smemoirs about turbulent times of the Civil War and the Russian Revolution.

Once she was asked about her first acquaintance with A. A. Andronov. According to her ownwords, Andronov was a handsome young man. (Legends are known about his stature and physicalforce.) He often visited E. A.’s brother (a physicist and future academician) at their Moscowapartment. E. A. was reluctant of the brother’s pal and prefered to refuge in her room. By chance,they met each other at the tram stop near her house; this happy journey was a life–long... (It wastouching to see how a 90-years old woman blushed and asked: ‘Well, what for I am telling youthat?’)

Many years devoted E. A. Leontovich–Andronova to the geometric (or qualitative, by her ownwords) theory of dynamical systems. We dedicate this volume to her memory.

Authors

References

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