# Rn-actions on n-dimensional manifolds

last updated: 18/jan/2012

This is a particular case of integrable non-Hamiltonian systems that my student Minh is working on with me for this thesis. We want to study such systems topologically. A real integrable system of type $(n,0)$ is nothing but a $\mathbb{R}^n$-action (generated by a family of $n$ commuting vector fields) on a $n$-dimensional manifold.

Commented bibliography for the problem:

E.L. Lima, Common singularities of commuting vector fields on $2$-manifolds. Comment. Math. Helv. 39 1964 97–110.

Review by B. L. Reinhart: The author proves that every continuous action of a real vector space on a compact 2-manifold with nonzero Euler number has a fixed point. This was previously announced [Bull. Amer. Math. Soc. 69 (1963), 366–368; MR0149499 (26 #6986)]. He also shows by example that the simplest nonabelian Lie group, the affine group of the line, can act on the two disc without fixed points. In proving the theorem, he uses some lemmas of independent interest concerning the limit of periodic points of an action of the line, and the maximum number of nontrivial minimal sets of an action of the line. (The latter result is attributed to Peixoto.)

This paper of Lima is freely available in electronic format (project SEALS, Switzerland: http://retro.seals.ch).

The Poincaré-Bendixon theorem says that a relatively compact orbit of a differentiable vector field on a 2-dimensional plane converges to either a fixed point or a periodic orbit called its limit cycle. Actually, Lime used the following result of Peixoto, which generalizes Poincaré-Bendixon theorem:

Lemma: A flow $\zeta: \mathbb{R} \times M \to M$ on a compact surface $M$ (with or without boundray) of genus $g$ has at most $2g-1$ distrinct non-trivial minimal sets. (Nontrivial means different from a point, a circle, and a torus).

Remark: In Lima’s case, no non-degeneracy condition is assumed, so there may be spiraling orbits (which converge to limit cycles or fixed points), and there may be also an uncountable number of orbits of the action (when the action is degenerate). Under the nondegeneracy condition, such phenomena cannot occure, so the picture will be much simpler.

FJ Turiel, An elementary proof of a Lima’s theorem for surfaces, Publicacions Matematiques, Vol. 33, Nº 3, 1989, 555-557.

Contains a simple proof of Lima’s theorem about the existence of a fixed point. Turiel’s proof is based on the construction of a regular 1-dim foliation (which is tangent to the orbits of the R2-action) if there is no fixed point. The existence of such a foliation implies that the Euler class is zero.

Available online at: http://dialnet.unirioja.es/servlet/articulo?codigo=2914745

C. Camacho, Morse-Smale R2 -actions on two manifolds. Dynamical Systems, Editor M. M. Peixoto, Academic Press, (1973), 71–75.

D. Martinez Torres, Global classification of generic multi-vector fields of top degree, Journal of London Math Soc 69 (2004), 751–766.

Why related ? Because if one takes the wedge product of $n$ vector fields of our family, then it becomes a multi-vector field (singular contravariant volume form), whose singular locus is the singular locus of our family (or of the corresponding Rn-action)

JL Arraut, C Maquera, On the orbit structure of Rn-actions on n-manifolds, Qualitative theory of dynamical systems 4 (2004), 169-180.

Review by Bis: The authors consider a locally free $C^2$-action of ${\Bbb R}^{n-1}$ on $T^{n-1}\times[0,1]$, tangent to the boundary. They prove that if there are no compact orbits in the interior, then all non-compact orbits have the same topological type. Moreover, they define a subset $A_n$ of $C^r$-actions of ${\Bbb R}^n$ on a manifold $N$ and show that if an action from $A_n$ has one orbit diffeomorphic to $T^{n-1}\times {\Bbb R}$ then every $n$-dimensional orbit is also diffeomorphic to $T^{n-1}\times {\Bbb R}$.

J. L. Arraut and Carlos Maquera, Local structural stability of actions of Rn on n-manifolds, Bol. Soc. Paran. Mat. (3s.) v. 24 1-2 (2006): 9–18.

Summary: “Let $M^m$ be a compact $m$-manifold and $\varphi\colon \bold{R}^n\times M^m\rightarrow M^m$ a $C^r$, $r\geq1$, be an action with infinitesimal generators of class $C^r$. We introduce the concept of transversally hyperbolic singular orbit for an action $\varphi$ and explore this concept in its relations to stability. Our main result says that if $m=n$ and $\scr{O}_p$ is a compact singular orbit of $\varphi$ that is transversally hyperbolic, then $\varphi$ is $C^1$ locally structurally stable at $\scr{O}_p$.”

Remark: this paper is available online.

Question: does “transversally hyperbolic” imply nondegeneracy ?

M Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Annali di Matematica pura ed applicata, 173 (1997), 213-232.

Review by Gary Meisters: The author discusses the local and global behavior of the orbits of a pair of commuting systems in open connected subsets $U$ of the plane ${\bold{R}}^{2}$ and also on 2-dimensional compact, connected, oriented manifolds (i.e., on 2-dimensional spheres with $p$ handles). The first 15 pages of the paper are devoted to the plane case, and in the last two-and-a-half pages the author determines which compact surfaces admit commuting vector fields, and discusses the possible global behavior of their orbits … The author then summarizes, subsumes, and extends some earlier related results of N. A. Lukaševič [Differencialʹnye Uravnenija 1 (1965), 295–302; MR0197863 (33 #6023)] and M. Villarini [Nonlinear Anal. 19 (1992), no. 8, 787–803; MR1186791 (93j:34061)]. Several interesting examples are given to illustrate various possibilities, and the author obtains many interesting results under the assumptions, which he collectively refers to as (CT), that the vector fields $V$ and $W$ are $C^{\infty}$, have the same (isolated) critical points, are transversal at noncritical points, and commute. His Theorems 3.1–3.3 paraphrased: The only compact (oriented) surfaces that can have commuting pairs of (CT) vector fields are the 2-sphere and the torus; every (CT) flow on the torus is the quotient of two parallelizable flows in ${\bold{R}}^{2}$; on the 2-sphere only three (CT) phase portraits are possible.

Remark: The (CT) condition of Sabatini is somewhat similar, but different, from the nondegeneracy condition. (CT) may be understood as “center type” (i.e. elliptic type, but can be degenerate)

S Firmo, A note on commuting diffeomorphisms on surfaces, Nonlinearity 18 (2005), 1511

(Discretized version of commuting flows)

Abstract: Let $\Sigma$ be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighbourhood V of the identity map of $\Sigma$ in the C1-topology with the following property: if G is an abelian subgroup of $Diff^1(\Sigma)$ generated by any family of elements in V, then the elements of G have common fixed points. This result generalizes a similar result of Bonatti (Bonatti C 1989 Difféomorphismes commutants des surfaces et stabilité des fibrations en tores Topology 29 101–26).

What to we know / want to know ?

We will assume that the nondegeneracy condition is satisfied. Then:

1) Local structure of singularities: = direct product of hyperbolic components with elliptic components with regular components (compare: the Hamiltonian nondegenerate case)

2)If O_1 and O_2 are 2 different orbits of the Rn action, and O_2 lies in the closure of O_1, then O_2 is more singular than O_1 (has higher corank). Proof: from the local structure of singularities (compare: my work on Hamiltonian singularities)

3) Minimal sets of the actions are closed orbits, i.e. tori (follows from 2: if not a closed orbit then the frontier will be “smaller” than itself)

4) Toric degree: the same toric degree at every point ! (Proof: together with point 2) What does it mean ? It means that there is a subaction which is toric, and a complementary action which is hyperbolic. For example, if there is an ellipti singular point then there is a global T1-action

5) Semi-local normal form around a closed orbit:

Direct product divided by finite group action (compare: topological decomposition theorem for Hamiltonian singularities, and also normal form theorem along a singular orbit in the Hamiltonian case)

6) classification in the 2D case ?

Toric degree = 2 –> only T2 without singularities (if elliptic singularity then toric degree =1, if hyperbolic singulariy then toric degree = 1 or 0)

Toric degree = 1 –> S2 (with 2 elliptic points, and maybe several circles of hyperbolic corank-1 points) or T2 (only circles of hyperbolic corank-1 points then)

Toric degree = 0 –> only hyperbolic singularities, and every 2-dimensional orbit is topologically a disk. Can the compact ambient surface have positive genus ?

7) Degenerate examples (in contrast with the nondegenerate case)

8) Dimension 3 ? Classify by toric degree ?

Toric degree = 3 –> only T3

Toric degree = 2 –> what manifolds can admit ?

Toric degree = 1 –> ?

Toric degree = 0 –> can say anything at all about the topology of the manifold ?

Toric degree 2 in dimension 3:

Action of R3 which descends to T2 X R action on a compact manifold, which is locally free almost everywhere.

Orbits of the T2 action –> T2, T1 and pt

The R-action will move T2 orbits to TR2 orbit, T1 orbits to T1 orbits, and points to points.

No fixed point possible, because a fixed point would ave 1 elliptic + 1 hyperbolic components –> toric degree = 1 and not 2.

Smallest orbits of the T2xR1 action are 1-dim or 2-dim.

The case of a 1-dim orbit of the 3-dim action: it must be transversally elliptic, otherwise it would have 2 hyperbolic components –> toric degree would be 1. So these 1-dim orbits are also (singular) orbits of the T2-action.

2-dim singular orbits are transversally hyperbolic (the only choice, because codimension = 1) –> the orbit = T2 (could be original T2 / finite groupe ?)

regular orbits = T2xR
2-dim singular orbits = T2
1-dim singular orbits = T1

Since orbits of type T2xR are not closed –> exist singular orbits (in the closure)

Remark: the action is free almost everywhere (if not free –> descends to the action of a factor-group)

Subcase: No orbit of type T2: 1 orbit of type T2xR + 2 orbits of type T1 at the 2 ends (each orbit of type T1 “closes” only 1 end so needs exactly 2) –> lense space (= gluing of 2 solid tori)

Subcase: No orbit of type T1 -> the action of T2 is locally free, and free almost everywhere. The isotropy group of each T2-orbit is at most Z2. If it’s Z2 then it’s a singular orbit for the T2xR action. Needs at least 2 singular T2 orbits of T2xR action (1 is acctracting and one is repelling for the R-action).

Sub-subcase: isotropy group = 0 for all singular orbits T2: then the only topological invariants are the number of singular orbits and the monodromy ? The ambiant manifold is a T2-bundle over T1 (each fiber is an orbit of the T2-action), with possibly non-trivial monodromy. The number of singular T2 is even (for each repelling there is an attracting), and they are placed in the circular order A-R-A-R-A-R-… There are some smooth invariants (because the normal hyperbolic vector field at one singular T2 ceases to be normal hyperbolic at the next singular T2).

Sub-subcase: isotropy group of a singular T2 is Z2. Then the manifold is non-orientable (already near the singular orbit it’s non-orientable). It will then have exactly 2 orbits T2 with isotrpy Z2 at the 2 ends of the fibration by T2. The number of singular T2 between these two ends is arbitrary. There is a discrete invariant coming from the comparison of the 2 isotropy groups (they are 2 subgroups of T2 isomorphic to Z2, which may coincide or be different).

Subcase: There are orbits of type T1.
Subsubcase: there is also an orbit of type T2 with isotropy Z2. Then this T2 + the orbit of type T1 are the 2 ends of the singular foliation by T2 (the orbit space of the T2-action in this case will be an interval). Between these 2 ends there may be a number of singular orbits of type T2

Remark: Can classify by first looking at the orbit space of the T2-action. This orbit space of 1-di, so it’s either a circle, or an interval. If it’s a circle, then each orbit is T2 with trivial isotropy. If it’s an interval, then the orbits at the 2 ends can be either T1 or T2 with isotropy Z2.

When the 2 ends are T1 –> lense space
When 1 end = T1 and 1 end = T2 –> non-orientable quotient of a lense space by Z2.

Remark: in any case, we can find a subgroup T1 in T2xR whose corresponding T1-action is locally free –> the ambient manifold is a closed Seifert space.

Need references on Seifert here !

On the notion of rank

For simple Lie algebras –> Cartan rank

for compact Lie group –> dimension of maximal torus

Milnor’s rank for manifold = max number of commuting vector fields which are linearly independent *everywhere*

Milnor’s notion seems a bit strange ?!

Toric degree = maximal dimension of a torus which can act on the manifold by an action which is free almost everywhere.

No direct reference for Milnor’s rank, but the definition is available in many places, e.g. the book by Camacho: C. Camacho, Geometric theory of foliations, 1985.

Why impose conditions on singularities ?

Because, if not, they can be ver degenerate, and will not give any interesting information about the ambient manifold.

Theorem: for any open set U in Rn there exist n commuting vector fields on Rn which are independent almost everywhere in U, and equal to zero outside U.

This theorem may be “folkloric” and I don’t know who first proved it, but at least one can find it in a paper by Alan Weinstein (about Poisson geometry of real simple  Lie algrebas)

Proof: Fill (a dense subset of U) by a union of disjoint balls –> solve the problem for each ball. Fix a diffeo F from Rn to ball , which is “very flat at infinity” (do it in a SO(n) equivariant way –> a function from R+ to bounded interval; function given by an intergal formula, with the integrand a function which decreases very fast at infinity). Project the constant vector fields on Rn to the ball by this map F –> get a set of commuting vector fields on the ball which are flat at the boundary of the ball.

Toric degree 0 in dimension 2

Question: can a surface of genus > 1 admit a nondegenerate pair of commuting vector fields ?

The answer seems to be yes, i.e. there is no obstruction if the system is hyperbolic. But need a concrete construction.

Toric degree 1 in dimension 2.

Proposition: If toric degree = 1 in dimension 3, then the orbits can have the following types:

dim 0: pt with 1 elliptic + 1 hyperbolic components

dim 1: T1 transversally hyperbolic; R1 transversally elliptic

dim 2: T1 x R1 transversally hyperbolic

dim 3: T1 x R2 regular orbit

(The type R2 is impossible. because it’s comdension is only 1 -> transversally has to be hyperbolic, but then toric degree = 0)

* pt: 2 R1, 1 T1xR1, 2 T1xR2

* T1: It can be twisted or non-twisted

Non-twisted T1: 4 T1xR1, 4 T1xR2

Twisted T1 (isotropy = Z2): 2 T1xR1, 2 T1xR2

* R1: 1 T1xR2

* T1xR1: It can be twisted (isotropy=Z2) or non-twisted

Non-twisted T1xR1: 2 T1xR2

Twisted T1xR1: 1 T1xR2

Observation 1:

If no fixed point, and the T1 action is free –> quotient by this T1-action –> reduction to the case of toric degree 0 in dimension 2 –> need result for dimension 2.

If no fixed point but the torus action has isotropy in some places –> quotient is not a manifold but an orbifold with a hyperbolic system on it –> the study is  a bit more complicated

In both of the above cases, the manifold is a Seifert fibration.

Observation 2:

Look at the closure of the union of the orbits of type T1xR1 ?

Smaller orbits = pt and T1.

It will be a union (maybe with transversal intersections) of S2, T2, their non-orientable versions. On each of these 2-surface we have a system of toric degree 1 in dimension 2.

Question:

What about the characteristic classes / topological invariants of the ambient manifold ? Rationally elliptic manifolds  ? (See Sullivan ? Taimanov’s similar result for Hamiltonian systems ?)

Toric degree 0 in dimension 3:

Too many possibilities ?

Conjecture/question: Any closed orientable 3-manifold admits a hyperbolic (toric degree = 0)  integrable system of type (3,0) ?

Idea of proof/disproof ? Knot surgery ? (Similar to the proof of the fact that any 3-manifold admits a contact structure ?)

Question:

Any results on manifolds which admit non-trivial Tk-actions ? See Moscow guys ?

References on torus actions:

Buchstaber/Panov / Orlik / …

Try this one: math.stanford.edu/~anton/TorusActions_BP.pdf

Orlik: Actions of the torus on 4-manifolds, I & II

Pao: topological structure of 4-manifolds which admit effective torus actions, Transactions AMS 1977

Haefliger, Actions of tori on orbifolds, Annals of Global analysis and geometry 1991.

Pak, Actions of Tn on (n+1) manifolds, PAcific J Math 1973

Melvin, On 4-manifolds with singular torus actions, Math Annalen 1981

Fintushel, Classification of circle actions on 4-manifolds, Transactions AMS 1977

Edmonds, A survey of group actions on 4-manifolds, Arxiv 2009

McGavran, Tn actions on simply-connected (n+2)-manifolds, PAcific J Math 1977

M. Ho Kim, Toral actions on 4-manifolds and their classification, Trans AMS 1993

etc. (lots of papers, need to browse them and make a quick overview of available results)

Hyperbolic case (toric degree= 0) in dimension 2

The closure of a 2-dimensional orbit in the universal covering:

Theorem: Let k be an arbitrary natural number greater or equal to 3, and (a1:b1), (a2:b2), …, (ak:bk) is an arbitrary family of non-zero pairs of real numbers such that their respective projective real numbers are distinct and are  in circular order such that the angle between any 2 consecutive pairs <\pi. Then, up to systsem-preserving diffeomorphisms, there exists a unique (2,0) system of hyperbolic type in a neighborhood of a k-gone with the following properties:

1) The interior of the polygone is a 2-dimensional orbit of the system

2) Each vertex is a fixed point, and each edge is an 1-dimensional orbit

3) a_iX1 + b_iX_2 vanishes on the edge numero i and has eigenvalue = 1 there

(1 and X_2 are the two commuting vector fields which generate the system).

Proof: can use surgery.

Global picture on a closed surface:

* Loops on the surface: each loop is smooth and simple (without self-intersections) and corresponds to a pair of numbers (a,b) such that aX_1 + bX_2 vanishes on the loop and has eigenvalue = 1 on the loop

* The intersections of the loops are transversal (an no 3 loops can have a common point)

* The domains (= connected components of the surface minus the loops) are polygones (edges are pices of loops) with at least 3 angles.

* The families of paires of numbers (a,b) for each polygone lie in a circular order as in the above theorem.

Remark: the manifold has an orientation, and the circular order has the same orientation ?

Theorem: Any such a picture can be realized, unique to up isomrphisms.

Question: What about the orientation ?

Remark: the orientation given by $X_1 \wedge X_2$ reverses everytime one crosses a loop. The ambient manifold can be non-orientable, it’s still OK.

Examples of construction

On S2: Cut S2 into 8 pieces by 3 loops (2 loops would not be sufficient). Each piece is a 3-gone.

Call the 3 loops A, B, C. For A: X_1 = 0 with eigenvalue 1, for B: X_2 = 0 with egenvalue 1, and for C: X1+X2 = 0 with eigenvalue -1 (i.e. -X1-X2 will have eiganvalue 1) . Everything is fine, the system can be constructed. (The corresponding pairs of numbers are: (0,1), (0,1), (-1:-1) which are in circular order, with the angle between 2 consecutive pairs < \pi).

********

Aritcle project:

NTZ & NVM

Topology of nondegenerate Rn-action on n-dimensional manifolds

1) Introduction

2) Degenerate singularities

3) Toric degree

4) Actions with hyperbolic singularities on surfaces

5) Actions in 3-manifolds

6) Actions in 4-manifolds

References:

(and reasons why they are included)

Bates & Cushman (2000 ?) -> non-Hamiltonian integrability

Bogoyavlenskii (1996 ?)  -> non-Hamiltonian integrability

Stolovitch (Publ. IHES 2001 ?) -> non-Hamiltonian integrability

NTZung (Math Res Lett 2002) -> non-Ham int

NTZung (arxiv 2011) -> local structure of non-degenerate singularities (general case, here use particular case of type (n,0) )

NTZung (Compositio 1996) –>  semi-local structure of singularities, construction of torus action, the idea that  the  toric degree invariant can be read off from the local structure of any point, the topological decomposition thm. –> non-Hamiltonian analogs of these results

Vey (1979) ? and Eliasson (1990 ?) –> Vey-Eliasson thm (the local structure thm is an analog of this thm for Rn-actions)

Miranda-Zung (2004) -> normal form of a neighborhood (semi-local normal form thm for Rn-action is an analog of MZ thm)

Bolsinov/Jovanovic/Fedorov/… -> examples of integrable non-Ham systems (used for motivation)

Lima 1964 -> mention his theorem about existence of fixed point (in our nondegenerate case the existence of such points becomes more obvious).Turiel (for a simple proof of Lima’s thm)

Sabatini 1997 -> his CT condition measn “center type”, which corresponds to toric degree 1 or 2 in our case (without being necessarily nondegenerate) without any hyperbolic dim 1 orbit -> his picture (3 possibilities on S2) can be seen also as a particular case of our classification on S2 with toric degree 1.

Seifert 1933 -> for Seifert manifolds (3-man foliated by T1)

Raymond & ? –> Book on Seifert fibrations (2010 ?) (may cite this instead of Seifert’s original paper)

Orlik & Raymond 1970 & 1974 -> T2-actions on 4 manifolds & topology of manifolds admitting such actions. (corresponds to the case of toric degree >= 2 in dimension 4)

Fintushel 1977 -> T1 actions on 4-manifolds (-> toric degree 1 on dimension 4, already gives topological obstructions on the ambient manifold)

### 4 comments to Rn-actions on n-dimensional manifolds

• Rooney

On the website of IMT http://www.math.univ-toulouse.fr/1-17731-Fiche-professionnelle.php?idFiche=288

your website ends with .com instead of .net. It need to be updated!!!

Reflection principle : near each hyperbolic submanifold (given by X=0 where X is a generator of the Rn action)there exists an involution which preserves the system, which is identity on this submanifold.

Nondegenerate actions of R2 with toric degree 0 on surfaces.

Theorem
1) On S2 there exists a nondegenerate R2 action of toric degree 0, which as 8 orbits of dimension 2. The number 8 is also the minimal number possible (i.e. any action of toric degree 0 will have at least 8 orbits of dimension 2)
2) For any g >= 1, on a closed orientable surface of genus g there exists an action of R2 of toric degree 0 which has exactly 2 orbits of dimension 2. The number 2 is also minimal possible.

Proof.
Existence:
1) Cut S2 into 8 pieces by 3 loops so that each piece is a trigone. According to the normal form theorem ???, there is a action on the closure of a trigone generated by two vector fields X1 & X2 such that X1, X2, and -X1-X2 correspond to the 3 edges. Extend X1 & X2 on the whole S2 by symmetry so that they become invariant wrt the 3 involutions on S2 (each loop is the fixed point set of an involution)

2) In the case of genus >= 1, just need 2 involutions to cut it into four (2g+2)-gones. Construct an action on one of these (2g+2)-gones, and extend it to the whole surface by the reflection principle (so that it becomes invariant with respect to the 2 involutions). Now make the same picture for genus 2g-1, then mod out by the free Z2 action (= product of the 2 involutions) -> get a surface of genus g, and the system on it has only 2 orbits of dimension 2.

Minimality:

1) 1 loop on S2 –> 2 disks with smooth boundary.
2 loops -> there are at least 4 pieces which are 2-gones. Each 2-gone needs to be cut in order to obtain polygones (with at least 3 edges), so in total we have at least 4×2=8 pieces, i.e. 8 orbits of dimension 2 of the action.

2) 2 is minimal, because when one crosses over an edge, the orientation given by X1 wdege X2 changes -> 2 regions which share an edge can not be the same on a orientable surface.

(CT) condition of Sabatini
2 commuing v.f. X, Y in dim 2 such that:
1) have only isolated singular points
2) X(p) = 0 if and only if Y(p) = 0
3) X and Y are independent at every non-zero point (i.e. no rank-1 point)

note that (CT) condition is very different from the condition that toric degree >=1

in fact, there exists system which satisfy (CT) condition, but which doesn’t admit a T1 action. example of S2 with just 2 orbits: a point and the rest (see Case 1 in the thm 3.3 of Sabatini 1997)

but:
if CT and nondegenerate then toric degree >=1

Morse-Smale condition

Morse-Smale $R^{2}$-actions on two-manifolds. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 71–74. Academic Press, New York, 1973.

Question: Morse-Smale condition for Rn-actions ?

Camacho defines Morse-Smale condition for R2 actions as follows:

1) the set of nonwandering points of the action is an embedded cell complex consisting of finitely many singular orbits

2) The compact orbits are hyperbolic

Camacho shows that Morse-Smale R2-actions exist on any compact orientable 2-manifold, and they are structurally stable. Moreover, Morse-Smale inequalities are given for such actions. Finally, it’s shown that the set of structurally stable R2-actions on any closed surface is not dense in the set of all R2-actions wrt the Whitney C-infinity topology. (source: Zentralblatt)

smooth vs topological classification ?

Definition: Two actions are smoothly equivalent if there is a smooth intertwining diffeomorphism $Phi$ which sends one action to the other. They are called topologically equivalent if $Phi$ is only required to be a homeomorphism.

Difference between smooth and topological ?

In the case of a closed 2-dim orbit in dimension 2 hyperbolic (polygone), smooth –> the whole set (a1,b1), … , (an,bn) up to a simultaneous transformation by a linear isomorphism of R2 (changes of basis of the R2-action). For topological equivalence, only the projective numbers (a1:b1), …, (an:bn) count.

Global invariants ?
Question: Any global smooth/topological incariants, besides the obvious ones ? Any “periods” ? Give a complete classification, at least in the 2-dimensional case ?