Actions of Baumslag-Solitar groups on surfaces

Let $BS(1,n) =$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $\TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= \TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $\TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.


Introduction and statements
An important question on group actions is existence and stability of global fixed points. For Lie group actions, it was shown by Lima [Lim64] that any action of the abelian Lie group R n on a surface with non-zero Euler characteristic has a global fixed point. This result was later extended by Plante [Pla86] to nilpotent Lie groups. On the other hand, Lima [Lim64] and Plante [Pla86] proved that the solvable Lie group GA(1, R) acts without fixed points on every compact surface.
For discrete group actions, Bonatti [Bon89] showed that any Z n action on surfaces with non-zero Euler characteristic generated by diffeomorphisms C 1 close to the identity has a global fixed point. Druck, Fang and Firmo [DF02] proved a discrete version of Plante's theorem.
This paper deals with the dynamics of actions of the solvable Baumslag-Solitar group, BS(1, n) =< a, b | aba −1 = b n >, where n ≥ 2, on closed surfaces.
In section 3, we explain the construction of these examples and exhibit a faithful smooth action of BS(1, n) on T 2 without finite orbits that can be considered as "the standard BS-action" on T 2 . More precisely, Definition 1.2. The standard BS(1, n)-action on T 2 is the action generated by : and h 0 (x, θ) = (nx, ln(n) + θ), where x ∈ R ∪ ∞ and θ ∈ S 1 .
Our first result is the following: Theorem 1.
The group < f 0 , h k > generated by f 0 and h k is isomorphic to BS(1, n).
If the rotation number of k is rational, there exist finite BS-orbits.
If the rotation number of k is irrational, there are no finite BS-orbits and the unique minimal set for the BS-action is included in ∞ × S 1 = f ix(f 0 ). Corollary 1. There exist C ∞ faithful BS-actions arbitrary C ∞ -close to the standard torus BS-action < f 0 , h 0 > that are not topologically conjugate to < f 0 , h 0 >. This implies that the standard BS-action on T 2 does not satisfy the rigidity properties described in the Farb-Franks-Shub theorem for the standard BS-action on S 1 . This property can also be compared to the rigidity result recently proved by Mc Carthy : "The trivial BS(1, n)-action on a compact manifold does not admit C 1 faithful perturbations" (see [MC10]).
Then we consider perturbed actions of the standard one. In particular, we prove that there exists either a finite orbit or a unique minimal set. Recall that a minimal set for an action of a group G on a compact metric space X is a non-empty closed G-invariant subset of X such that if K ⊂ M is a closed G-invariant set then either K = M or K = ∅.
Let C 1 and C 2 be the circles defined by C 1 = ∞ × S 1 and C 2 = 0 × S 1 . Note that both circles are h 0 -invariant.
Theorem 2. Let us consider a BS-action < f, h > on T 2 generated by f and h sufficiently C 1 -close to f 0 and h 0 respectively. Then: (1) there exists two circles C ′ 1 and C ′ 2 close to C 1 and C 2 respectively which are hinvariant. Moreover, the ω h -limit set of any point in T 2 \ C ′ 2 is included in C ′ 1 and the α h -limit set of any point in T 2 \ C ′ 1 is included C ′ 2 .
(2) the set of f -fixed points is not empty and it is contained in the circle C ′ 1 .
(3) either : (a) there exist finite BS-orbits contained in C ′ 1 , or (b) the action has a unique minimal set M which is included in C ′ 1 (and in the set of f -fixed points). Moreover, M is either C ′ 1 or a Cantor set.
We check that the "standard action" on T 2 satisfies item 3(b) but in the proof of Corollary 1 we exhibit C ∞ -perturbations of it that have a different dynamical behavior: they satisfy item 3(a). In section 6, we exhibit an example of an action with a C 1 persistent global fixed point. More precisely, we construct an action with fixed point satisfying that any C 1 -perturbation of it also has fixed point.
On the other hand, we develop a general dynamical study for faithful BS(1, n)-actions on closed surfaces. From now on, let us consider f and h two homeomorphisms that generate a BS(1, n)-action, that is, Our first "dynamical" result on the torus concerns the rotation set of f (for the definition see Section 2).
Theorem 3. Let < f, h > be a faithful action of BS(1, n) on T 2 . Then there exists a positive integer N, such that f N is isotopic to identity and has a lift whose rotation set is the single point {(0, 0)}. Moreover, the set of f N -fixed points denoted by f ix(f N ) is non-empty.
Remark 1.1. In section 3, we exhibit two diffeomorphisms F and H generating a faithful action of BS(1, n) on T 2 , where F admits periodic orbits but it does not have fixed points.
Since the group < f N , h > is isomorphic to BS(1, nN), Theorem 3 allows us to restrict our study on the torus to the case where f is isotopic to identity, the rotation set of a lift of f is {(0, 0)} and f has fixed points. In this situation we prove that there exists a BS-minimal set included in the set of f -fixed points. More precisely, we prove the more general following statement.
Theorem 4. Let X be a compact metric space and < f, h > be a representation of BS(1, n) in Homeo(X).
(a) If f ix(f ) is non-empty, then: (2) There exists an BS-minimal set included in f ix(f ). Moreover, this BS-minimal set coincides with a h-minimal set in f ix(f ). (3) If the set of f -fixed points is finite then the action admits a global finite orbit.
(b) If the set of periodic points of f , per(f ), is non-empty, then there exist a positive integer N and a BS-minimal set, M, such that M ⊂ f ixf N .
As a consequence of item (b) of Theorem 4, Theorem 3 and the fact that < f, h > is a faithful representation of BS(1, n), we have the following: Corollary 2. Let X be a compact metric space and < f, h > be a faithful representation of BS(1, n) in Homeo(X) such that P er(f ) is non-empty. Then the action of < f, h > is not minimal. In particular: (1) There is no faithful minimal action of BS(1, n) by homeomorphisms on T 2 .
(2) Let Σ be a compact surface of non zero Euler characteristic. A faithful topological BS(1, n)-action < f, h > on Σ is not minimal, provided that f is isotopic to identity.
When < f, h > is a topological action of BS(1, n) on a closed surface S satisfying that any f -invariant probability has support included in the set of f -fixed points, we prove the following: Theorem 5. Let S be a closed orientable surface and < f, h > be a representation of BS(1, n) in Homeo(S). Suppose that for any f -invariant probability measure µ, supp(µ) ⊂ f ix(f ). Then: (1) Any f -minimal set is a fixed point. The set of periodic points of f , per(f ), coincides with the set f ix(f ). For next corollary, that we prove using Theorem 1.3 of [FH06], we need the following: Definition 1.3. Let g ∈ Dif f 1 (S), an N-periodic point x 0 is called elliptic if the eigenvalues of the differential of g at x 0 , Dg N (x 0 ), have module 1.
Corollary 3. Let S be a closed orientable surface and < f, h > be a representation of BS(1, n) in Dif f 1 (S) such that: • If S has genus at least 1, f is isotopic to identity.
• If S = S 2 , some iterate of f has at least three fixed points. Then there exists a positive integer N such that : (1) Any f -minimal set is a periodic point. The set of periodic points of f , per(f ), coincides with the set f ix(f N ).
(2) Any BS-minimal set is included in f ix(f N ). In fact, any BS-minimal set is included in a subset of f -elliptic points in f ix(f N ). (1) Does it exist a faithful action of < f, h >= BS(1, n) on T 2 with h non isotopic to identity? We know that there does not exist representation of BS(1, n) into Af f (T 2 ) = SL(2, Z) ⋉ R 2 the group consisting of maps :g(x, y) = A.(x, y) + V , where A ∈ SL(2, Z) and V ∈ R 2 . (2) Does it exist a faithful continuous action of < f, h >= BS(1, n) on T 2 with minimal sets outside per(f ) ? (3) Is the product action on (R ∪ ∞) × S 1 generated by f 0 (x, θ) = (x + 1, θ) and h 0 (x, θ) = (nx, k(θ)), where k is a circle north-south diffeomorphism topologically rigid ?
In Section 2, we give definitions, properties and basic tools that we use in the rest of the paper. We exhibit examples of BS(1, n) acting on T 2 , and Theorem 1 and Corollary 1 are proved in Section 3. The goal in Section 4 is proving Theorem 3. In Section 5 we prove Theorems 4, 5 and Corollaries 2 and 3. In Section 6, we consider perturbations of the standard BS(1, n)-action on T 2 : we describe their minimal sets by proving Theorem 2. We also construct an action with a persistent global fixed point.

Isotopy class of torus homeomorphims.
We denote by Homeo Z 2 (R 2 ) the set of homeomorphisms F : R 2 → R 2 such that F (Z 2 ) ⊆ Z 2 and Homeo 0 Z 2 (R 2 ) the set of homeomorphisms F : R 2 → R 2 such that F (Z 2 ) ⊆ Z 2 and F (x + P ) = F (x) + P , for all x ∈ R 2 and P ∈ Z 2 . Note that, a lift of a 2-torus homeomorphism isotopic to identity belongs to Homeo 0 Z 2 (R 2 ). Conversely, if a 2-torus homeomorphism admits a lift F ∈ Homeo 0 Z 2 (R 2 ) then it is isotopic to identity.
Let g : T 2 → T 2 be a homeomorphism and let G : R 2 → R 2 be a lift of g. We can associate to G a linear map A G defined by : This map satisfy the following properties : (1) A G does not depend neither on the integers m and n nor on the lift G of g. In fact, A G is the morphism induced by g on the first homology group of T 2 . So we can also denote A g for A G and we will use both notations.

Definitions.
Let f be a 2-torus homeomorphism isotopic to identity. We denote byf a lift of to R 2 . We callf -rotation set the subset of R 2 defined by From now on, we use bothf or F for a lift of f to R 2 .
2.2.2. Some classical properties and results on the rotation set. Let f be a 2-torus homeomorphism isotopic to the identity and f be a lift of f to R 2 .
• Misiurewicz and Ziemian (see [MZ89]) have proved that: if there exist neighborhoods of f 1 and h 1 in the C 1 -topology such that whenever f and h are C r maps chosen from these neighborhoods and the group generated by < f, h > is isomorphic to BS(1, n), then the perturbed action is C r conjugate to the original one, that is there exists a 2.4. BS(1, n)-actions. Consequence of the conjugation between f n and f . As consequences of the group-relation h • f • h −1 = f n , we get easily the following two propositions : Proposition 2.2.
Let f and h be as in the previous proposition, then Proof of (4). Since ent(f n ) = n.ent(f ) and ent(h • f • h −1 ) = ent(f ) the possible values for ent(f ) are 0 or ∞.

Examples of BS(1, n)-actions on T 2
In this section we will exhibit examples of BS(1, n)-actions on T 2 .

Product of faithful actions on S 1 .
Let < f i , h i >, i = 1, 2 be two C 1 actions of BS(1, n) on S 1 , we construct an action of BS(1, n) on T 2 by setting : According to [GL11], there exists a finite < f i , h i >-orbit at some point y i ∈ S 1 , hence the < f, h >-orbit of the point y = (y 1 , y 2 ) is finite.
The following two sections show examples of BS-actions on T 2 without finite orbits.
3.2. Product of non faithful actions on S 1 .
We construct faithful BS(1, n)-actions without finite orbits as product of a faithful circle action and a non faithful one.
Let < f 1 , h 1 > be a faithful action of BS(1, n) on S 1 and k be a circle homeomorphism. We construct a faithful action of BS(1, n) on T 2 by setting: f = (f 1 , Id) and h = (h 1 , k). Clearly, if k has no finite orbit, there is no global finite orbit.
3.3. Actions that come from actions of the affine group of the real line. Identifying the affine real map x → ax + b with (a, b), the affine group of the real line, The Baumslag-Solitar group BS(1, n) can be seen as the subgroup generated by the elements (1, 1) and (n, 0). Let Φ : GA(1, R) → Dif f r (M) be an action of GA(1, R), the induced BS(1, n)-action is the restriction of Φ to < (1, 1), (n, 0) >.

The standard actions.
Definition 3.1. The standard action of GA(1, R) on the circle is the action by Moebius maps on the projective line, that is : This action is faithful and has a global fixed point at ∞.
Definition 3.2. The standard action of BS(1, n) on the circle is the induced BS(1, n)action, it is generated by the two Moebius maps It is faithful and has a global fixed point at ∞. Moreover f 0 has a unique fixed point at ∞ that is elliptic and h 0 has two hyperbolic fixed points : ∞ that is an attractor and 0 that is a repeller.
The orbit of a point x is explicit : All orbits are dense except the orbit of the global fixed point ∞.
Remark 3.1. Applying the change of coordinate x = tan( u 2 ) the standard GA(1, R)-action is given by : Non faithful actions.
A family of non faithful action is given by : where (ϕ t ) is any flow on the circle.
Remark 3.2. There exist actions that do not come from actions of the affine group of the real line: There exist (even orientation preserving) circle homeomorphisms which do not embed in a continuous flow (see [Zdu85]). However, the family < f (θ) = θ, h(θ) = ϕ ln n (θ) >, where ϕ is a flow, extends to actions < f (θ) = θ, h(θ) = k(θ) >, where k is any circle homeomorphism. It is easy to see that h • f • h −1 = f n and that these actions are not faithful and have the dynamics of k : 3.3.2. Actions of GA(1, R) and induced BS(1, n) on the 2-torus.
Taking the product of the standard action with a non faithful action of GA(1, R) on the circle, we get a family of faithful GA(1, R)-actions on the 2-torus : ) and (ϕ t ) is any flow on the circle.
The "extended" induced BS(1, n)-actions are the actions generated by f 0 (x, θ) = (x + 1, θ) and h k (x, θ) = (nx, k(θ)), where k is any circle orientation preserving homeomorphism. They are faithful since they are products of two actions, and one of them is faithful. Their dynamics depend on k.
The standard action of GA(1, R) on the 2-torus is the action Φ ϕ , where ϕ t (θ) = θ+t is the flow of the circle rotations, that is given by The standard action of BS(1, n) on the 2-torus is the induced action, that is the action generated by f 0 (x, θ) = (x + 1, θ) and h 0 (x, θ) = (nx, θ + ln n).
This GA(1, R)-action has no global fixed point and it has an 1-dimension circular orbit {∞} × S 1 .
This BS(1, n)-action has no finite orbit, the restriction of h 0 to ∞ × S 1 is the irrational rotation by ln n. The unique minimal set is ∞ × S 1 .
In the previous section, we have seen that these actions are faithful BS(1, n)-actions, since they are products of two BS(1, n)-actions on S 1 , where one of them is faithful.
Note that the set of f 0 -fixed points, f ix(f 0 ) is the circle C 1 := ∞×S 1 and any horizontal circle (R ∪ {∞}) × θ 0 is f 0 -invariant. The circles C 1 = ∞ × S 1 and C 2 := 0 × S 1 are h kinvariant. The restriction of h k to these circles is the homeomorphism k.
• If the rotation number of k is rational, then there exists a point in ∞ × S 1 with a h k -finite orbit.
As it is f -fixed, its BS-orbit is finite. • If the rotation number of k is irrational (for example for the standard action), there are neither fixed points nor periodic points of h k . Therefore there is no global fixed point for this action. Moreover, there is no finite orbit. The circle C 1 contains the ω h k -limit set of any point in T 2 \ C 2 and C 2 contains the α h k -limit set of any point in T 2 \ C 1 . Hence, the unique minimal set M for this action is contained C 1 . If k is minimal, M coincides with C 1 , the set of f 0 -fixed points. If k is a Denjoy homeomorphism, M is strictly contained in C 1 , the set of f 0 -fixed points.
Proof of Corollary 1.
Consider BS-actions generated by f 0 and h ǫ given by h ǫ (x, θ) = (nx, θ + ln(n) + ǫ). If ln(n) + ǫ is rational, then the restriction of h ǫ to ∞ × S 1 is of finite order and every point in ∞ × S 1 has a finite BS-orbit, this action is clearly not topologically conjugate to the standard one. But, this can occur with ǫ arbitrary small, so for a BS-action arbitrary C ∞ -close to the standard action. BS(1, n).

Other examples of actions of
In this part, we construct diffeomorphisms f , h [resp. F and H] generating a faithful BS(1, n) action on the circle [resp. the torus] where f [resp. F ] has not fixed points but it has periodic points.
It is easy to see that the group generated byf and h is isomorphic to BS(1, n).
n−1 (mod1). We claim that the group generated by f and h is isomorphic to BS(1, n).
commutes with h (and also withf ). Then commutes withf and has order n − 1. Hence, f and h generate an action of BS(1, n).
This action is faithful, since it is a well known fact that for a non faithful action of BS(1, n), f has finite order. By construction f admits exactly n − 1 periodic points of period n − 1, so f is not of finite order.
This construction provides an example of two circle diffeomorphisms f and h generating a faithful action of BS(1, n), where f has no fixed points but periodic ones.
3.5.2. On the Torus. Let f and h be the circle diffeomorphisms as below. We define two torus diffeomorphisms : The diffeomorphisms F and H generate a faithful action of BS(1, n) on the torus, F admits periodic points but not fixed points.

Isotopy class of f and rotation set.
The aim of this section is proving Theorem 3. Proof. For proving the proposition, it is enough to prove that there exists N ∈ N such that A N f = Id. As A f ∈ GL(2, Z) and f is conjugated to f n we have : • the linear maps A f and A f n = A n f are conjugated by A h ∈ GL(2, Z), • the modulus of the eigenvalues of A f are 1.
• the product of the eigenvalues is +1 or −1.
• the trace of A f is an integer.
Case 1: A f admits a real eigenvalue.
In this case, the possible eigenvalues are +1 or −1 and A f is conjugated to one of the following applications: It is clear that A 2 1 = Id. We are going to prove that A 2 cannot occur.
If ε 1 = 1 then A n 2 = 1 n 0 1 . One can see that A n 2 can not be conjugated to A 2 in GL(2, Z). More precisely, one can compute the conjugating matrix in GL(2, R), it is of the form : This matrix does not belong to GL(2, Z).
This case is not possible.
For n even, one can easily see that A n 2 can not be conjugated to A 2 , since T r(A n 2 ) = 2 = T r(A 2 ).
For n odd, one can see that A n 2 can not be conjugated to A 2 in GL(2, Z): one compute the conjugating matrix in GL(2, R), it is of the form: This matrix does not belong to GL(2, Z). This case is not possible.
Hence, A f is conjugated to a rotation of angle θ. The trace of A f is 2 cos θ and it is an integer. Then the possible values for cos θ are : 0, 1, −1, According to the previous proposition, given an action of BS(1, n) =< f, h > on T 2 there exists an integer N such that f N is isotopic to identity. From now on, we assume that f is isotopic to identity : this is not a restrictive hypothesis since the action of < f N , h > on T 2 is an action of the Baumslag-Solitar group BS(1, nN).
For proving Theorem 3, we begin by proving the following: Proposition 4.2. If f is isotopic to identity andf is a lift of f , then ρ(f ) is a rational point.
For proving this proposition we need the following lemmas: Let (a, b) be a vector in the rotation set of H • F • H −1 . By definition, As A H = Id, the map (H − Id) is bounded (periodic) so the limit: By definition of the rotation set, the limit : As A H −1 = Id, the map (H −1 − Id) is bounded so the limit : Let (a, b) be a vector in the rotation set of H • F • H −1 . By definition,

Proof. As two lifts of a torus map differ by an integer vector, we have that
for some integer vector P . By iterating this formula we have: Then Hence, by properties of the rotation set we have ρ( h • f • h −1 ) = ρ( h • f • h −1 ) + P = Ah(ρ(f )) + P , because of the previous lemma.
This implies that ρ( f ) has empty interior, so since it is a convex set, it is either a segment or a point.
In As B 2 is an affine map, its linear part has 1 as eigenvalue, its trace is 1 n 2 T r(A 2 h ) so it has the form p n 2 with p ∈ Z. Its determinant is 1 n 4 so its other eigenvalue is 1 n 4 . Therefore its trace is 1 + 1 n 4 so has not the form p n 2 with p ∈ Z, this is a contradiction.
Consequently, the rotation set ρ( f ) is a single point which is the unique fixed point of the affine map B. Since B has rational coefficients, then ρ( f ) has rational coordinates.

Proof of the Theorem 3.
According to Proposition 4.1, there is an integer N such that f N is isotopic to identity. By Proposition 4.2, the rotation number of any lift f N is a rational vector.
According to Corollary 3.5 of [MZ89], Hence {(0, 0)} = ρ erg ( f N q ). Then, using Theorem 3.5 of [Fra89], f N q has a fixed point and therefore f ix(f N q ) is non empty.

Existence of a BS-minimal set in per(f ).
The aim of this section is to show the existence of a minimal set for the action included in the set of f -periodic points. In the case that < f, h > is a representation of BS(1, n) in Homeo(X), where X is a compact metric space (Theorem 4), we ask for the existence of fixed or periodic points of f and in Theorem 5 we assume that any f -invariant probability measure has support included in the set of f -fixed points. In this case, we also study fminimal sets and the topological entropy of f . In this section we also prove Corollaries 2 and 3.
We are going to prove Theorem 4.
Let M be an f -invariant set, then Let P = f ix(f ). It holds that h −1 (P ) ⊂ P so h −j (P ) is a closed f -invariant set for any j ∈ N. If the action were minimal, its unique minimal set would be X and would be contained in f ixf N , according to item (b) of Theorem 4. This implies that X = f ixf N and so f N = Id, the action would not be faithful. This is a contradiction.
The following is the proof of Theorem 5.
Proof of (1). Let M f be an f -minimal set and x 0 ∈ M f . Let µ k = 1 k k−1 i=0 δ f i (x 0 ) and µ a weak limit of µ k . It is known that µ is an f -invariant probability measure and its support is included in M f = O f (x 0 ), the closure of the f -orbit of x 0 . In addition, by hypotheses It follows that M f is reduced to a fixed point. Since a periodic orbit is a minimal set we have that per(f ) coincides with the set f ix(f ). Proof of (3). Recall that ent top (f ) = sup{ent ν (f )} where ν is an f -invariant probability measure, the supremum of all metric entropies. Since ent ν (f ) = ent ν (f | supp(ν) ) and f | supp(ν) = Id, we have that ent top (f ) = 0.
We finish this section by proving Corollary 3.
If S = T 2 , let N be the positive integer given by Theorem 1, then the set f ix(f N ) = ∅. If S = S 2 , let N be the smallest positive integer such that f N has at least three fixed points and it is orientation preserving. Otherwise, N = 1.
In addition, f is distortion element of BS(1, n), so according to Theorem 1.3 of [FH06] for any f -invariant probability measure, µ, it holds that supp(µ) ⊆ f ix(f N ) so Theorem 5 implies the claim of this corollary except the f -ellipticity of the points in minimal sets.
For simplicity, we will prove the ellipticity in the case where f has fixed points. The general case is analogous.
Let x 0 be a point in a BS-minimal set M BS . Since M BS is also an h-minimal set, the h-orbit of x 0 is recurrent. Then there exists a subsequence (n k ) (n k → ∞) such that From h n k • f • h −n k = f n n k , we deduce that : As the points x 0 and h −n k (x 0 ) are fixed by f and (Dh −n k (x 0 )) −1 = Dh n k (h −n k (x 0 )), then: ( We conclude that Df (x 0 ) and (Df (x 0 )) n n k have the same eigenvalues, finally the eigenvalues of Df (x 0 ) have module 1.
Before proving Theorem 2, we prove the following Lemma 6.1. Let us consider a BS-action < f, h > on T 2 generated by f and h sufficiently C 0 -close to homeomorphismsf 0 andh 0 which generate a BS-action. If bothf 0 andh 0 are isotopic to identity and the rotation set of a lift off 0 is (0, 0), then the rotation set of a lift of f is (0, 0).

Proof of the lemma.
For (f, h) sufficiently close to (f 0 ,h 0 ), f and h are isotopic to identity. By Lemma 4.2 the rotation set of any liftf of f satisfy nρ(f ) = ρ(f ) + (p, q), where (p, q) is an integer vector. Then the rotation set off is a rational vector ( p n−1 , q n−1 ), with p, q integers. It is proved in [MZ89] that the rotation set map ρ : Homeo Z 2 (R 2 ) → K(R 2 ), the set of compact subsets of R 2 is upper semi-continuous with respect to the compact-open topology on Homeo Z 2 (R 2 ) and the Haussdorff topology on K(R 2 ). In other words, if G is an element of Homeo Z 2 (R 2 ) and U is a neighborhood of ρ(G) in R 2 , then for F sufficiently close to G, we have ρ(F ) ⊂ U.
The rotation set of a lift off 0 is (0, 0), consider a neighborhood U of (0, 0) in R 2 that contains no points of the form ( p n−1 , q n−1 ), with p, q integers and (p, q) = (0, 0). According to the previous result of [MZ89], for f sufficiently close to f 0 , the rotation set of a lift of f is included in U and since it has form ( p n−1 , q n−1 ), it must be (0, 0).
Proof of Theorem 2.
(1) The circles C 1 and C 2 are h 0 -normally hyperbolic in the sense of [HPS77]. Consider a neighborhood U 1 of C 1 where C 1 is h 0 -attractive and a neighborhood U 2 of C 2 where C 2 is h 0 -repulsive. Obviously, there exists some integer k 0 such that h k According to Theorem 4.1 of [HPS77], there exists a C 1 -neighborhood V of h 0 in Dif f 1 (T 2 ) such that for all h ∈ V there exist two circles C ′ 1 and C ′ 2 which are C 1 -closed to C 1 and C 2 respectively and they are h-invariant. Moreover (2) Obviously, the rotation set of a lift of f 0 is (0, 0). According to previous lemma, the rotation set of a lift of f must be (0, 0). Then f ix(f ) is not empty. Let and consists of f -fixed points, according to Theorem 4. In other words, C ′ 2 intersects f ix(f ). But for f sufficiently close to f 0 we have that f j (C ′ 2 ) ∩ C ′ 2 = ∅, for any j = 0. Hence x 0 ∈ C ′ 1 .
(3) We first prove that any minimal set of BS intersects C ′ 1 . Let us consider M a BS-minimal set : Since M ⊂ C ′ 2 , there is x 0 ∈ M \ C ′ 2 . Then ω h (x 0 ) ⊂ C ′ 1 ∩ M, so we are done. The circle C ′ 1 is h-invariant, we can consider the rotation number ρ of the restriction of h to C ′ 1 : There is a BS-minimal set M included in f ix(f ) so in C ′ 1 . This set M contains an h-minimal set in C ′ 1 . Moreover h has periodic orbit and any minimal set of h |C ′ 1 is an h-periodic orbit. Then there is an h-periodic orbit contained in M ⊂ f ix(f ). So this h-periodic orbit is a finite BS-orbit. Case 2: ρ / ∈ Q. Case 2a: h |C ′ 1 is conjugated to an irrational rotation. We claim that C ′ 1 = f ix(f ). Let x 0 ∈ f ix(f ), then α h (x 0 ) = C ′ 1 and it is contained in f ix(f ). Hence f ix(f ) = C ′ 1 . Now, we prove that C ′ 1 is a minimal set for the BS-action: Let x be in C ′ 1 = f ixf . The closure of h-orbit of x is BS-invariant and coincide with C ′ 1 . Consequently, the circle C ′ 1 is a minimal set for the BS-action.
Case 2b: h |C ′ 1 is semi-conjugated (not conjugated) to an irrational rotation. Then h |C ′ 1 admits a unique minimal set K that is homeomorphic to a Cantor set. Let x 0 be a fixed point of f , then K = α h (x 0 ) ⊂ f ix(f ). So K is BS-invariant, so it contains a BS-minimal set.
Since any BS-minimal set intersects C ′ 1 and α h (x) = K for all x ∈ C ′ 1 , any BS-minimal set contains K.
Finally, K is the unique BS-minimal set and it is a Cantor set.
In the case that the action is C 2 we have the following Corollary 4. If the action is C 2 and sufficiently C 1 -close to < f 0 , h 0 > then either : (1) C ′ 1 = f ix(f ) is the unique minimal set for the action and the minimal sets of f are its fixed points or (2) there exists a finite BS-orbit contained in C ′ 1 .

Proof.
According to theorem 2 (3), either there exists a finite BS-orbit in C ′ 1 or the action has an unique minimal set M which is the unique h |C ′ 1 -minimal set. In the second case, since h is C 2 , the circle map h |C ′ 1 is C 2 and according to Denjoy's theorem, M is the whole circle C ′ 1 = f ix(f ).
Proposition 6.1. Let us consider a BS-action < f, h > on T 2 generated by f and h sufficiently C 1 -close tof 0 andh 0 , wheref 0 andh 0 are isotopic to identity. If the rotation set of a lift off 0 is (0, 0) andh 0 is a Morse Smale diffeomorphism satisfying that any periodic point ish 0 -fixed, then < f, h > admits fixed point.

Proof.
Any h sufficiently C 1 -close toh 0 is a Morse Smale diffeomorphism where any h-periodic point is fixed. In particular, any h-minimal set is an h-fixed point.
By lemma 6.1, rotation set of a lift f is (0, 0), so f ix(f ) is not empty. As a consequence of Theorem 4, there is a BS-minimal set included in f ix(f ), this minimal set contains an h-minimal set, that is a fixed point of h. This point is a global fixed point.