Observable rotation sets

6.2.1 Definitions

As said before, from the notion of observable measure, it is easy to define a notion of observable ergodic rotation set. Another definition, more topologic, seemed reasonable to us for observable rotation sets:

Definition 6.9.

ρ^obs(F) =

v∈R²| ∀ε>0,λn

x| ∃u∈ρ(x) :d(u, v)<εo

>0

.

4. The set of Lyapunov stable periodic orbits is a Cantor set.

ρ^obs_mes(F) =

These two sets are non-empty compact subsets of the classical rotation set, and the first one is even a subset of ρ_pts(F). The next lemma states that these two definitions coincide:

Lemma 6.10. ρ^obs_mes(F) =ρ^obs(F).

Proof of Lemma6.10. We first prove that ρ^obs_mes(F) ⊂ ρ^obs(F). Let v ∈ ρ^obs_mes(F) and ε > 0.

Then there existsµ∈Obs(f) such thatv =R

T²D(F) dµ, in particularλ(A_ε/2(µ))>0. But if x∈ A_ε/2(µ), then there exists a strictly increasing sequence of integers (n_i(x))_i such that for everyi≥0,

holds for everyiand on aλ-positive measure set of pointsx.

For the other inclusion, letv∈ρ^obs(F) and set

A˜_ε(v) ={x∈T²| ∃u∈ρ(x) :d(u, v)<ε}.

By hypothesis,λ( ˜A_ε(v))>0 for everyε>0. To eachx∈A˜_ε(v) we associate the setpω_ε^v(x) of limit points of the sequence of measures

1

where (n_i(x))_iis a strictly increasing sequence such that

By compactness ofP, the setpω^v_ε(x) is non-empty and compact. In the sequel we will use the following easy remark: if 0<ε<ε⁰andx∈A˜_ε, thenpω^v_ε(x)⊂pω_ε^v0(x).

By contradiction, suppose that for everyµ∈P, there existsε_µ>0 such that λn

thus, as balls B(µ_j,ε_µj) coverP,

Integrating this estimation according to the function D(F), we obtain:

6.2.2 Properties of the observable rotation set

We begin by giving two lemmas which state the dynamical behaviour of the observ-able rotation sets.

Lemma 6.11. For everyq∈N,ρ^obs(F^q) =qρ^obs(F).

Proof of Lemma6.11. It suffices to remark thatρ_Fq(x) = qρ_F(x) (one inclusion is trivial and the other is easily obtained by Euclidean division).

Remark6.12. In general ρ^obs(F⁻¹),−ρ^obs(F): see for instance the point3. of Example 6.14.

Lemma 6.13. If H is a homeomorphism of R² commuting with integral translations and preserving null sets, thenρ^obs(H◦F◦H⁻¹) =ρ^obs(F).

Proof of Lemma6.13. It follows easily from the fact that the notion of observable mea-sure is stable by conjugacy (see Remark6.5).

We now give a few simple examples of calculation of observable rotation sets.

Example6.14.1. Iff = Id, thenρ^obs(F) ={(0,0)}. 2. If

F(x, y) = (x+ cos(2πy), y), thenρ_pts(F) =ρ^obs(F) = [−1,1]× {0}.

3. If

F(x, y) =

x+ cos(2πy), y+ 1

100sin(2πy)

,

thenρ_pts(F) ={(0,−1),(0,1)}, butρ^obs(F) ={(0,−1)}andρ^obs(F⁻¹) ={(0,1)}. 4. Let

P x y

!

= x+¹₂cos(2πy) + 1 y

!

and Q x y

!

= x

y+¹₂cos(2πx) + 1

! .

Then the rotation set of the (conservative) homeomorphism F = P◦Q is equal to [0,1]². Moreover, we can perturb F into a (conservative) homeomorphism ˜F such that

˜F is the identity on the neighbourhoods of the points whose coordinates belong to 1/2Z(applying for example Theorem 5.40). Then, the vertices of the square [0,1]² belong to the observable rotation set of ˜F.

5. Let P be a convex polygon with rational vertices. In [Kwa92], J. Kwapisz has con-structed an axiom A diffeomorphismf ofT²whose rotation set is the polygon P. It is possible to modify slightly Kwapisz’s construction so that all the sinks off are fixed points, and so that the union of the basins of these sinks haveλ-full measure. Hence, the observable rotation set off_Pis reduced to{(0,0)}.

We now give the results about the link between the rotation set and the observable rotation set in the generic setting. We begin by the dissipative case.

Proposition 6.15. Iff is generic amongHomeo(T²), thenρ(F) = conv(ρ^obs(F)). If moreover f is generic with a non-empty interior rotation set, thenρ(F) =ρ^obs(F).

To prove this proposition, we will use the following lemma, which is a direct conse-quence of Proposition6.7.

Lemma 6.16. Iff is generic amongHomeo(T²), then

ρ^obs(F) = Cl{ρ( ˜x)|xis a Lyapunov stable periodic point}.

We will also need a theorem of realization of rotation vectors by periodic points.

Theorem 6.17(J. Franks, [Fra89]). For everyf ∈Homeo(T²), every rational point of the interior ofρ(F)is realized as the rotation vector of a periodic point of the homeomorphismf. Proof of Proposition6.15. Theorem6.1 states that for an open dense set of homeomor-phisms, the rotation set is a rational polygon. Then, a theorem of realization of J. Franks [Fra88, Theorem 3.5] implies that every vertex of this polygon is realized as the rotation vector of a periodic point of the homeomorphism, which can be made attractive by a little perturbation of the homeomorphism. Then generically we can find a Lyapunov stable periodic point which shadows the previous periodic point (by Lemma 4.4), in particular it has the same rotation vector. Thus every vertex ofρ(F) belongs to ρ^obs(F) andρ(F) = conv(ρ^obs(F)).

For ε >0, we can find a finite ε-dense subset R_ε of ρ(F) made of rational points.

Thus, Theorem6.17associates to each of these rational vectors a periodic point of the homeomorphism which realizes this rotation vector; we can even make these periodic points of the homeomorphism attractive. Thus, for everyε>0, the set O_ε made of the homeomorphisms such that every vector of R_ε is realized by a strictly periodic open subset ofT²is open and dense in the set of homeomorphisms with non-empty interior rotation set. Applying Lemma 4.4 we find a G_δ dense subset of O_ε on which every strictly periodic open subset ofT² contains a Lyapunov stable periodic point; on this set the Hausdorff distance betweenρ(F) =ρ^obs(F) is smaller thanε. The conclusion of the proposition then easily follows from Baire theorem.

Remark6.18. It is not true thatρ(F) =ρ^obs(F) holds for a generic homeomorphism: see for instance the point3of Example6.14, where on a neighbourhood off the setρ^obsis contained in a neighbourhood of the points (0,−1) and (0,1).

For the conservative case, we recall the result of Proposition6.2: the rotation set of a generic conservative homeomorphism has non-empty interior. The following result states that in this case the observable rotation set is much smaller, more precisely it consists in a single vector, namely the mean rotation vector.

Proposition 6.19. Iff is generic amongHomeo(T²,λ), thenρ^obs(F) ={ρ_λ(F)}, whereρ_λ(F) is the mean rotation vector with respect to the measureλ.

Thus, for almost everyx∈T²(with respect to the measureλ), the setρ(x) is reduced to a single point which is the mean rotation vector.

Proof of Proposition6.19. It is easily implied by the fact that the measureλis the only observable measure (Lemma6.8, which easily follows from Oxtoby-Ulam theorem).