Chaos and quasi-periodicity in diffeomorphisms of the solid torus

that the two manifolds W u (C α,ε ), W s (C α,ε ) still intersect transversally. To apply Lemma 8
we restrict to two suitable compact subsets A u ⊂ W u (C α ) and A s ⊂ W s (C α ) as follows.
Consider the segments pc ⊂ W u (p) and pd ⊂ W s (p) in Figure 7. Define
A u = pc × ¡ 1 , A s = pd × ¡ 1 .
In this way, the circle H is the intersection of the manifolds A u and A s , bounded away from
their boundaries. Consider the inclusions i : A u → M and j : A s → M. By the closeness of
W u (C α ) to W u (C α,ε ) there exists a C r -diffeomorphism h : A u → A u ε ⊂ W u (C α,ε ) such that
the map i is C r -close to i ε ◦ h, where i ε : A u ε → M is the inclusion [30, Section 2.6]. Similarly,
there exists a diffeomorphism k : A s → A s ε ⊂ W s (C α,ε ) such that the map j is C r -close j ε ◦ k,
where j ε : A s ε → M is the inclusion. By Lemma 9 the map i × j : Au × A s → M × M is
transversal to the diagonal ∆. For ε small, the map (i ε ◦ h) × (j ε ◦ k) : A u × A s → M × M is
C r -close to i × j:
A u × A s i×j
−−−→ M × M
⏐
h×k↓
A u ε × A s i ε×j ε
ε −−−→ M × M.
Since ∆ is closed and A u × A s is compact, Lemma 8 implies that there exists an ε ∗ , with
0 < ε ∗ < ε r , such that (i ε ◦ h) × (j ε ◦ k) ⋔ ∆ for ε < ε ∗ . Furthermore, the submanifolds
(i × j) −1 (∆) and
(
(iε ◦ h) × (j ε ◦ k) ) −1
(∆)
are diffeomorphic. We also have that ( (i ε ◦ h) × (j ε ◦ k) ) −1 (∆) is diffeomorphic to A u ε ∩ As ε ,
and (i × j) −1 (∆) = A u ∩ A s = H .
This shows that the intersection H ε = A u ε ∩ As ε is diffeomorphic to H . Define Du ε as the
part of W u (C α,ε ) bounded by the invariant circle C α,ε and the circle of homoclinic points H ε .
Define Dε s = k(D s ) similarly. Then the manifolds Dε u ⊂ W u (C α,ε ) and Dε s ⊂ W s (C α,ε ) form
the boundary of an open region U ⊂ M homeomorphic to a torus. By the closeness of the
perturbed manifolds W s (C α,ε ) and W u (C α,ε ) to the unperturbed W s (C ) and W u (C ), both U
and W u (C α,ε ) are bounded. Also notice that P α,ε is dissipative: by taking ε ∗ small enough,
we ensure that |det(DF (x, y, θ))| < ˜c < 1 for all ε < ε ∗ and (x, y, θ) in U. Therefore, all
forward evolutions beginning at points (x, y, θ) ∈ U remain bounded. Like in the first part
of the proof, one has
ω(x, y, θ) ⊂ Cl (W u (C α,ε ))
for all (x, y, θ) ∈ U, α ∈ [0, 1] and ε < ε ∗ .
3.1.2 An application of kam theory
So far, we did not discuss the dynamics in the saddle invariant circle C α,ε of map P α,ε
in (6). Generically, the dynamics on C α,ε is of Morse-Smale type. In this case, the circle
consists of the union of the unstable manifold of some periodic saddle. Theorem 3 describes
a complementary case, for which the dynamics is quasi-periodic. Fix τ > 2 and define the
set of Diophantine frequencies D γ by
{
D γ = α ∈ [0, 1] |
∣ α − p }
q ∣ ≥ γq−τ for all ¡ p, q ∈ , q ≠ 0 , (18)
where γ > 0. Since we will apply a version of the KAM Theorem holding for non-conservative,
finitely differentiable systems (see [5, Chap. 5] and [6]), a certain amount of smoothness of
the circle C α,ε is needed, depending on the Diophantine condition specified in (18). Therefore
we require that the perturbed family of maps P α,ε is C n , for n large enough.
18