Chaos and quasi-periodicity in diffeomorphisms of the solid torus

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PSfrag replacements p U ′ ∂ s q c ∂ u d Figure 7: Segments ∂ s and ∂ u of the stable and unstable manifold, respectively, of a saddle fixed point p bound a region U, see text for more explanation. Remark 4. 1. We used the family (7) for the figures, expecting that it is sufficiently rich for our purposes. 2. When comparing the Figures 1 (C) and 6, the main difference is that in the second case there is no significant occurrence of Λ 2 = 0. Still we expect that in all cases the attractor is the closure Cl (W u (C )) of a quasi-periodic invariant circle C of saddle-type. 3. It seems that in the skew case (2), the phenomenon of an attractor with Λ 1 = 0 and Λ 2 < 0, which is not an invariant circle is somewhat related to ‘nonchaotic strange attractors’, compare with [28, 15, 17, 22, 29]. See also Section 4.3 for further discussion on this topic. Interestingly, tiny perturbations away from the skew case seemingly give rise to a quasiperiodic Hénon-like attractor, so with Λ 1 > 0. 3 Proofs 3.1 Basins of attraction and quasi-periodic invariant circles In this section we give proofs of Theorem 2 (next section) and Theorem 3 (Section 3.1.2). 3.1.1 The Tangerman-Szewc argument generalised Let K : 2 → 2 be a dissipative diffeomorphism having a saddle fixed point p = (x 0 , y 0 ). Suppose the stable and unstable manifolds W s (p) and W u (p) intersect transversally at the homoclinic point q ∈ W s (p) ∩ W u (p), see Figure 7. Also assume that W u (p) is bounded as a subset of 2 . The Tangerman-Szewc Theorem (see e.g. [31, Appendix 3]) states that the basin of attraction of the closure of W u (p) contains the open region U ′ bounded by the two arcs ∂ s ⊂ W s (p) and ∂ u ⊂ W u (p) with extremes p and q, see Figure 7. This argument is used to prove existence of strange attractors (in particular, with non-trivial basin of attraction) near homoclinic tangencies of a saddle fixed point of a dissipative diffeomorphism, cf. [26, 39, 42]. We first prove Theorem 2 for ε = 0. This is a straightforward generalisation of the above Tangerman-Szewc Theorem. For small ε, the result is obtained by using persistence of normally hyperbolic invariant manifolds [20, Theorem 1.1] and two transversality lemmas. 16
Proof of Theorem 2. Consider the circle C α = C α,0 , invariant under map P α,0 in (6). The manifolds W u (C α ) and W s (C α ) are given by W u (p) × ¡ 1 and W s (p) × ¡ 1 , respectively. They intersect transversally at a circle H = {q} × ¡ 1 , consisting of points homoclinic to C α . Consider the two arcs ∂ s ⊂ W s (p) and ∂ u ⊂ W u (p) with extremes p and q (Figure 7).They bound an open set U ′ ⊂ 2 . Define D s and D u to be the portions of stable, and unstable manifold of C α , respectively, given by D s = ∂ s × ¡ 1 ⊂ W s (C α ) and D u = ∂ u × ¡ 1 ⊂ W u (C α ). Both surfaces D s and D u are compact, and their union forms the boundary of the open region U = U ′ × 1 , which ¡ is topologically a solid torus. The volume of U decreases under iteration of P α,0 . Denoting by meas(·) the Lebesgue measure both on 2 and on 2 × 1 , due to condition 2 ¡ in Theorem 2 we have ∫ ∫ meas(Pα,0 n (U)) = 2π dxdy = 2π |det DK n | dxdy ≤ 2πκ n meas(U ′ ). K n (U ′ ) U ′ This implies that the forward evolution of every point (x, y, θ) ∈ U approaches the boundary of P n α,0 (U): dist ( P n α,0(x, y, θ), ∂P n α,0(U) ) → 0 as n → +∞. Indeed, suppose that this does not hold. Then there exists a ϱ > 0 such that for all n there exists N > n such that the ball with centre Pα,0(x, N y, θ) and radius ϱ > 0 is contained inside Pα,0 N (U). But this would contradict the fact that meas(P α,0 n (U)) → 0 as n → +∞. The boundary of Pα,0 n (U) also consists of two portions of stable and unstable manifold of C : ∂Pα,0 n (U) = P α,0 n (Ds ) ∪ Pα,0 n (Du ). The diameter of P n α,0(D s ) tends to zero as n → +∞, because all points in D s are attracted to the circle C α . Since W u (C α ) is bounded, all evolutions starting in U are bounded and approach W u (C α ), that is, dist(P n α,0 (x, y, θ), P n α,0 (Du )) → 0 as n → +∞ for all (x, y, θ) ∈ U. This implies that ω(x, y, θ) ⊂ Cl (W u (C α )) for all (x, y, θ) ∈ U. To extend this result to small perturbations P α,ε of P α,0 , the following transversality lemmas are used. Lemma 8. [33, Chap. 7] Consider a map f : V → M, where V and M are C r - differentiable manifolds and f is C r . Suppose V is compact, W ⊂ M is a closed C r - submanifold and f is transversal to W at V (notation: f ⋔ W ). Then f −1 (W ) is a C r - submanifold of codimension codim V (f −1 (W )) = codim M (W ). Further suppose that there is a neighbourhood of f(∂ V ) ∪ ∂ W disjoint from f(V ) ∩ W , where ∂ V and ∂ W are the boundaries of V and W . Then any map g : V → M, sufficiently C r -close to f, is also transversal to W , and the two submanifolds g −1 (W ) and f −1 (W ) are diffeomorphic. Lemma 9. [19, Section 3.2] Let V 1 , V 2 , and M be C r -differentiable manifolds and consider two diffeomorphisms f i : V i → M, i = 1, 2. Then f 1 ⋔ f 2 if and only if f 1 × f 2 ⋔ ∆, where f 1 × f 2 : V 1 × V 2 → M × M is the product map and ∆ ⊂ M × M is the diagonal: ∆ = {(y, y) | y ∈ M}. Fix r ¡ ∈ and take ε < ε r , where ε r is given in Proposition 1. Then the map P α,ε has an r-normally hyperbolic invariant circle C α,ε of saddle type. Furthermore, the manifolds W u (C α,ε ), W s (C α,ε ), and C α,ε are C r -close to W u (C α ), W s (C α ), and C α . We now show 17
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Chaos and quasi-periodicity in diffeomorphisms of the solid torus

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